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Introduction
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection0
Tue, 12 Apr 2016 23:00:00 GMT
<p>This course is designed for those who are employed in the health services, perhaps as a paramedic or as operating theatre staff. If you are a student, you will have a tutor to help you, and perhaps a workbased mentor supplied by the employer  normally the NHS. The aim is to use the workplace as a teaching arena that helps provide relevance and meaning to the activities you undertake, and it is especially designed to be relevant to students' current or future employment in health areas.</p><p></p><p>This OpenLearn course provides a sample of level 1 study in <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/courses/find/healthandwellbeing?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Health & Wellbeing</a></span></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection0
IntroductionS110_1<p>This course is designed for those who are employed in the health services, perhaps as a paramedic or as operating theatre staff. If you are a student, you will have a tutor to help you, and perhaps a workbased mentor supplied by the employer  normally the NHS. The aim is to use the workplace as a teaching arena that helps provide relevance and meaning to the activities you undertake, and it is especially designed to be relevant to students' current or future employment in health areas.</p><p></p><p>This OpenLearn course provides a sample of level 1 study in <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/courses/find/healthandwellbeing?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Health & Wellbeing</a></span></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Learning outcomes
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsectionlearningoutcomes
Tue, 12 Apr 2016 23:00:00 GMT
<p>After studying this course, you should be able to:</p><ul><li><p>understand the decimal system of numbering (hundreds, tens, units)</p></li><li><p>explain the best way to write down decimal numbers and associated units of measurement in the healthcare workplace, in a manner that avoids confusion</p></li><li><p>understand the concepts of discrete and continuous variables and the best types of graphs used to represent these data</p></li><li><p>analyse, construct and extract information from graphs.</p></li></ul>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsectionlearningoutcomes
Learning outcomesS110_1<p>After studying this course, you should be able to:</p><ul><li><p>understand the decimal system of numbering (hundreds, tens, units)</p></li><li><p>explain the best way to write down decimal numbers and associated units of measurement in the healthcare workplace, in a manner that avoids confusion</p></li><li><p>understand the concepts of discrete and continuous variables and the best types of graphs used to represent these data</p></li><li><p>analyse, construct and extract information from graphs.</p></li></ul>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.1 Introducing the decimal system of numbers
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Many different systems for writing numbers have been developed over the history of humankind.</p><p>The easiest way of counting small numbers is to use your fingers, and for this reason many numerical systems, such as the decimal system, are based around the number ten. But what happens when you run out of fingers to count on?</p><p>Numbering systems get round this problem by using a system of <i>scale</i> in which many small units are represented by a single larger unit, and many of these larger units are represented by a single even bigger unit, and so on.</p><p>For instance, the number <b>five</b> can be represented as five fingers, or by a single hand. In the same way, if you were writing numbers, you might want to use one symbol to represent single units and another to represent a larger collection of these units. How about a line <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/d4739dbc/s110_1_ie002i.jpg" alt="" width="7" height="16" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span> to represent an individual unit and a star <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/df2aefa0/s110_1_ie001i.jpg" alt="" width="18" height="17" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> to represent a collection of five individual units? Using this system, the number <b>eleven</b> could be represented as two stars and a line <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4ca3eff2/s110_1_ie003i.jpg" alt="" width="46" height="17" style="maxwidth:46px;" class="oucontentinlinefigureimage"/></span> (i.e. five plus five plus one).</p><p>The decimal system uses a similar concept of scale, but it is arranged such that <i>ten</i> of any particular unit makes <i>one</i> unit of the next size up. This also works the other way round: each larger unit consists of <i>ten</i> units of the next size down.</p><p>For instance, ten pennies (ten 'units') are the same as one 10p piece (one 'ten'). Similarly, ten 10p pieces (in other words one hundred pennies) are the same as one pound.</p><p>These different groups differ by a <b>factor of ten</b> (each group is ten times larger or smaller than the one that precedes or follows it), which is also known as an <b>order of magnitude</b> (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.1#fig001001">Figure 1</a>).</p><div class="oucontentfigure" style="width:511px;" id="fig001_001"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm11030240" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/7601507b/s110_1_001i.small.jpg" alt="Figure 1" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm11030240">View larger image</a></div><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 Three orders of magnitude: 100, 10 and 1</span></div></div><a id="back_thumbnailfigure_idm11030240"></a></div><p>Groups of decimal numbers start from zero, so within a group of ten, any particular unit may be numbered between 0 and 9. Unlike many earlier systems of numbering, the decimal system doesn't rely on having to use different symbols to indicate larger and larger groupings of numbers. Instead, the order in which the symbols 0 to 9 are written provides the information that tells you the size of a number. In other words the decimal system is a <b>positional system</b> of numbering  the numbers are read and written from left to right, and the order in which they are written or spoken is used to tell you how many different factors of ten are present in the number. For example, the number 125 (one hundred and twenty five) contains three different orders of magnitude. On the left are the numbers of hundreds (one hundred), then the numbers of tens (two tens), and finally the numbers of units (five units).</p><p>The name of the <b>unit of measurement</b> is usually written to the right of the number, after a space, and this information tells you what individual objects are being counted (people, grams, metres, etc.). As such, a distance of one hundred and twenty five metres would be written as 125 m, where 'm' represents the unit of measurement.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.1
1.1 Introducing the decimal system of numbersS110_1<p>Many different systems for writing numbers have been developed over the history of humankind.</p><p>The easiest way of counting small numbers is to use your fingers, and for this reason many numerical systems, such as the decimal system, are based around the number ten. But what happens when you run out of fingers to count on?</p><p>Numbering systems get round this problem by using a system of <i>scale</i> in which many small units are represented by a single larger unit, and many of these larger units are represented by a single even bigger unit, and so on.</p><p>For instance, the number <b>five</b> can be represented as five fingers, or by a single hand. In the same way, if you were writing numbers, you might want to use one symbol to represent single units and another to represent a larger collection of these units. How about a line <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/d4739dbc/s110_1_ie002i.jpg" alt="" width="7" height="16" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span> to represent an individual unit and a star <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/df2aefa0/s110_1_ie001i.jpg" alt="" width="18" height="17" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> to represent a collection of five individual units? Using this system, the number <b>eleven</b> could be represented as two stars and a line <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4ca3eff2/s110_1_ie003i.jpg" alt="" width="46" height="17" style="maxwidth:46px;" class="oucontentinlinefigureimage"/></span> (i.e. five plus five plus one).</p><p>The decimal system uses a similar concept of scale, but it is arranged such that <i>ten</i> of any particular unit makes <i>one</i> unit of the next size up. This also works the other way round: each larger unit consists of <i>ten</i> units of the next size down.</p><p>For instance, ten pennies (ten 'units') are the same as one 10p piece (one 'ten'). Similarly, ten 10p pieces (in other words one hundred pennies) are the same as one pound.</p><p>These different groups differ by a <b>factor of ten</b> (each group is ten times larger or smaller than the one that precedes or follows it), which is also known as an <b>order of magnitude</b> (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.1#fig001001">Figure 1</a>).</p><div class="oucontentfigure" style="width:511px;" id="fig001_001"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm11030240" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/7601507b/s110_1_001i.small.jpg" alt="Figure 1" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm11030240">View larger image</a></div><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 Three orders of magnitude: 100, 10 and 1</span></div></div><a id="back_thumbnailfigure_idm11030240"></a></div><p>Groups of decimal numbers start from zero, so within a group of ten, any particular unit may be numbered between 0 and 9. Unlike many earlier systems of numbering, the decimal system doesn't rely on having to use different symbols to indicate larger and larger groupings of numbers. Instead, the order in which the symbols 0 to 9 are written provides the information that tells you the size of a number. In other words the decimal system is a <b>positional system</b> of numbering  the numbers are read and written from left to right, and the order in which they are written or spoken is used to tell you how many different factors of ten are present in the number. For example, the number 125 (one hundred and twenty five) contains three different orders of magnitude. On the left are the numbers of hundreds (one hundred), then the numbers of tens (two tens), and finally the numbers of units (five units).</p><p>The name of the <b>unit of measurement</b> is usually written to the right of the number, after a space, and this information tells you what individual objects are being counted (people, grams, metres, etc.). As such, a distance of one hundred and twenty five metres would be written as 125 m, where 'm' represents the unit of measurement.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.2 Decimal points
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>Suppose you have less than one of any particular unit: how would you represent that using the decimal system?</p><p>Well, we've already seen that decimal numbers rely on a positional system, in which values get smaller by factors of ten as you read from left to right. If we continue doing this, then the number to the right of a single unit represents tenths of that unit. A <b>decimal point</b> is then used to mark the boundary between the whole units and tenths of that unit.</p><p>For instance, I look in my pocket and find I have one pound ten pence. If my units of measurement were pounds, then I would write this amount as £1.1 (one and one tenth of a pound). However, if my units of measurement were pence, then I would write this as 110p (one hundred and ten pence).</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2#tbl001">Table 1</a>, you can see how the position of each digit in the sequence is used to tell you the overall size of the number.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl001"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 1 Decimal numbers described in terms of their orders of magnitude</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
<b>Description</b>
</th><th scope="col">
<b>Tens</b>
</th><th scope="col">
<b>Units</b>
</th><th scope="col">
<b>Decimal point</b>
</th><th scope="col">
<b>Tenths</b>
</th><th scope="col">
<b>Hundredths</b>
</th></tr><tr><td>
<b>4.5</b> is 4 units and 5 tenths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">4</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">5</td><td class="oucontenttablemiddle ">0</td></tr><tr><td>
<b>6.87</b> is 6 units, 8 tenths and 7 hundredths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">6</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">8</td><td class="oucontenttablemiddle ">7</td></tr><tr><td>
<b>98.04</b> is 9 tens, 8 units, 0 tenths and 4 hundredths</td><td class="oucontenttablemiddle ">9</td><td class="oucontenttablemiddle ">8</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">4</td></tr><tr><td>
<b>0.06</b> is 6 hundredths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">6</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>Now try placing decimal points appropriately and reading the values of decimal numbers for yourself with practice questions 1, 2 and 3.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2
1.2 Decimal pointsS110_1<p>Suppose you have less than one of any particular unit: how would you represent that using the decimal system?</p><p>Well, we've already seen that decimal numbers rely on a positional system, in which values get smaller by factors of ten as you read from left to right. If we continue doing this, then the number to the right of a single unit represents tenths of that unit. A <b>decimal point</b> is then used to mark the boundary between the whole units and tenths of that unit.</p><p>For instance, I look in my pocket and find I have one pound ten pence. If my units of measurement were pounds, then I would write this amount as £1.1 (one and one tenth of a pound). However, if my units of measurement were pence, then I would write this as 110p (one hundred and ten pence).</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2#tbl001">Table 1</a>, you can see how the position of each digit in the sequence is used to tell you the overall size of the number.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl001"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 1 Decimal numbers described in terms of their orders of magnitude</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
<b>Description</b>
</th><th scope="col">
<b>Tens</b>
</th><th scope="col">
<b>Units</b>
</th><th scope="col">
<b>Decimal point</b>
</th><th scope="col">
<b>Tenths</b>
</th><th scope="col">
<b>Hundredths</b>
</th></tr><tr><td>
<b>4.5</b> is 4 units and 5 tenths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">4</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">5</td><td class="oucontenttablemiddle ">0</td></tr><tr><td>
<b>6.87</b> is 6 units, 8 tenths and 7 hundredths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">6</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">8</td><td class="oucontenttablemiddle ">7</td></tr><tr><td>
<b>98.04</b> is 9 tens, 8 units, 0 tenths and 4 hundredths</td><td class="oucontenttablemiddle ">9</td><td class="oucontenttablemiddle ">8</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">4</td></tr><tr><td>
<b>0.06</b> is 6 hundredths</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">•</td><td class="oucontenttablemiddle ">0</td><td class="oucontenttablemiddle ">6</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>Now try placing decimal points appropriately and reading the values of decimal numbers for yourself with practice questions 1, 2 and 3.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.2.1 Study Note 1
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Simple rules for dealing with orders of magnitude and decimal points in decimal numbers: values ten times bigger than the order of magnitude you are looking at go to the left, ten times smaller go to the right, and less than 1 to the right of the decimal point.</p><p><i>Note:</i> in many European countries, a comma is used instead of a decimal point. For instance in France and Germany two and a half (in other words 2.5) can be written as 2,5. This is important to bear in mind, for example, if the drug you are measuring out comes from a European supplier, or you are following a laboratory protocol that was developed abroad.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.2.1
1.2.1 Study Note 1S110_1<p>Simple rules for dealing with orders of magnitude and decimal points in decimal numbers: values ten times bigger than the order of magnitude you are looking at go to the left, ten times smaller go to the right, and less than 1 to the right of the decimal point.</p><p><i>Note:</i> in many European countries, a comma is used instead of a decimal point. For instance in France and Germany two and a half (in other words 2.5) can be written as 2,5. This is important to bear in mind, for example, if the drug you are measuring out comes from a European supplier, or you are following a laboratory protocol that was developed abroad.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.3 Marking decimals on a scale
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3
Tue, 12 Apr 2016 23:00:00 GMT
<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3#fig001002">Figure 2</a> shows a picture of a ruler. The major units are marked in centimetres (1 to 11 cm), whilst the intervals between the centimetres have each been split into ten equal, smaller units. These minor units are therefore <i>tenths</i> of a centimetre, commonly known as 'millimetres'. (There are 10 millimetres in 1 centimetre.) A similar type of decimal scale is used on many devices that you will work with, such as syringes, gas pressure gauges and pH meters, even though the physical quantities they measure, and therefore their units of measurement, are different.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3#fig001002">Figure 2</a> several numbers have been highlighted to allow you to relate the way a decimal number is written to the quantity that number represents. For instance, if our units of measurement are centimetres, then we can see that 0.3 cm is less than 1 cm (it is actually 3 millimetres). Similarly, 2 cm is less than 3.5 cm. However, 70 mm is less than 10.2 cm, because 70 mm means the same as 7 cm.</p><div class="oucontentfigure" style="width:432px;" id="fig001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4b171e7e/s110_1_002i.jpg" alt="Figure 2" width="432" height="71" style="maxwidth:432px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Units and tenths of units represented on a ruler</span></div></div></div><p>Now practise placing decimal values on a similar scale for yourself with practice questions 4 to 8.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3
1.3 Marking decimals on a scaleS110_1<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3#fig001002">Figure 2</a> shows a picture of a ruler. The major units are marked in centimetres (1 to 11 cm), whilst the intervals between the centimetres have each been split into ten equal, smaller units. These minor units are therefore <i>tenths</i> of a centimetre, commonly known as 'millimetres'. (There are 10 millimetres in 1 centimetre.) A similar type of decimal scale is used on many devices that you will work with, such as syringes, gas pressure gauges and pH meters, even though the physical quantities they measure, and therefore their units of measurement, are different.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.3#fig001002">Figure 2</a> several numbers have been highlighted to allow you to relate the way a decimal number is written to the quantity that number represents. For instance, if our units of measurement are centimetres, then we can see that 0.3 cm is less than 1 cm (it is actually 3 millimetres). Similarly, 2 cm is less than 3.5 cm. However, 70 mm is less than 10.2 cm, because 70 mm means the same as 7 cm.</p><div class="oucontentfigure" style="width:432px;" id="fig001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4b171e7e/s110_1_002i.jpg" alt="Figure 2" width="432" height="71" style="maxwidth:432px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Units and tenths of units represented on a ruler</span></div></div></div><p>Now practise placing decimal values on a similar scale for yourself with practice questions 4 to 8.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.4 Decimal places
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4
Tue, 12 Apr 2016 23:00:00 GMT
<p>If you have less than one unit you should put a zero before the decimal point to make it easier for yourself and others to read the value (e.g. you should write 0.4 rather than just .4, as will be explained later in this course). However, how many zeros should you put <i>after</i> the last whole number in the series? For instance, is 0.4 the same as 0.40?</p><p>The short answer is that on one level, it is. However, by writing 0.40 we are saying that there are four tenths and zero hundredths, and importantly we are saying that we can actually measure to an accuracy of an individual hundredth of a unit; in other words to two <b>decimal places</b>. In contrast, by writing 0.4 we are only claiming an accuracy to the level of individual tenths of a unit, or to one decimal place.</p><p>One way of getting a more accurate measurement is to use an instrument with a more finely divided scale.
</p><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4#fig001003">Figure 3</a> shows a closeup of two thermometers, labelled A and B, that were placed side by side to record the air temperature in a room.</p><div class="oucontentfigure oucontentmediamini" id="fig001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/9a351365/s110_1_003i.jpg" alt="Figure 3" width="284" height="147" style="maxwidth:284px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 3 Thermometers A and B measuring the same temperature in a room. The finer the scale, the more certain you can be of the reading, and the more decimal places you can quote with confidence</span></div></div></div><p>In terms of accuracy the scale on thermometer A is quite coarse, as the markings represent individual degrees Celsius (°C). Using this scale, we can see that the room temperature is somewhere between 21 °C and 22 °C. On closer inspection, someone might estimate it at 21.7 °C, but someone else could easily record it as 21.6 °C or 21.8 °C. There is some uncertainty in the first decimal place, and there is certainly no way we could accurately state the temperature to two decimal places using this thermometer.</p><p>In order to give someone an idea of how confident we are about the measurements we make with thermometer A we should quote the <b>range</b> of possible values (i.e. the highest and lowest values) that the actual temperature could be. Since we estimate the reading to be between 21.6 °C and 21.8 °C we would choose the midpoint of these values, and say that the temperature was within 0.1 °C of 21.7 °C. As we will see later, another way to write this would be 21.7 °C 'plus or minus' 0.1 °C, or 21.7 ± 0.1 °C. This gives us a measure of the <b>uncertainty</b> of thermometer A.</p><p>Now look at thermometer B. This thermometer has a finer scale, with divisions marked every 0.1 °C. Now we can clearly see that the room temperature is between 21.6 °C and 21.7 °C. This is within the range of possible values that we estimated from thermometer A, but the finer scale of thermometer B allows us to be more certain of the temperature. Nevertheless, even though thermometer B allows us to read the temperature to within 0.1 °C we cannot be so sure about the second decimal place; someone might read it as 21.63 °C, whilst another person might read it as 21.61 °C or 21.65 °C. With this scale, we can be sure of the first decimal place, but not the second.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4
1.4 Decimal placesS110_1<p>If you have less than one unit you should put a zero before the decimal point to make it easier for yourself and others to read the value (e.g. you should write 0.4 rather than just .4, as will be explained later in this course). However, how many zeros should you put <i>after</i> the last whole number in the series? For instance, is 0.4 the same as 0.40?</p><p>The short answer is that on one level, it is. However, by writing 0.40 we are saying that there are four tenths and zero hundredths, and importantly we are saying that we can actually measure to an accuracy of an individual hundredth of a unit; in other words to two <b>decimal places</b>. In contrast, by writing 0.4 we are only claiming an accuracy to the level of individual tenths of a unit, or to one decimal place.</p><p>One way of getting a more accurate measurement is to use an instrument with a more finely divided scale.
</p><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4#fig001003">Figure 3</a> shows a closeup of two thermometers, labelled A and B, that were placed side by side to record the air temperature in a room.</p><div class="oucontentfigure oucontentmediamini" id="fig001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/9a351365/s110_1_003i.jpg" alt="Figure 3" width="284" height="147" style="maxwidth:284px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 3 Thermometers A and B measuring the same temperature in a room. The finer the scale, the more certain you can be of the reading, and the more decimal places you can quote with confidence</span></div></div></div><p>In terms of accuracy the scale on thermometer A is quite coarse, as the markings represent individual degrees Celsius (°C). Using this scale, we can see that the room temperature is somewhere between 21 °C and 22 °C. On closer inspection, someone might estimate it at 21.7 °C, but someone else could easily record it as 21.6 °C or 21.8 °C. There is some uncertainty in the first decimal place, and there is certainly no way we could accurately state the temperature to two decimal places using this thermometer.</p><p>In order to give someone an idea of how confident we are about the measurements we make with thermometer A we should quote the <b>range</b> of possible values (i.e. the highest and lowest values) that the actual temperature could be. Since we estimate the reading to be between 21.6 °C and 21.8 °C we would choose the midpoint of these values, and say that the temperature was within 0.1 °C of 21.7 °C. As we will see later, another way to write this would be 21.7 °C 'plus or minus' 0.1 °C, or 21.7 ± 0.1 °C. This gives us a measure of the <b>uncertainty</b> of thermometer A.</p><p>Now look at thermometer B. This thermometer has a finer scale, with divisions marked every 0.1 °C. Now we can clearly see that the room temperature is between 21.6 °C and 21.7 °C. This is within the range of possible values that we estimated from thermometer A, but the finer scale of thermometer B allows us to be more certain of the temperature. Nevertheless, even though thermometer B allows us to read the temperature to within 0.1 °C we cannot be so sure about the second decimal place; someone might read it as 21.63 °C, whilst another person might read it as 21.61 °C or 21.65 °C. With this scale, we can be sure of the first decimal place, but not the second.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.4.1 Study Note 2
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>An important point to remember when writing down measurements from a scale is never to quote more decimal places than you can reliably read from the measuring device you are using.</p><div class="oucontentfigure oucontentmediamini" id="fig001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/744053e7/s110_1_004i.jpg" alt="Figure 4" width="180" height="257" style="maxwidth:180px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 The right tools for the job? A 5 ml syringe and a 20 ml syringe. Match the syringe size to the volume of liquid to be administered</span></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 1</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1#fig001004">Figure 4</a> shows a photo of a 5 ml syringe and a 20 ml syringe. Which is the most appropriate syringe for measuring to an accuracy of 0.5 ml?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Looking at the scales given on the sides of the syringes the markings on the 5 ml syringe go up in 0.5 ml jumps, whilst the marking on the 20 ml syringe go up in 2 ml jumps. Therefore, you could only be confident about measuring to an accuracy of 0.5 ml using the 5 ml syringe.</p></div></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What would you do if you needed to inject 0.6 ml?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p> </p><p>You would need to select a smaller syringe than either of those shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1#fig001004">Figure 4</a> in order to be confident of measuring and administering the correct volume of liquid. A 1 ml syringe, marked in 0.1 ml increments, would be ideal.</p></div></div></div></div><p>In general, the smaller the syringe size the more finely the scale is marked and therefore the easier it is to distinguish small changes in volume. For this reason (as well as reasons of cost; large syringes are generally more expensive than small ones), it's generally easier to read and more accurate if you use the smallest syringe that can accommodate the volume you need to inject.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1
1.4.1 Study Note 2S110_1<p>An important point to remember when writing down measurements from a scale is never to quote more decimal places than you can reliably read from the measuring device you are using.</p><div class="oucontentfigure oucontentmediamini" id="fig001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/744053e7/s110_1_004i.jpg" alt="Figure 4" width="180" height="257" style="maxwidth:180px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 The right tools for the job? A 5 ml syringe and a 20 ml syringe. Match the syringe size to the volume of liquid to be administered</span></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 1</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1#fig001004">Figure 4</a> shows a photo of a 5 ml syringe and a 20 ml syringe. Which is the most appropriate syringe for measuring to an accuracy of 0.5 ml?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Looking at the scales given on the sides of the syringes the markings on the 5 ml syringe go up in 0.5 ml jumps, whilst the marking on the 20 ml syringe go up in 2 ml jumps. Therefore, you could only be confident about measuring to an accuracy of 0.5 ml using the 5 ml syringe.</p></div></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What would you do if you needed to inject 0.6 ml?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p> </p><p>You would need to select a smaller syringe than either of those shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.4.1#fig001004">Figure 4</a> in order to be confident of measuring and administering the correct volume of liquid. A 1 ml syringe, marked in 0.1 ml increments, would be ideal.</p></div></div></div></div><p>In general, the smaller the syringe size the more finely the scale is marked and therefore the easier it is to distinguish small changes in volume. For this reason (as well as reasons of cost; large syringes are generally more expensive than small ones), it's generally easier to read and more accurate if you use the smallest syringe that can accommodate the volume you need to inject.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.5 Rounding to decimal places
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.5
Tue, 12 Apr 2016 23:00:00 GMT
<p>Sometimes the result of a calculation gives a number with lots of decimal places  far more than you need or could reliably measure. For instance, suppose a patient is required to receive 5 ml of medicine a day, evenly spaced in three injections. How much medicine should they be given in each dose?</p><p>To divide the 5 ml of medicine into three equal parts would mean measuring out 5 ÷ 3 = 1.6666 ml (where the 6s keep repeating, or <b>recurring</b> indefinitely). It's not realistic or feasible to measure out the medicine to this kind of accuracy. Instead, you first need to think about what level of accuracy is needed. An injection of this volume would be most accurately dispensed using a 2 ml syringe, marked in 0.1 ml increments. In this case then, the accuracy of your measurement would be limited to 0.1 ml, or in other words to one decimal place. To administer the amount calculated above you would need to round the figure to the nearest decimal place.</p><p>The rule to remember with rounding to a particular decimal place is that if the next number to the right of that decimal place is 5 or more, you round the figure up to the next highest number, and if it's 4 or less it remains the same. For instance, to correct 1.6666 ml to one decimal place, find the first decimal place and then look at the next (smaller) decimal place to its right, which we've highlighted here as 1.6<b>6</b>66 ml. As this number is greater than 5 we have to round up, and the amount becomes 1.7 ml corrected to one decimal place (1 dp). If the original number had been 1.6<b>4</b>66 ml then the value corrected to one decimal place would be 1.6 ml (1 dp).</p><p>Here are some more worked examples for you to practise with:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/66127a3a/s110_1_ue001i.gif" alt=""/></div><p>Now try rounding numbers to specific decimal places for yourself with practice question 9.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Box 1 General rules for numbers in healthcare
</h2><div class="oucontentinnerbox"><ul class="oucontentbulleted"><li>
<p>
<i>Try to avoid the need for a decimal point</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Use 500 mg not 0.5 g</p>
</li><li>
<p>Use 125 mcg not 0.125 mg</p>
</li></ul>
</li><li>
<p>
<i>Never leave a decimal point 'naked'</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Paracetamol 0.5 mg not Paracetamol .5 mg</p>
</li></ul>
</li><li>
<p>
<i>Avoid using a terminal zero</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Diazepam 2 mg not Diazepam 2.0 mg</p>
</li></ul>
</li><li>
<p>
<i>Put a space between the drug name and dose</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Apresoline 55 mg not Apresoline55 mg</p>
</li></ul>
</li></ul></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.5
1.5 Rounding to decimal placesS110_1<p>Sometimes the result of a calculation gives a number with lots of decimal places  far more than you need or could reliably measure. For instance, suppose a patient is required to receive 5 ml of medicine a day, evenly spaced in three injections. How much medicine should they be given in each dose?</p><p>To divide the 5 ml of medicine into three equal parts would mean measuring out 5 ÷ 3 = 1.6666 ml (where the 6s keep repeating, or <b>recurring</b> indefinitely). It's not realistic or feasible to measure out the medicine to this kind of accuracy. Instead, you first need to think about what level of accuracy is needed. An injection of this volume would be most accurately dispensed using a 2 ml syringe, marked in 0.1 ml increments. In this case then, the accuracy of your measurement would be limited to 0.1 ml, or in other words to one decimal place. To administer the amount calculated above you would need to round the figure to the nearest decimal place.</p><p>The rule to remember with rounding to a particular decimal place is that if the next number to the right of that decimal place is 5 or more, you round the figure up to the next highest number, and if it's 4 or less it remains the same. For instance, to correct 1.6666 ml to one decimal place, find the first decimal place and then look at the next (smaller) decimal place to its right, which we've highlighted here as 1.6<b>6</b>66 ml. As this number is greater than 5 we have to round up, and the amount becomes 1.7 ml corrected to one decimal place (1 dp). If the original number had been 1.6<b>4</b>66 ml then the value corrected to one decimal place would be 1.6 ml (1 dp).</p><p>Here are some more worked examples for you to practise with:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/66127a3a/s110_1_ue001i.gif" alt=""/></div><p>Now try rounding numbers to specific decimal places for yourself with practice question 9.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Box 1 General rules for numbers in healthcare
</h2><div class="oucontentinnerbox"><ul class="oucontentbulleted"><li>
<p>
<i>Try to avoid the need for a decimal point</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Use 500 mg not 0.5 g</p>
</li><li>
<p>Use 125 mcg not 0.125 mg</p>
</li></ul>
</li><li>
<p>
<i>Never leave a decimal point 'naked'</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Paracetamol 0.5 mg not Paracetamol .5 mg</p>
</li></ul>
</li><li>
<p>
<i>Avoid using a terminal zero</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Diazepam 2 mg not Diazepam 2.0 mg</p>
</li></ul>
</li><li>
<p>
<i>Put a space between the drug name and dose</i>
</p>
<ul class="oucontentunnumbered"><li>
<p>Apresoline 55 mg not Apresoline55 mg</p>
</li></ul>
</li></ul></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Study Note 3 'Dos and don'ts' with decimals in the healthcare workplace
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.5.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>As suggested in Box 1 above, there are a number of common 'dos and don'ts' that you need to remember and apply whenever you are dealing with decimals in your workplace.</p><ul class="oucontentbulleted"><li>
<p>
<b>Look carefully!</b> Because a decimal point is just a dot on the page it is sometimes easy to miss when reading, especially on lined paper or in faxed documents. For this reason if there are no whole units, always place a zero before the decimal point when writing decimal numbers, e.g. seven tenths should be written as 0.7 and not as .7.</p>
</li><li>
<p>Similarly, don't add additional zeros after a decimal point as this may indicate a degree of accuracy to which you are unable to measure, e.g. one and a half should be written as 1.5 and not as 1.50 (unless, for a specific reason, you need to quote the value to 2 decimal places).</p>
</li><li>
<p>In general, try to avoid the need for decimal places by changing the scale to use a different unit of measurement. For example, half a gram can be written as 0.5 g. , This is the same as 500 mg (500 milligrams). Similarly, 0.125 mg can be rewritten as 125 micrograms. Note that although the accepted scientific symbol for 'micrograms' is 'μg', when this is handwritten it can often be confused with 'mg', the symbol for milligrams. To avoid confusion, and help to reduce the risk of error, many hospitals prefer to use the symbol 'mcg' for micrograms. We will be covering scales and units of measurement in the next section.</p>
</li></ul><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox oucontentsnoheading " id="act001_005_001"><div class="oucontentouterbox"><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Listen to the audio track below. It contains information that reinforces what you have just learned here.</p><p>Click to listen to the track [3 minutes 30 seconds, 4.02MB]</p><div id="mp3_001" class="oucontentmedia oucontentaudiovideo ompversion1" style="width:342px;"><div class="oucontentdefaultfilter"><span class="oumediafilter"><a href="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/f7430b8c/s110_1_001s.mp3?forcedownload=1" class="oumedialinknoscript ompspacer">Download this audio clip.</a><span class="accesshide">Audio player: s110_1_001s.mp3</span><a href="#" class="ompentermedia ompaccesshide" tabindex="1">
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</span></div><div class="oucontentfiguretext"><div class="oucontentmediadownload"><a href="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/f7430b8c/s110_1_001s.mp3?forcedownload=1" title="Download this audio clip">Download</a></div><div class="oucontentcaption oucontentnonumber oucontentcaptionplaceholder"> </div></div></div><div class="oucontentinteractionprint"><div class="oucontentinteractionunavailable">Interactive feature not available in single page view (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.5.1#mp3001">see it in standard view</a>).</div></div></div></div></div></div> <script>
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https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.5.1
Study Note 3 'Dos and don'ts' with decimals in the healthcare workplaceS110_1<p>As suggested in Box 1 above, there are a number of common 'dos and don'ts' that you need to remember and apply whenever you are dealing with decimals in your workplace.</p><ul class="oucontentbulleted"><li>
<p>
<b>Look carefully!</b> Because a decimal point is just a dot on the page it is sometimes easy to miss when reading, especially on lined paper or in faxed documents. For this reason if there are no whole units, always place a zero before the decimal point when writing decimal numbers, e.g. seven tenths should be written as 0.7 and not as .7.</p>
</li><li>
<p>Similarly, don't add additional zeros after a decimal point as this may indicate a degree of accuracy to which you are unable to measure, e.g. one and a half should be written as 1.5 and not as 1.50 (unless, for a specific reason, you need to quote the value to 2 decimal places).</p>
</li><li>
<p>In general, try to avoid the need for decimal places by changing the scale to use a different unit of measurement. For example, half a gram can be written as 0.5 g. , This is the same as 500 mg (500 milligrams). Similarly, 0.125 mg can be rewritten as 125 micrograms. Note that although the accepted scientific symbol for 'micrograms' is 'μg', when this is handwritten it can often be confused with 'mg', the symbol for milligrams. To avoid confusion, and help to reduce the risk of error, many hospitals prefer to use the symbol 'mcg' for micrograms. We will be covering scales and units of measurement in the next section.</p>
</li></ul><div class="
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</script>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.6.1 Getting comfortable with factors of ten
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Moving a decimal point by one place changes the value of the number by a factor of ten. For instance, to multiply a value by ten you can just move the decimal point one place to the right:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4143d9e9/s110_1_ue002i.gif" alt=""/></div><p>Notice that if the starting number doesn't have a decimal point shown we can place an imaginary decimal point after the last digit, and a zero to the right of this, in order to help us see the change in the order of magnitude.</p><p>To multiply a number by 100, move the decimal point 2 places to the right:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e83e9eb1/s110_1_ue003i.gif" alt=""/></div><p>The same principle applies to dividing by 10, 100, 1000 etc., except you move the decimal place to the left instead, thus making the original number smaller. For instance:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/57d0f5f9/s110_1_ue004i.gif" alt=""/></div><p>Look at the animated examples in the second 'Info' box of your <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/">practice questions</a></span>. These animations illustrate how sequentially moving a decimal point by one place at a time changes that number by a factor of ten each time the point is moved.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a>, each number in the second column differs from the one in the row immediately above or below it by a factor of ten, or in other words by one order of magnitude.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl002"><h3 class="oucontenth3 oucontentheading oucontentnonumber">
Table 2 Names, units and symbols for factors of ten found within the decimal system of numbering</h3><div class="oucontenttablewrapper"><table><tr><th scope="col">
Name
</th><th scope="col">
Number
</th><th scope="col">
Order of magnitude
</th><th scope="col">
Power
</th><th scope="col">
Unit
</th><th scope="col">
Symbol
</th></tr><tr><td>Million</td><td>1 000 000</td><td class="oucontenttablemiddle ">6</td><td>10<sup>6</sup>
</td><td>
<b>mega</b>
</td><td class="oucontenttablemiddle ">M</td></tr><tr><td>Hundred thousand</td><td>100 000</td><td class="oucontenttablemiddle ">5</td><td>10<sup>5</sup>
</td><td></td><td>
</td></tr><tr><td>Ten thousand</td><td>10 000</td><td class="oucontenttablemiddle ">4</td><td>10<sup>4</sup>
</td><td></td><td>
</td></tr><tr><td>Thousand</td><td>1 000</td><td class="oucontenttablemiddle ">3</td><td>10<sup>3</sup>
</td><td>
<b>kilo</b>
</td><td class="oucontenttablemiddle ">k</td></tr><tr><td>Hundred</td><td>100</td><td class="oucontenttablemiddle ">2</td><td>10<sup>2</sup>
</td><td>hecto</td><td class="oucontenttablemiddle ">h</td></tr><tr><td>Ten</td><td>10</td><td class="oucontenttablemiddle ">1</td><td>10<sup>1</sup>
</td><td></td><td>
</td></tr><tr><td>One</td><td>1</td><td class="oucontenttablemiddle ">0</td><td>10<sup>0</sup>
</td><td></td><td>
</td></tr><tr><td>Tenth</td><td>0.1</td><td class="oucontenttablemiddle ">−1</td><td>10<sup>−1</sup>
</td><td>deci</td><td class="oucontenttablemiddle ">d</td></tr><tr><td>Hundredth</td><td>0.01</td><td class="oucontenttablemiddle ">−2</td><td>10<sup>−2</sup>
</td><td>centi</td><td class="oucontenttablemiddle ">c</td></tr><tr><td>Thousandth</td><td>0.001</td><td class="oucontenttablemiddle ">−3</td><td>10<sup>−3</sup>
</td><td>
<b>milli</b>
</td><td class="oucontenttablemiddle ">m</td></tr><tr><td>Ten thousandth</td><td>0.000 1</td><td class="oucontenttablemiddle ">−4</td><td>10<sup>−4</sup>
</td><td></td><td>
</td></tr><tr><td>Hundred thousandth</td><td>0.000 01</td><td class="oucontenttablemiddle ">−5</td><td>10<sup>−5</sup>
</td><td></td><td>
</td></tr><tr><td>Millionth</td><td>0.000 001</td><td class="oucontenttablemiddle ">−6</td><td>10<sup>−6</sup>
</td><td>
<b>micro</b>
</td><td class="oucontenttablemiddle ">μ</td></tr><tr><td>Ten millionth</td><td>0.000 000 1</td><td class="oucontenttablemiddle ">−7</td><td>10<sup>−7</sup>
</td><td></td><td>
</td></tr><tr><td>Hundred millionth</td><td>0.000 000 01</td><td class="oucontenttablemiddle ">−8</td><td>10<sup>−8</sup>
</td><td></td><td>
</td></tr><tr><td>Billionth</td><td>0.000 000 001</td><td class="oucontenttablemiddle ">−9</td><td>10<sup>−9</sup>
</td><td>nano</td><td class="oucontenttablemiddle ">n</td></tr></table></div><div class="oucontentsourcereference"></div><div class="oucontenttablefootnote">The units shown in <b>bold</b> are those that you are likely to encounter on a daily basis in your work.</div></div><p>In the column marked 'Power', there is a 10 and a small, raised (superscript) number next to it for each of the names of the factors of ten mentioned. This superscript number is called the <b>power</b> or the <b>exponent</b>, and it indicates how many times the first number or 'base' (in this case a 10) is multiplied by itself in order to give the actual amount being signified. For instance, 10<sup>2</sup> ('ten squared'), means 10 × 10 = 100. Similarly, 10<sup>4</sup> means 10 × 10 × 10 × 10 = 10 000.</p><p>Notice also that numbers less than zero can be shown by a negative exponent. For instance, 10<sup>−1</sup> means '1 divided by 10', where 1 ÷ 10 = 0.1. Similarly, 10<sup>−4</sup> means '1 divided by 10<sup>4</sup>', which is 1 ÷ 10 000, or 0.000 1. Note that any number to the power zero means that it is divided by itself and is therefore 1 (e.g. 10 ÷ 10 = 1), which explains why <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a> states that 10<sup>0</sup> = 1.</p><p>An exponent will often have a number and a multiplication sign written before it, for example 2.4 × 10<sup>3</sup>, which also means 2.4 × 1000, or 2400. An easy way to work out the value of numbers expressed as exponents is to move the decimal point the same number of places as the exponent.</p><ul class="oucontentbulleted"><li>
<p>If the exponent is positive, then move the decimal point to the right by the same number of places as the exponent.</p>
</li><li>
<p>If the exponent is negative, then move the decimal point to the left by the same number of places as the exponent.</p>
</li></ul><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/eda2b254/s110_1_ue005i.gif" alt=""/></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1
1.6.1 Getting comfortable with factors of tenS110_1<p>Moving a decimal point by one place changes the value of the number by a factor of ten. For instance, to multiply a value by ten you can just move the decimal point one place to the right:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4143d9e9/s110_1_ue002i.gif" alt=""/></div><p>Notice that if the starting number doesn't have a decimal point shown we can place an imaginary decimal point after the last digit, and a zero to the right of this, in order to help us see the change in the order of magnitude.</p><p>To multiply a number by 100, move the decimal point 2 places to the right:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e83e9eb1/s110_1_ue003i.gif" alt=""/></div><p>The same principle applies to dividing by 10, 100, 1000 etc., except you move the decimal place to the left instead, thus making the original number smaller. For instance:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/57d0f5f9/s110_1_ue004i.gif" alt=""/></div><p>Look at the animated examples in the second 'Info' box of your <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/">practice questions</a></span>. These animations illustrate how sequentially moving a decimal point by one place at a time changes that number by a factor of ten each time the point is moved.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a>, each number in the second column differs from the one in the row immediately above or below it by a factor of ten, or in other words by one order of magnitude.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl002"><h3 class="oucontenth3 oucontentheading oucontentnonumber">
Table 2 Names, units and symbols for factors of ten found within the decimal system of numbering</h3><div class="oucontenttablewrapper"><table><tr><th scope="col">
Name
</th><th scope="col">
Number
</th><th scope="col">
Order of magnitude
</th><th scope="col">
Power
</th><th scope="col">
Unit
</th><th scope="col">
Symbol
</th></tr><tr><td>Million</td><td>1 000 000</td><td class="oucontenttablemiddle ">6</td><td>10<sup>6</sup>
</td><td>
<b>mega</b>
</td><td class="oucontenttablemiddle ">M</td></tr><tr><td>Hundred thousand</td><td>100 000</td><td class="oucontenttablemiddle ">5</td><td>10<sup>5</sup>
</td><td></td><td>
</td></tr><tr><td>Ten thousand</td><td>10 000</td><td class="oucontenttablemiddle ">4</td><td>10<sup>4</sup>
</td><td></td><td>
</td></tr><tr><td>Thousand</td><td>1 000</td><td class="oucontenttablemiddle ">3</td><td>10<sup>3</sup>
</td><td>
<b>kilo</b>
</td><td class="oucontenttablemiddle ">k</td></tr><tr><td>Hundred</td><td>100</td><td class="oucontenttablemiddle ">2</td><td>10<sup>2</sup>
</td><td>hecto</td><td class="oucontenttablemiddle ">h</td></tr><tr><td>Ten</td><td>10</td><td class="oucontenttablemiddle ">1</td><td>10<sup>1</sup>
</td><td></td><td>
</td></tr><tr><td>One</td><td>1</td><td class="oucontenttablemiddle ">0</td><td>10<sup>0</sup>
</td><td></td><td>
</td></tr><tr><td>Tenth</td><td>0.1</td><td class="oucontenttablemiddle ">−1</td><td>10<sup>−1</sup>
</td><td>deci</td><td class="oucontenttablemiddle ">d</td></tr><tr><td>Hundredth</td><td>0.01</td><td class="oucontenttablemiddle ">−2</td><td>10<sup>−2</sup>
</td><td>centi</td><td class="oucontenttablemiddle ">c</td></tr><tr><td>Thousandth</td><td>0.001</td><td class="oucontenttablemiddle ">−3</td><td>10<sup>−3</sup>
</td><td>
<b>milli</b>
</td><td class="oucontenttablemiddle ">m</td></tr><tr><td>Ten thousandth</td><td>0.000 1</td><td class="oucontenttablemiddle ">−4</td><td>10<sup>−4</sup>
</td><td></td><td>
</td></tr><tr><td>Hundred thousandth</td><td>0.000 01</td><td class="oucontenttablemiddle ">−5</td><td>10<sup>−5</sup>
</td><td></td><td>
</td></tr><tr><td>Millionth</td><td>0.000 001</td><td class="oucontenttablemiddle ">−6</td><td>10<sup>−6</sup>
</td><td>
<b>micro</b>
</td><td class="oucontenttablemiddle ">μ</td></tr><tr><td>Ten millionth</td><td>0.000 000 1</td><td class="oucontenttablemiddle ">−7</td><td>10<sup>−7</sup>
</td><td></td><td>
</td></tr><tr><td>Hundred millionth</td><td>0.000 000 01</td><td class="oucontenttablemiddle ">−8</td><td>10<sup>−8</sup>
</td><td></td><td>
</td></tr><tr><td>Billionth</td><td>0.000 000 001</td><td class="oucontenttablemiddle ">−9</td><td>10<sup>−9</sup>
</td><td>nano</td><td class="oucontenttablemiddle ">n</td></tr></table></div><div class="oucontentsourcereference"></div><div class="oucontenttablefootnote">The units shown in <b>bold</b> are those that you are likely to encounter on a daily basis in your work.</div></div><p>In the column marked 'Power', there is a 10 and a small, raised (superscript) number next to it for each of the names of the factors of ten mentioned. This superscript number is called the <b>power</b> or the <b>exponent</b>, and it indicates how many times the first number or 'base' (in this case a 10) is multiplied by itself in order to give the actual amount being signified. For instance, 10<sup>2</sup> ('ten squared'), means 10 × 10 = 100. Similarly, 10<sup>4</sup> means 10 × 10 × 10 × 10 = 10 000.</p><p>Notice also that numbers less than zero can be shown by a negative exponent. For instance, 10<sup>−1</sup> means '1 divided by 10', where 1 ÷ 10 = 0.1. Similarly, 10<sup>−4</sup> means '1 divided by 10<sup>4</sup>', which is 1 ÷ 10 000, or 0.000 1. Note that any number to the power zero means that it is divided by itself and is therefore 1 (e.g. 10 ÷ 10 = 1), which explains why <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a> states that 10<sup>0</sup> = 1.</p><p>An exponent will often have a number and a multiplication sign written before it, for example 2.4 × 10<sup>3</sup>, which also means 2.4 × 1000, or 2400. An easy way to work out the value of numbers expressed as exponents is to move the decimal point the same number of places as the exponent.</p><ul class="oucontentbulleted"><li>
<p>If the exponent is positive, then move the decimal point to the right by the same number of places as the exponent.</p>
</li><li>
<p>If the exponent is negative, then move the decimal point to the left by the same number of places as the exponent.</p>
</li></ul><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/eda2b254/s110_1_ue005i.gif" alt=""/></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Common units of measurement
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>In practice, the most commonly used units differ from each other by a factor of 1000, and the names of some of these have been highlighted previously in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a>. As an example of this, you can see in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.2#fig001005">Figure 5</a> how the units of weight  tonne (t), kilogram (kg), gram (g), milligram (mg), and microgram (μg)  differ from each other by a factor of 1000.</p><p>Figure 5: Common units of measurement differ by a factor of 1000</p><div id="fig001_005" class="oucontentmedia" style="width:480px;"><div id="mediaidm5063552" class="oucontentactivecontent"><div class="oucontentflashjswarning">Active content not displayed. This content requires JavaScript to be enabled, and a recent version of <a href="https://helpx.adobe.com/flashplayer.html">Flash Player</a> to be installed and enabled.</div></div><script type="text/javascript">
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</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.2
Common units of measurementS110_1<p>In practice, the most commonly used units differ from each other by a factor of 1000, and the names of some of these have been highlighted previously in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.1#tbl002">Table 2</a>. As an example of this, you can see in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.2#fig001005">Figure 5</a> how the units of weight  tonne (t), kilogram (kg), gram (g), milligram (mg), and microgram (μg)  differ from each other by a factor of 1000.</p><p>Figure 5: Common units of measurement differ by a factor of 1000</p><div id="fig001_005" class="oucontentmedia" style="width:480px;"><div id="mediaidm5063552" class="oucontentactivecontent"><div class="oucontentflashjswarning">Active content not displayed. This content requires JavaScript to be enabled, and a recent version of <a href="https://helpx.adobe.com/flashplayer.html">Flash Player</a> to be installed and enabled.</div></div><script type="text/javascript">
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</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.6.3 Litres and kilograms
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.3
Tue, 12 Apr 2016 23:00:00 GMT
<p>The two physical units of measurement that you will probably come across most often in your workplace concern volumes of liquids and weight measurements. It's important to get a feeling for what various factors of ten look like, so that you can spot when there seems to be a mistake in a value that you've calculated or have been given by someone else.</p><p>The litre is the main unit of measurement for liquid volumes (written as liter in America), but what does a litre of fluid look like? What about a <i>milli</i>litre (ml; one thousandth of a litre) or a <i>micro</i>litre (μl; one millionth of a litre)?</p><p>A litre is the volume of liquid or gas that would fit into a cube measuring 10 cm on each side (10 cm × 10 cm × 10 cm = 1000 cubic centimetres (cm<sup>3</sup>) = 1 litre). A millilitre is the volume of a cube measuring 1 cm on each side (1 cm × 1 cm × 1 cm = 1 cm<sup>3</sup> = 1 ml). A microlitre is the volume of a cube measuring 1 mm on each side (1 mm × 1 mm × 1 mm = 1mm<sup>3</sup> = 1μl)</p><ul class="oucontentbulleted"><li>
<p>A typical carton of fruit juice has a capacity of a litre.</p>
</li><li>
<p>A teaspoon holds about 5 millilitres of liquid.</p>
</li><li>
<p>One raindrop is about thirty microlitres.</p>
</li></ul><p>Look around your workplace to find out the typical volumes used for various applications: e.g. what's the capacity of a blood bag or a drip bag? How about the various syringes or pipettes you might use? If your work is labbased, then you will probably be measuring volumes down to a smaller scale of microlitres. Again, get used to the volumes of typical containers (e.g. 7 ml and 20 ml specimen tubes; 2 ml, 1.5 ml and 0.5 ml microcentrifuge tubes) and the equipment appropriate to measure out each of these various volumes (measuring cylinders 11000 ml; pipettes 0.130 ml; micropipettes 0.11000 μl).</p><p>The kilogram is the basic unit of measurement for weight. Again, what does a kilogram feel like? What about a gram (g; one thousandth of a kilogram)? What about a <i>milli</i>gram (mg; one thousandth of a gram)?</p><ul class="oucontentbulleted"><li>
<p>A litre of water weighs one kilogram.</p>
</li><li>
<p>A £20 note weighs about one gram, whilst a pound coin is almost 10 grams. Originally, a 'pound' was the monetary value given to a pound weight of sterling silver (an alloy of 92.5% silver and 7.5% copper)  hence 'pound sterling'.</p>
</li><li>
<p>About 12 grains of salt weigh one milligram.</p>
</li></ul><p>Once more, look around your workplace and find the typical weights that you might deal with. Get used to the different sizes of various tablets and medications and the typical doses that patients are given. This will help you spot when a calculation might be wrong and you need to doublecheck with someone.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.6.3
1.6.3 Litres and kilogramsS110_1<p>The two physical units of measurement that you will probably come across most often in your workplace concern volumes of liquids and weight measurements. It's important to get a feeling for what various factors of ten look like, so that you can spot when there seems to be a mistake in a value that you've calculated or have been given by someone else.</p><p>The litre is the main unit of measurement for liquid volumes (written as liter in America), but what does a litre of fluid look like? What about a <i>milli</i>litre (ml; one thousandth of a litre) or a <i>micro</i>litre (μl; one millionth of a litre)?</p><p>A litre is the volume of liquid or gas that would fit into a cube measuring 10 cm on each side (10 cm × 10 cm × 10 cm = 1000 cubic centimetres (cm<sup>3</sup>) = 1 litre). A millilitre is the volume of a cube measuring 1 cm on each side (1 cm × 1 cm × 1 cm = 1 cm<sup>3</sup> = 1 ml). A microlitre is the volume of a cube measuring 1 mm on each side (1 mm × 1 mm × 1 mm = 1mm<sup>3</sup> = 1μl)</p><ul class="oucontentbulleted"><li>
<p>A typical carton of fruit juice has a capacity of a litre.</p>
</li><li>
<p>A teaspoon holds about 5 millilitres of liquid.</p>
</li><li>
<p>One raindrop is about thirty microlitres.</p>
</li></ul><p>Look around your workplace to find out the typical volumes used for various applications: e.g. what's the capacity of a blood bag or a drip bag? How about the various syringes or pipettes you might use? If your work is labbased, then you will probably be measuring volumes down to a smaller scale of microlitres. Again, get used to the volumes of typical containers (e.g. 7 ml and 20 ml specimen tubes; 2 ml, 1.5 ml and 0.5 ml microcentrifuge tubes) and the equipment appropriate to measure out each of these various volumes (measuring cylinders 11000 ml; pipettes 0.130 ml; micropipettes 0.11000 μl).</p><p>The kilogram is the basic unit of measurement for weight. Again, what does a kilogram feel like? What about a gram (g; one thousandth of a kilogram)? What about a <i>milli</i>gram (mg; one thousandth of a gram)?</p><ul class="oucontentbulleted"><li>
<p>A litre of water weighs one kilogram.</p>
</li><li>
<p>A £20 note weighs about one gram, whilst a pound coin is almost 10 grams. Originally, a 'pound' was the monetary value given to a pound weight of sterling silver (an alloy of 92.5% silver and 7.5% copper)  hence 'pound sterling'.</p>
</li><li>
<p>About 12 grains of salt weigh one milligram.</p>
</li></ul><p>Once more, look around your workplace and find the typical weights that you might deal with. Get used to the different sizes of various tablets and medications and the typical doses that patients are given. This will help you spot when a calculation might be wrong and you need to doublecheck with someone.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.7 SI units and conversions
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7
Tue, 12 Apr 2016 23:00:00 GMT
<p>The international system of units (<i>Le Système International d'Unités</i>: abbreviated to SI) was developed in France during the 18th century in an effort to create a unified and rational system of weights and measures. The SI system became adopted as the world standard in 1960.</p><p>There are seven basic units (or base units) to the SI system and these are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl003">Table 3</a>. All other units of measurement can be derived from combinations of these base SI units.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl003"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 3 The seven base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
Quantity
</th><th scope="col">
Name
</th><th scope="col">
Symbol
</th></tr><tr><td>Length</td><td>metre</td><td class="oucontenttablemiddle ">m</td></tr><tr><td>Mass</td><td>kilogram</td><td class="oucontenttablemiddle ">kg</td></tr><tr><td>Time</td><td>second</td><td class="oucontenttablemiddle ">s</td></tr><tr><td>Temperature</td><td>kelvin</td><td class="oucontenttablemiddle ">K</td></tr><tr><td>Electric current</td><td>ampere</td><td class="oucontenttablemiddle ">A</td></tr><tr><td>Luminous intensity</td><td>candela</td><td class="oucontenttablemiddle ">cd</td></tr><tr><td>Amount of substance</td><td>mole</td><td class="oucontenttablemiddle ">mol</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>Many units of measurement arise from a combination of different base SI units, and some of these are given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl004">Table 4</a>. For instance, you will see from the table that speed is defined in 'metres per second'. This can be written as metres divided by seconds, i.e. m/s, where the slanted line indicates the act of division. Another way you may see this expressed is as m s<sup>−1</sup> where the negative exponent indicates the division.
</p><p>Similarly, the newton is a SIderived unit of force. It is defined as the amount of force needed to accelerate a mass of 1 kg at a rate of 1 metre per second per second, and is expressed in terms of the following SI units: m kg s<sup>−2</sup>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl004"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 4</b> Commonly used units that are derived from base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
Quantity
</th><th scope="col">
Derived unit
</th><th scope="col">
Symbol
</th><th scope="col">
SI units
</th></tr><tr><td>Speed</td><td>metres per second</td><td class="oucontenttablemiddle ">m s<sup>−1</sup>
</td><td>m s<sup>−1</sup>
</td></tr><tr><td>Force</td><td>newton</td><td class="oucontenttablemiddle ">N</td><td>m kg s<sup>−2</sup>
</td></tr><tr><td>Energy</td><td>joule</td><td class="oucontenttablemiddle ">J</td><td>m<sup>2</sup> kg s<sup>−2</sup>
</td></tr><tr><td>Volume</td><td>cubic metre</td><td class="oucontenttablemiddle ">m<sup>3</sup>
</td><td>m<sup>3</sup>
</td></tr><tr><td>Pressure</td><td>pascal</td><td class="oucontenttablemiddle ">Pa</td><td>kg <i>m</i>
<sup>−1</sup> s<sup>−2</sup>
</td></tr><tr><td>Absorbed dose (radiation)</td><td>gray</td><td class="oucontenttablemiddle ">Gy</td><td>m<sup>2</sup> s<sup>−2</sup>
</td></tr><tr><td>Equivalent dose (radiation)</td><td>sievert</td><td class="oucontenttablemiddle ">Sv</td><td>m<sup>2</sup> s<sup>−2</sup>
</td></tr><tr><td>Radioactivity</td><td>becquerel</td><td class="oucontenttablemiddle ">Bq</td><td>s<sup>−1</sup>
</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>In addition, some nonstandard SI units are in common usage, and a selection of these is given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl005">Table 5</a>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl005"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 5</b> Commonly used units that are not derived from base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
<b>Quantity</b>
</th><th scope="col">
<b>Symbol</b>
</th><th scope="col">
<b>NonSI unit</b>
</th><th scope="col">
<b>Conversion factors</b>
</th></tr><tr><td>Time</td><td>min</td><td>minute</td><td>1 min = 60 s</td></tr><tr><td>Time</td><td>h</td><td>hour</td><td>1 h = 60 min = 3600 s</td></tr><tr><td>Time</td><td>d</td><td>day</td><td>1 d = 24 h = 1440 min = 86400 s</td></tr><tr><td>Volume</td><td>1</td><td>litre</td><td>1 l = 0.001 m<sup>3</sup>
</td></tr><tr><td>Mass</td><td>t</td><td>tonne</td><td>1 t = 1000 kg</td></tr><tr><td>Energy</td><td>cal</td><td>calorie</td><td>1 cal = 4.18 J</td></tr><tr><td>Temperature</td><td>°C</td><td>Celsius</td><td>1 °C = 274.15 K</td></tr><tr><td>Pressure</td><td>mmHg</td><td>millimetres of mercury</td><td>1 mmHg = 133.3 Pa</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>When you are dealing with particularly large or small quantities, the SI system is combined with the decimal system, such that you write the decimal notation first then follow it with the relevant SI unit.</p><p>For instance, 1000 metres is 1 <i>kilo</i>metre and this is abbreviated to 1 km. One hundredth of a metre is one <i>centi</i>metre, abbreviated as 1 cm.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 2 A note about temperature</h2><div class="oucontentinnerbox"><p>Although the basic SI unit of temperature is the kelvin, most people use and are familiar with degrees Celsius (°C). (Note that when using kelvins, the 'degrees' symbol '°' is not used.) These two temperature scales are equivalent, so a temperature change of 1 °C is the same actual increase or decrease as a temperature change of 1 K. The only difference is that these temperature scales don't start at the same place. The Celsius scale takes its starting point to be the temperature at which water freezes (0 °C), whilst the Kelvin scale starts from absolute zero (0 K; the coldest temperature theoretically possible, where even molecules stop moving). Zero kelvin is the same as −273.15 °C.</p></div></div></div><p>It is vitally important that you are comfortable with how the SI and decimal systems interact and how to make conversions within these units of measurement. In order to help you achieve this, now test yourself with practice questions 10 and 11.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p><p>Activity 1 will allow you to practise comparing quantities (in this case weights and temperatures) that are in different units of measurement. You can practise similar calculations to those you will be asked to perform in Part 1 for yourself using many common foodstuffs that list the quantities of vitamins, fats, fibre, etc. that they contain. The time you should allow to complete the three parts of the activity is 45 minutes. Please try working through the questions first before looking at the answers.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="act001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.1 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 25 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 1 Vitamin tablets</h3><p>A multivitamin tablet contains various vitamins and minerals. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl006">Table 6</a> shows the recommended daily allowance (RDA) for a selection of them, as shown on the packet.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl006"><h4 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 6</b> Label from packet of vitamin tablets</h4><div class="oucontenttablewrapper"><table><tr><th scope="col">
Component
</th><th scope="col">
%RDA
</th><th scope="col">
Amount per tablet
</th></tr><tr><td>Vitamin A</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">800 μg</td></tr><tr><td>Vitamin B6</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">2 mg</td></tr><tr><td>Vitamin C</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">60 mg</td></tr><tr><td>Vitamin D</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">5 μg</td></tr><tr><td>Vitamin E</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">10 mg</td></tr><tr><td>Magnesium</td><td class="oucontenttableright ">50</td><td class="oucontenttablemiddle ">150 mg</td></tr></table></div><div class="oucontentsourcereference"></div></div><p><i>Note: </i>although denoting micrograms as 'μg' is scientifically correct, remember that in the healthcare workplace it's generally better to write it as 'mcg' to avoid possible confusion with milligrams (mg).</p><p>Using this list, can you order the components, from the highest to the lowest amount, by the weight of the substance that equates to 100% RDA?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The first thing to notice is that not all of the units of measurement are the same, some are in milligrams (mg) and others are in micrograms (μg). It helps if you convert all the units of measurement to be the same before attempting to put them in order.</p><p>Only Vitamin A and D are expressed in micrograms, so let's convert them to milligrams. Because a microgram is 1000 times smaller than a milligram, to convert μg into mg, divide by 1000. (Remember, to divide by 1000 you can move the decimal point 3 places to the left: 1 μg = 0.001 mg).</p><p>The amount of Vitamin A present is 800 μg, or 800 ÷ 1000 = 0.8 mg. This is the full RDA (100%), so the RDA is 0.8 mg.</p><p>The amount of Vitamin D present is 5 μg, or 5 ÷ 1000 = 0.005 mg. Again, this is the full RDA, so the RDA is 0.005 mg.</p><p>The amount of magnesium present is 150 mg. However, this is only half of the RDA (50%), so the full RDA is twice this amount:</p><p>150 mg x 2 = 300 mg</p><p>Therefore, in order of decreasing RDA, the components should read: magnesium (300 mg), vitamin C (60 mg), vitamin E (10 mg), vitamin B6 (2 mg), vitamin A (0.8 mg), and vitamin D (0.005 mg).</p></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="act001_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.2 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 5 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 2 Temperature</h3><p>What is the freezing point of water on the Kelvin scale?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The freezing point of water is 0 °C, which is +273.15 K.</p></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="act001_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.3 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 15 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 3 Temperature</h3><p>What is the normal range of human body temperature in degrees Celsius? Why is there a range of values?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Body temperature varies with age, time of the day, menstrual cycle and location in the body: in a 6yearold the temperature can vary by around 1 °C per day; your temperature is generally highest in the evening; ovulation can increase the temperature by about 0.6 °C; the body core (gut, liver) is generally warmer than the periphery (arms, legs). Bearing all this in mind, the range 36.537.2 °C is considered normal for a healthy person. 38 °C or above is a significant fever.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7
1.7 SI units and conversionsS110_1<p>The international system of units (<i>Le Système International d'Unités</i>: abbreviated to SI) was developed in France during the 18th century in an effort to create a unified and rational system of weights and measures. The SI system became adopted as the world standard in 1960.</p><p>There are seven basic units (or base units) to the SI system and these are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl003">Table 3</a>. All other units of measurement can be derived from combinations of these base SI units.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl003"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 3 The seven base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
Quantity
</th><th scope="col">
Name
</th><th scope="col">
Symbol
</th></tr><tr><td>Length</td><td>metre</td><td class="oucontenttablemiddle ">m</td></tr><tr><td>Mass</td><td>kilogram</td><td class="oucontenttablemiddle ">kg</td></tr><tr><td>Time</td><td>second</td><td class="oucontenttablemiddle ">s</td></tr><tr><td>Temperature</td><td>kelvin</td><td class="oucontenttablemiddle ">K</td></tr><tr><td>Electric current</td><td>ampere</td><td class="oucontenttablemiddle ">A</td></tr><tr><td>Luminous intensity</td><td>candela</td><td class="oucontenttablemiddle ">cd</td></tr><tr><td>Amount of substance</td><td>mole</td><td class="oucontenttablemiddle ">mol</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>Many units of measurement arise from a combination of different base SI units, and some of these are given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl004">Table 4</a>. For instance, you will see from the table that speed is defined in 'metres per second'. This can be written as metres divided by seconds, i.e. m/s, where the slanted line indicates the act of division. Another way you may see this expressed is as m s<sup>−1</sup> where the negative exponent indicates the division.
</p><p>Similarly, the newton is a SIderived unit of force. It is defined as the amount of force needed to accelerate a mass of 1 kg at a rate of 1 metre per second per second, and is expressed in terms of the following SI units: m kg s<sup>−2</sup>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl004"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 4</b> Commonly used units that are derived from base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
Quantity
</th><th scope="col">
Derived unit
</th><th scope="col">
Symbol
</th><th scope="col">
SI units
</th></tr><tr><td>Speed</td><td>metres per second</td><td class="oucontenttablemiddle ">m s<sup>−1</sup>
</td><td>m s<sup>−1</sup>
</td></tr><tr><td>Force</td><td>newton</td><td class="oucontenttablemiddle ">N</td><td>m kg s<sup>−2</sup>
</td></tr><tr><td>Energy</td><td>joule</td><td class="oucontenttablemiddle ">J</td><td>m<sup>2</sup> kg s<sup>−2</sup>
</td></tr><tr><td>Volume</td><td>cubic metre</td><td class="oucontenttablemiddle ">m<sup>3</sup>
</td><td>m<sup>3</sup>
</td></tr><tr><td>Pressure</td><td>pascal</td><td class="oucontenttablemiddle ">Pa</td><td>kg <i>m</i>
<sup>−1</sup> s<sup>−2</sup>
</td></tr><tr><td>Absorbed dose (radiation)</td><td>gray</td><td class="oucontenttablemiddle ">Gy</td><td>m<sup>2</sup> s<sup>−2</sup>
</td></tr><tr><td>Equivalent dose (radiation)</td><td>sievert</td><td class="oucontenttablemiddle ">Sv</td><td>m<sup>2</sup> s<sup>−2</sup>
</td></tr><tr><td>Radioactivity</td><td>becquerel</td><td class="oucontenttablemiddle ">Bq</td><td>s<sup>−1</sup>
</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>In addition, some nonstandard SI units are in common usage, and a selection of these is given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl005">Table 5</a>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl005"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 5</b> Commonly used units that are not derived from base SI units</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">
<b>Quantity</b>
</th><th scope="col">
<b>Symbol</b>
</th><th scope="col">
<b>NonSI unit</b>
</th><th scope="col">
<b>Conversion factors</b>
</th></tr><tr><td>Time</td><td>min</td><td>minute</td><td>1 min = 60 s</td></tr><tr><td>Time</td><td>h</td><td>hour</td><td>1 h = 60 min = 3600 s</td></tr><tr><td>Time</td><td>d</td><td>day</td><td>1 d = 24 h = 1440 min = 86400 s</td></tr><tr><td>Volume</td><td>1</td><td>litre</td><td>1 l = 0.001 m<sup>3</sup>
</td></tr><tr><td>Mass</td><td>t</td><td>tonne</td><td>1 t = 1000 kg</td></tr><tr><td>Energy</td><td>cal</td><td>calorie</td><td>1 cal = 4.18 J</td></tr><tr><td>Temperature</td><td>°C</td><td>Celsius</td><td>1 °C = 274.15 K</td></tr><tr><td>Pressure</td><td>mmHg</td><td>millimetres of mercury</td><td>1 mmHg = 133.3 Pa</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>When you are dealing with particularly large or small quantities, the SI system is combined with the decimal system, such that you write the decimal notation first then follow it with the relevant SI unit.</p><p>For instance, 1000 metres is 1 <i>kilo</i>metre and this is abbreviated to 1 km. One hundredth of a metre is one <i>centi</i>metre, abbreviated as 1 cm.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 2 A note about temperature</h2><div class="oucontentinnerbox"><p>Although the basic SI unit of temperature is the kelvin, most people use and are familiar with degrees Celsius (°C). (Note that when using kelvins, the 'degrees' symbol '°' is not used.) These two temperature scales are equivalent, so a temperature change of 1 °C is the same actual increase or decrease as a temperature change of 1 K. The only difference is that these temperature scales don't start at the same place. The Celsius scale takes its starting point to be the temperature at which water freezes (0 °C), whilst the Kelvin scale starts from absolute zero (0 K; the coldest temperature theoretically possible, where even molecules stop moving). Zero kelvin is the same as −273.15 °C.</p></div></div></div><p>It is vitally important that you are comfortable with how the SI and decimal systems interact and how to make conversions within these units of measurement. In order to help you achieve this, now test yourself with practice questions 10 and 11.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p><p>Activity 1 will allow you to practise comparing quantities (in this case weights and temperatures) that are in different units of measurement. You can practise similar calculations to those you will be asked to perform in Part 1 for yourself using many common foodstuffs that list the quantities of vitamins, fats, fibre, etc. that they contain. The time you should allow to complete the three parts of the activity is 45 minutes. Please try working through the questions first before looking at the answers.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="act001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.1 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 25 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 1 Vitamin tablets</h3><p>A multivitamin tablet contains various vitamins and minerals. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.7#tbl006">Table 6</a> shows the recommended daily allowance (RDA) for a selection of them, as shown on the packet.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl006"><h4 class="oucontenth3 oucontentheading oucontentnonumber">
<b>Table 6</b> Label from packet of vitamin tablets</h4><div class="oucontenttablewrapper"><table><tr><th scope="col">
Component
</th><th scope="col">
%RDA
</th><th scope="col">
Amount per tablet
</th></tr><tr><td>Vitamin A</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">800 μg</td></tr><tr><td>Vitamin B6</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">2 mg</td></tr><tr><td>Vitamin C</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">60 mg</td></tr><tr><td>Vitamin D</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">5 μg</td></tr><tr><td>Vitamin E</td><td class="oucontenttableright ">100</td><td class="oucontenttablemiddle ">10 mg</td></tr><tr><td>Magnesium</td><td class="oucontenttableright ">50</td><td class="oucontenttablemiddle ">150 mg</td></tr></table></div><div class="oucontentsourcereference"></div></div><p><i>Note: </i>although denoting micrograms as 'μg' is scientifically correct, remember that in the healthcare workplace it's generally better to write it as 'mcg' to avoid possible confusion with milligrams (mg).</p><p>Using this list, can you order the components, from the highest to the lowest amount, by the weight of the substance that equates to 100% RDA?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The first thing to notice is that not all of the units of measurement are the same, some are in milligrams (mg) and others are in micrograms (μg). It helps if you convert all the units of measurement to be the same before attempting to put them in order.</p><p>Only Vitamin A and D are expressed in micrograms, so let's convert them to milligrams. Because a microgram is 1000 times smaller than a milligram, to convert μg into mg, divide by 1000. (Remember, to divide by 1000 you can move the decimal point 3 places to the left: 1 μg = 0.001 mg).</p><p>The amount of Vitamin A present is 800 μg, or 800 ÷ 1000 = 0.8 mg. This is the full RDA (100%), so the RDA is 0.8 mg.</p><p>The amount of Vitamin D present is 5 μg, or 5 ÷ 1000 = 0.005 mg. Again, this is the full RDA, so the RDA is 0.005 mg.</p><p>The amount of magnesium present is 150 mg. However, this is only half of the RDA (50%), so the full RDA is twice this amount:</p><p>150 mg x 2 = 300 mg</p><p>Therefore, in order of decreasing RDA, the components should read: magnesium (300 mg), vitamin C (60 mg), vitamin E (10 mg), vitamin B6 (2 mg), vitamin A (0.8 mg), and vitamin D (0.005 mg).</p></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="act001_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.2 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 5 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 2 Temperature</h3><p>What is the freezing point of water on the Kelvin scale?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The freezing point of water is 0 °C, which is +273.15 K.</p></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="act001_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Activity 1.3 Comparing SI units</h2><div class="oucontentinnerbox"><div class="oucontentsaqtiming">0 hours 15 minutes</div><div class="oucontentsaqquestion"><h3 class="oucontenth4 oucontentbasic">Part 3 Temperature</h3><p>What is the normal range of human body temperature in degrees Celsius? Why is there a range of values?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Body temperature varies with age, time of the day, menstrual cycle and location in the body: in a 6yearold the temperature can vary by around 1 °C per day; your temperature is generally highest in the evening; ovulation can increase the temperature by about 0.6 °C; the body core (gut, liver) is generally warmer than the periphery (arms, legs). Bearing all this in mind, the range 36.537.2 °C is considered normal for a healthy person. 38 °C or above is a significant fever.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.8 Adding and subtracting decimal numbers
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.8
Tue, 12 Apr 2016 23:00:00 GMT
<p>By now you should be familiar with decimal numbers, the different ways in which they can be represented (written in full, or simplified by using an exponent), and understand that by moving the decimal point to the left or right of its original position you can change the value of a decimal number by various factors of ten. In this section, we're going to explore how to add and subtract decimal numbers without using a calculator.</p><p>We use addition and subtraction all the time in our daily lives (for example, when estimating if we've enough money to buy a round of drinks, or comparing prices whilst out shopping). However, addition and subtraction are skills that you need to master, because you cannot rely on there always being a calculator to hand. Although this may sound like a trivial task that anyone can do, surprisingly few people are confident in these two skills, especially when under pressure! (It has been estimated that in England there are 14.9 million people aged between 1665 who lack the skills to pass a maths GCSE qualification (DfES, 2003).)</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.8
1.8 Adding and subtracting decimal numbersS110_1<p>By now you should be familiar with decimal numbers, the different ways in which they can be represented (written in full, or simplified by using an exponent), and understand that by moving the decimal point to the left or right of its original position you can change the value of a decimal number by various factors of ten. In this section, we're going to explore how to add and subtract decimal numbers without using a calculator.</p><p>We use addition and subtraction all the time in our daily lives (for example, when estimating if we've enough money to buy a round of drinks, or comparing prices whilst out shopping). However, addition and subtraction are skills that you need to master, because you cannot rely on there always being a calculator to hand. Although this may sound like a trivial task that anyone can do, surprisingly few people are confident in these two skills, especially when under pressure! (It has been estimated that in England there are 14.9 million people aged between 1665 who lack the skills to pass a maths GCSE qualification (DfES, 2003).)</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.8.1 Study Note 4
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.8.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>If you have difficulty with this section, you might find it helpful to investigate some of the Government schemes aimed at improving maths skills. More information about such schemes can be found at http://www.direct.gov.uk/en/EducationAndLearning/AdultLearning/ImprovingYourSkills/index.htm (accessed 5 March 2008).</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Box 3 The basics
</h2><div class="oucontentinnerbox"><p>When adding or subtracting two or more decimal numbers you need to be sure that they are lined up so that you can compare units with units, tens with tens, and hundreds with hundreds, etc. Even if the numbers contain a decimal point, they can still be lined up into columns of the same magnitude.</p></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.8.1
1.8.1 Study Note 4S110_1<p>If you have difficulty with this section, you might find it helpful to investigate some of the Government schemes aimed at improving maths skills. More information about such schemes can be found at http://www.direct.gov.uk/en/EducationAndLearning/AdultLearning/ImprovingYourSkills/index.htm (accessed 5 March 2008).</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box001_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Box 3 The basics
</h2><div class="oucontentinnerbox"><p>When adding or subtracting two or more decimal numbers you need to be sure that they are lined up so that you can compare units with units, tens with tens, and hundreds with hundreds, etc. Even if the numbers contain a decimal point, they can still be lined up into columns of the same magnitude.</p></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.9 Addition of decimal numbers
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.9
Tue, 12 Apr 2016 23:00:00 GMT
<p>If we add 109.8 ml of one liquid to 6.5 ml of another liquid, what would be the total volume of liquid in ml?</p><p>To compare 109.8 with 6.5, you need to remember that</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e124d4e5/s110_1_i001i.jpg" alt="" width="342" height="52" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>Place the two numbers in a grid on top of each other and make sure that columns representing the <i>same magnitude</i> line up with each other, and add an extra line at the bottom where you can put the result of the addition. You should end up with something that looks like this:</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/860fb600/s110_1_i002i.jpg" alt="" width="245" height="75" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>To add these numbers start from the right, with the smallest sized group and compare the numbers in the Tenths column.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/787b4150/s110_1_i003i.jpg" alt="" width="262" height="119" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the decimal system, ten of one group must be expressed as one of the next sized order of magnitude, so 13 tenths is expressed as <i>one</i> unit and <i>three</i> tenths.</p><p>Put your addition into the results line at the bottom.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/2e0aebcd/s110_1_i004i.jpg" alt="" width="245" height="77" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Then move up one order of magnitude to the next column to the left (Units) and add these values together.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ba9054cc/s110_1_i005i.jpg" alt="" width="342" height="100" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>Add the five units here to the one that's already in the result line, giving six units, and put the single ten into the Tens box in the results line.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4b17834f/s110_1_i006i.jpg" alt="" width="250" height="84" style="maxwidth:250px;" class="oucontentinlinefigureimage"/></span>
</p><p>Moving up one order of magnitude and adding the tens together we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/9bf6de2f/s110_1_ue006i.jpg" alt=""/></div><p>There are no tens to add to the result line, but you keep the 1 that you've already put there from the addition of the units.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/b19df822/s110_1_i007i.jpg" alt="" width="245" height="76" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Move up to the final order of magnitude and add the hundreds together.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c30de0d2/s110_1_i008i.jpg" alt="" width="262" height="119" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>So, as you can see from your results line, the sum of 109.8 ml and 6.5 ml = 116.3 ml</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/1b530776/s110_1_i009i.jpg" alt="" width="245" height="77" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Now practise some additions for yourself with practice question 12.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.9
1.9 Addition of decimal numbersS110_1<p>If we add 109.8 ml of one liquid to 6.5 ml of another liquid, what would be the total volume of liquid in ml?</p><p>To compare 109.8 with 6.5, you need to remember that</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e124d4e5/s110_1_i001i.jpg" alt="" width="342" height="52" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>Place the two numbers in a grid on top of each other and make sure that columns representing the <i>same magnitude</i> line up with each other, and add an extra line at the bottom where you can put the result of the addition. You should end up with something that looks like this:</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/860fb600/s110_1_i002i.jpg" alt="" width="245" height="75" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>To add these numbers start from the right, with the smallest sized group and compare the numbers in the Tenths column.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/787b4150/s110_1_i003i.jpg" alt="" width="262" height="119" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the decimal system, ten of one group must be expressed as one of the next sized order of magnitude, so 13 tenths is expressed as <i>one</i> unit and <i>three</i> tenths.</p><p>Put your addition into the results line at the bottom.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/2e0aebcd/s110_1_i004i.jpg" alt="" width="245" height="77" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Then move up one order of magnitude to the next column to the left (Units) and add these values together.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ba9054cc/s110_1_i005i.jpg" alt="" width="342" height="100" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>Add the five units here to the one that's already in the result line, giving six units, and put the single ten into the Tens box in the results line.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4b17834f/s110_1_i006i.jpg" alt="" width="250" height="84" style="maxwidth:250px;" class="oucontentinlinefigureimage"/></span>
</p><p>Moving up one order of magnitude and adding the tens together we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/9bf6de2f/s110_1_ue006i.jpg" alt=""/></div><p>There are no tens to add to the result line, but you keep the 1 that you've already put there from the addition of the units.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/b19df822/s110_1_i007i.jpg" alt="" width="245" height="76" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Move up to the final order of magnitude and add the hundreds together.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c30de0d2/s110_1_i008i.jpg" alt="" width="262" height="119" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>So, as you can see from your results line, the sum of 109.8 ml and 6.5 ml = 116.3 ml</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/1b530776/s110_1_i009i.jpg" alt="" width="245" height="77" style="maxwidth:245px;" class="oucontentinlinefigureimage"/></span>
</p><p>Now practise some additions for yourself with practice question 12.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.10 Subtraction of decimal numbers
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.10
Tue, 12 Apr 2016 23:00:00 GMT
<p>Subtraction of numbers can be used to answer questions such as 'what's the difference between two values?' or 'if something has decreased by a certain amount, what's its new value?' Subtraction can also be thought of as undoing the process of addition. For instance, instead of saying '£10 take away £7.85 leaves how much?' you could say, 'what do I have to add to £7.85 to get back to £10?'</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/eccf4839/s110_1_i010i.jpg" alt="Source: www.CartoonStock.com" width="288" height="299" style="maxwidth:288px;" class="oucontentinlinefigureimage"/></span>
</p><p>Let's work through an example: how much is 25.18  16.87?</p><p>The way to write down decimal numbers for a subtraction is the same as for an addition in terms of arranging them into columns that contain numbers of the same size category (hundreds, tens, units, etc.). The only difference is that the number that you want to subtract <i>must go below the number you are subtracting it from</i>. Just like addition, you should work through your calculation from right to left.</p><p>The other important rule to remember with subtractions is that, for each column, if the number you are subtracting from is smaller than the number you want to subtract, then you need to make that number larger by 'giving it' one of the units from the next highest order of magnitude.</p><p>Let's see how that works out in practice with our example of 25.18  16.87.</p><p>In the first column from the right (Hundredths), we need to subtract 7 from 8. Eight take away seven leaves us with one hundredth.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e8ae3790/s110_1_i011i.jpg" alt="" width="258" height="73" style="maxwidth:258px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the next column (Tenths), we need to subtract 8 from 1. In order to do this, remove one unit from the next highest order of magnitude (here, units) and add this to the tenths. One unit is the same as ten tenths, so we add ten to the Tenths column: 1 tenth plus 10 tenths, gives 11 tenths in this column.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/3d44c48d/s110_1_i012i.jpg" alt="" width="262" height="88" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>Now we can subtract 8 from the resulting 11 tenths, to leave a total of 3 tenths.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/527334c6/s110_1_i013i.jpg" alt="" width="258" height="73" style="maxwidth:258px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the next column (Units) we need to subtract 6 from 4, so once again we need to give this column a unit from the next highest order of magnitude (tens). 1 ten is 10 units and if we add this to the 4 units we already have, this gives 14 units in this column. Now we can subtract 6 from 14, to leave 8 units.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/039b9633/s110_1_i014i.jpg" alt="" width="235" height="79" style="maxwidth:235px;" class="oucontentinlinefigureimage"/></span>
</p><p>Finally, in the Tens column, we're left with 1 ten to subtract from 1 ten, which leaves 0 tens.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e661a77b/s110_1_i015i.jpg" alt="" width="256" height="74" style="maxwidth:256px;" class="oucontentinlinefigureimage"/></span>
</p><p>So, the result of 25.18  16.87 = 8.31.</p><p>You can doublecheck you're right by adding 8.31 to 16.87 in order to confirm you get back to the original 25.18.</p><p>Now practise some subtractions for yourself with practice question 13.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.10
1.10 Subtraction of decimal numbersS110_1<p>Subtraction of numbers can be used to answer questions such as 'what's the difference between two values?' or 'if something has decreased by a certain amount, what's its new value?' Subtraction can also be thought of as undoing the process of addition. For instance, instead of saying '£10 take away £7.85 leaves how much?' you could say, 'what do I have to add to £7.85 to get back to £10?'</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/eccf4839/s110_1_i010i.jpg" alt="Source: www.CartoonStock.com" width="288" height="299" style="maxwidth:288px;" class="oucontentinlinefigureimage"/></span>
</p><p>Let's work through an example: how much is 25.18  16.87?</p><p>The way to write down decimal numbers for a subtraction is the same as for an addition in terms of arranging them into columns that contain numbers of the same size category (hundreds, tens, units, etc.). The only difference is that the number that you want to subtract <i>must go below the number you are subtracting it from</i>. Just like addition, you should work through your calculation from right to left.</p><p>The other important rule to remember with subtractions is that, for each column, if the number you are subtracting from is smaller than the number you want to subtract, then you need to make that number larger by 'giving it' one of the units from the next highest order of magnitude.</p><p>Let's see how that works out in practice with our example of 25.18  16.87.</p><p>In the first column from the right (Hundredths), we need to subtract 7 from 8. Eight take away seven leaves us with one hundredth.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e8ae3790/s110_1_i011i.jpg" alt="" width="258" height="73" style="maxwidth:258px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the next column (Tenths), we need to subtract 8 from 1. In order to do this, remove one unit from the next highest order of magnitude (here, units) and add this to the tenths. One unit is the same as ten tenths, so we add ten to the Tenths column: 1 tenth plus 10 tenths, gives 11 tenths in this column.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/3d44c48d/s110_1_i012i.jpg" alt="" width="262" height="88" style="maxwidth:262px;" class="oucontentinlinefigureimage"/></span>
</p><p>Now we can subtract 8 from the resulting 11 tenths, to leave a total of 3 tenths.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/527334c6/s110_1_i013i.jpg" alt="" width="258" height="73" style="maxwidth:258px;" class="oucontentinlinefigureimage"/></span>
</p><p>In the next column (Units) we need to subtract 6 from 4, so once again we need to give this column a unit from the next highest order of magnitude (tens). 1 ten is 10 units and if we add this to the 4 units we already have, this gives 14 units in this column. Now we can subtract 6 from 14, to leave 8 units.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/039b9633/s110_1_i014i.jpg" alt="" width="235" height="79" style="maxwidth:235px;" class="oucontentinlinefigureimage"/></span>
</p><p>Finally, in the Tens column, we're left with 1 ten to subtract from 1 ten, which leaves 0 tens.</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/e661a77b/s110_1_i015i.jpg" alt="" width="256" height="74" style="maxwidth:256px;" class="oucontentinlinefigureimage"/></span>
</p><p>So, the result of 25.18  16.87 = 8.31.</p><p>You can doublecheck you're right by adding 8.31 to 16.87 in order to confirm you get back to the original 25.18.</p><p>Now practise some subtractions for yourself with practice question 13.</p><p>Right click and open the <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://students.open.ac.uk/openmark/ols110.practice1/"> practice questions</a></span> in a separate window, then you can switch easily between the course text and the questions.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

1.11 Addition and subtraction in practice  fluid balance
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.11
Tue, 12 Apr 2016 23:00:00 GMT
<p>A common healthcare example that uses addition and subtraction involves calculating the fluid balance of a patient.</p><p>Fluid balance is a simple but very useful way to estimate whether a patient is either becoming dehydrated or overfilled with liquids. It is calculated, on a daily basis, by adding up the total volume of liquid that has gone into their body (drinks, oral liquid medicines, intravenous drips, transfusions), then adding up the total volume of liquid that has come out of their body (urine, wound drains, blood lost during surgery, vomit). The fluid balance is then calculated by subtracting the total output from the total input, and is generally quoted in millilitres.</p><p>Ideally, the total volume of liquids that goes into a person ought to balance the total volume that eventually comes out of them, so the difference in the total input and output should be almost zero. However, if the fluid balance is positive, then this indicates that more liquid is going in than is coming out (i.e. they are swelling up: not necessarily a bad thing if, for instance, they were admitted suffering from dehydration). A negative fluid balance indicates that more fluid is coming out than is going in and the patient is at risk of becoming dehydrated.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq001_011_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Calculate the fluid balance for the patient described in the following scenario.</p><p>Over the course of a day, a patient who has just undergone chest surgery receives an intravenous saline drip of 1 litre and another of 900 ml. They drink 550 ml of water and 200 ml of fruit juice. During the same day, they produce 2.5 litres of urine, and they lose 110 ml of fluid from a tube that is draining their chest wound.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The fluid balance is the difference between the total fluid input and the total fluid output during a day.</p><p>To calculate this, first, check whether the units of measurement are in millilitres; convert them to millilitres, if necessary.</p><p>Then add together all the fluid inputs in millilitres.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/846d5d2e/s110_1_ue007i.gif" alt=""/></div><p>Then add together all the fluid outputs in millilitres.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/051cac59/s110_1_ue008i.gif" alt=""/></div><p>The fluid balance is the daily total inputs minus the daily total outputs.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/40e417ac/s110_1_ue009i.gif" alt=""/></div><p>This is a positive fluid balance, indicating the patient is taking on more fluid than they are losing, although only by a tiny amount (40 ml is about half a tea cup). If the output had been larger than the input, then the fluid balance would be a negative number (e.g. input 2000 ml  output 2500 ml = 500 ml fluid balance).</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection1.11
1.11 Addition and subtraction in practice  fluid balanceS110_1<p>A common healthcare example that uses addition and subtraction involves calculating the fluid balance of a patient.</p><p>Fluid balance is a simple but very useful way to estimate whether a patient is either becoming dehydrated or overfilled with liquids. It is calculated, on a daily basis, by adding up the total volume of liquid that has gone into their body (drinks, oral liquid medicines, intravenous drips, transfusions), then adding up the total volume of liquid that has come out of their body (urine, wound drains, blood lost during surgery, vomit). The fluid balance is then calculated by subtracting the total output from the total input, and is generally quoted in millilitres.</p><p>Ideally, the total volume of liquids that goes into a person ought to balance the total volume that eventually comes out of them, so the difference in the total input and output should be almost zero. However, if the fluid balance is positive, then this indicates that more liquid is going in than is coming out (i.e. they are swelling up: not necessarily a bad thing if, for instance, they were admitted suffering from dehydration). A negative fluid balance indicates that more fluid is coming out than is going in and the patient is at risk of becoming dehydrated.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq001_011_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Calculate the fluid balance for the patient described in the following scenario.</p><p>Over the course of a day, a patient who has just undergone chest surgery receives an intravenous saline drip of 1 litre and another of 900 ml. They drink 550 ml of water and 200 ml of fruit juice. During the same day, they produce 2.5 litres of urine, and they lose 110 ml of fluid from a tube that is draining their chest wound.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>The fluid balance is the difference between the total fluid input and the total fluid output during a day.</p><p>To calculate this, first, check whether the units of measurement are in millilitres; convert them to millilitres, if necessary.</p><p>Then add together all the fluid inputs in millilitres.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/846d5d2e/s110_1_ue007i.gif" alt=""/></div><p>Then add together all the fluid outputs in millilitres.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/051cac59/s110_1_ue008i.gif" alt=""/></div><p>The fluid balance is the daily total inputs minus the daily total outputs.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/40e417ac/s110_1_ue009i.gif" alt=""/></div><p>This is a positive fluid balance, indicating the patient is taking on more fluid than they are losing, although only by a tiny amount (40 ml is about half a tea cup). If the output had been larger than the input, then the fluid balance would be a negative number (e.g. input 2000 ml  output 2500 ml = 500 ml fluid balance).</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

2.1 Differences between accuracy and precision
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Accuracy is a measure of how close a result is to the true value. Precision is a measure of how repeatable the result is. For instance, a group of three friends tried the shooting gallery at a fair and their targets are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.1#fig003001">Figure 6</a>. The first person was an expert marksman, but they were using a rifle with sights that had not been set properly. Although they aimed the sights at the bull'seye they consistently hit the target off to the left side instead. They were not accurate, but they were precise. The second person was also an expert marksman, but noticed the incorrectly set sights and compensated by aiming to the right of the bull'seye. Consequently, all their shots hit the centre of the target  they were both accurate and precise and their results were good. The third person was hopeless with the rifle and their shots landed all over the target  they were neither accurate nor precise.</p><div class="oucontentfigure" style="width:460px;" id="fig003_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/da24c186/s110_1_006i.jpg" alt="Figure 6" width="460" height="196" style="maxwidth:460px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 Target practice analogy, demonstrating the difference between accuracy and precision. Low accuracy but high precision = systematic error. High accuracy and high precision = good results. Low accuracy and low precision = bad results</span></div></div></div><p>From the example given you can see how it is possible to be very precise, but not at all accurate. This is called a <i>systematic error</i> (sometimes also called <i>bias</i>) and can normally be corrected.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.1
2.1 Differences between accuracy and precisionS110_1<p>Accuracy is a measure of how close a result is to the true value. Precision is a measure of how repeatable the result is. For instance, a group of three friends tried the shooting gallery at a fair and their targets are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.1#fig003001">Figure 6</a>. The first person was an expert marksman, but they were using a rifle with sights that had not been set properly. Although they aimed the sights at the bull'seye they consistently hit the target off to the left side instead. They were not accurate, but they were precise. The second person was also an expert marksman, but noticed the incorrectly set sights and compensated by aiming to the right of the bull'seye. Consequently, all their shots hit the centre of the target  they were both accurate and precise and their results were good. The third person was hopeless with the rifle and their shots landed all over the target  they were neither accurate nor precise.</p><div class="oucontentfigure" style="width:460px;" id="fig003_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/da24c186/s110_1_006i.jpg" alt="Figure 6" width="460" height="196" style="maxwidth:460px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 Target practice analogy, demonstrating the difference between accuracy and precision. Low accuracy but high precision = systematic error. High accuracy and high precision = good results. Low accuracy and low precision = bad results</span></div></div></div><p>From the example given you can see how it is possible to be very precise, but not at all accurate. This is called a <i>systematic error</i> (sometimes also called <i>bias</i>) and can normally be corrected.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

2.1.1 Accuracy
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.2.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>The way to ensure that equipment is accurate is to use a series of known standards against which to calibrate the equipment. Calibrating should be done at least each day and sometimes more frequently (such as before using the equipment to measure unknown samples). Many types of measuring equipment go through an automatic calibration when they are switched on, but others require the user to provide a series of known calibration standards.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.2.1
2.1.1 AccuracyS110_1<p>The way to ensure that equipment is accurate is to use a series of known standards against which to calibrate the equipment. Calibrating should be done at least each day and sometimes more frequently (such as before using the equipment to measure unknown samples). Many types of measuring equipment go through an automatic calibration when they are switched on, but others require the user to provide a series of known calibration standards.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

2.2.2 Precision
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.2.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>Measuring the same sample should give the same result every time if the equipment is precise. In practice, the information displayed by a measuring device can depend on several factors (such as temperature and humidity) and can drift slightly over time. Nevertheless, during the time it takes to complete a measurement sequence, all measurements ought to remain within a specified, small margin of error, often marked on the equipment. We will see later on, in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5">Section 3.5</a>, how to quantify the precision of a series of measurements.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.2.2
2.2.2 PrecisionS110_1<p>Measuring the same sample should give the same result every time if the equipment is precise. In practice, the information displayed by a measuring device can depend on several factors (such as temperature and humidity) and can drift slightly over time. Nevertheless, during the time it takes to complete a measurement sequence, all measurements ought to remain within a specified, small margin of error, often marked on the equipment. We will see later on, in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5">Section 3.5</a>, how to quantify the precision of a series of measurements.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

2.3 Common maths problems and errors in the workplace
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.3
Tue, 12 Apr 2016 23:00:00 GMT
<p>In a busy, hospital environment mistakes with medicines and other treatments can happen at any time. Some of these are caused by communication/administrative problems, whilst others are due to mathematical errors (the news stories shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.3#fig003002">Figure 7</a> are sadly typical).</p><div class="oucontentfigure" style="width:469px;" id="fig003_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/6bc41d26/s110_1_007i.jpg" alt="Figure 7" width="469" height="647" style="maxwidth:469px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 Mistakes with placing a decimal point can cost lives; articles available from http://news.bbc.co.uk and http://news.scotsman.com (websites accessed 6 March 2008)</span></div></div></div><p>However, many of these types of errors can be avoided by pausing and thinking, 'is this right'?</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.3
2.3 Common maths problems and errors in the workplaceS110_1<p>In a busy, hospital environment mistakes with medicines and other treatments can happen at any time. Some of these are caused by communication/administrative problems, whilst others are due to mathematical errors (the news stories shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.3#fig003002">Figure 7</a> are sadly typical).</p><div class="oucontentfigure" style="width:469px;" id="fig003_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/6bc41d26/s110_1_007i.jpg" alt="Figure 7" width="469" height="647" style="maxwidth:469px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 Mistakes with placing a decimal point can cost lives; articles available from http://news.bbc.co.uk and http://news.scotsman.com (websites accessed 6 March 2008)</span></div></div></div><p>However, many of these types of errors can be avoided by pausing and thinking, 'is this right'?</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

2.4 Sources of errors
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.4
Tue, 12 Apr 2016 23:00:00 GMT
<p>The following is a list of common problems that can lead to medication errors. They fall into three broad categories according to where they occur in the sequence from a drug being prescribed to it being administered to a patient. As you can see, the same types of mistake can occur in each category. Those errors that involve maths are highlighted in <i>italics</i>:
</p><p><b>Prescription errors</b></p><ul class="oucontentbulleted"><li>
<p>Wrong drug prescribed (contraindicated, or allergy, or interferes with existing drug therapy).</p>
</li><li>
<p>Prescription illegible/<i>misread (number, decimal point, units of measurement)</i> or unsigned.</p>
</li><li>
<p>Wrong dosage form selected (tablets vs. liquid).</p>
</li><li>
<p>Wrong administration route selected (oral, intravenous, etc.).</p>
</li><li>
<p>
<i>Wrong prescribed drug concentration, or drug quantity/volume, or drug rate of administration</i>.</p>
</li><li>
<p>Verbal (orally communicated) orders.</p>
</li></ul><p><b>Dispensing errors</b></p><ul class="oucontentbulleted"><li>
<p>Drug not available.</p>
</li><li>
<p>Drug preparation error (wrong drug selected or <i>misreading/miscalculation of drug concentration, or drug quantity/volume</i>).</p>
</li><li>
<p>Equipment failure/malfunction.</p>
</li><li>
<p>Labelling error (e.g. wrongly labelled syringe).</p>
</li></ul><p><b>Administration/omission errors</b></p><ul class="oucontentbulleted"><li>
<p>Misreading/<i>miscalculating drug quantity/volume or drug rate of administration, or dose of radiation</i>, or route of administration.</p>
</li><li>
<p>Wrong drug administration technique/equipment.</p>
</li><li>
<p>Equipment failure/malfunction.</p>
</li><li>
<p>Drug chart not kept up to date/needs rewriting.</p>
</li><li>
<p>Drug not given at correct time, or correct frequency, or not given at all.</p>
</li><li>
<p>Inadequate patient ID or drug given to wrong patient.</p>
</li></ul><p>In addition, the following factors can all contribute to medication errors:</p><ul class="oucontentbulleted"><li>
<p>excessive workload;</p>
</li><li>
<p>lapses in individual performance;</p>
</li><li>
<p>inadequate training;</p>
</li><li>
<p>inadequate communication.</p>
</li></ul><p>(Adapted from ASHP, 1993)</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box003_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 4 How can you help to minimise errors?</h2><div class="oucontentinnerbox"><p>Take a proactive role in minimising errors by keeping yourself up to date with the latest alerts and safe medical practices.</p><p>The <i>Medicines and Healthcare Products Regulatory Agency</i> website (http://www.mhra.gov.uk/) is run by the UK Government (website accessed 13 March 2014). It has uptodate safety information and alerts about problems with medicines, especially problems concerned with the labelling and packaging of medicines that might lead to drug errors.</p><p>The <i>Institute for Safe Medication Practices</i> (in the USA) also has a good website (http://www.ismp.org/) that you can browse for more information (website accessed 13 March 2014). Of particular interest are the newsletters that highlight potential risks to be aware of and good practice to adopt. You may wish to look at some of the articles and extracts that are available from the current issue and past issues of the newsletter, or search for a particular topic of interest (note: for the purposes of this course it is not necessary to subscribe to the newsletters). At present (2007), this website only covers America and Spain, so whilst there's much useful and practical advice here, please be aware that procedural differences may occur in the UK.</p></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox oucontentsnoheading " id="act002_001"><div class="oucontentouterbox"><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>The four audio tracks linked below contain information that reinforces what you have just learned here. Listen to them now.</p><p>Click to listen to the track [3 minutes 56 seconds, 4.50MB]</p><div id="mp3_002" class="oucontentmedia oucontentaudiovideo ompversion1" style="width:342px;"><div class="oucontentdefaultfilter"><span class="oumediafilter"><a href="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ed0fbb55/s110_1_002s.mp3?forcedownload=1" class="oumedialinknoscript ompspacer">Download this audio clip.</a><span class="accesshide">Audio player: s110_1_002s.mp3</span><a href="#" class="ompentermedia ompaccesshide" tabindex="1">
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</span></div><div class="oucontentfiguretext"><div class="oucontentmediadownload"><a href="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/f7430b8c/s110_1_005s.mp3?forcedownload=1" title="Download this audio clip">Download</a></div><div class="oucontentcaption oucontentnonumber oucontentcaptionplaceholder"> </div></div></div><div class="oucontentinteractionprint"><div class="oucontentinteractionunavailable">Interactive feature not available in single page view (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.4#mp3005">see it in standard view</a>).</div></div></div></div></div></div><p>As you have just seen from the errors list above, it is not always safe to assume that a prescription has been filled out correctly. The most common numerical mistakes are differences by a factor of 10, 100 or 1000, either because of a mistake with a calculation or because the wrong unit of measurement has been written down. For instance, as you have seen, when handwritten, the symbols for '<i>milli</i>' (m) and '<i>micro</i>' (μ) can often be confused, but are different by a factor of 1000, which is why 'mcg' is routinely used in hospitals to indicate micrograms.</p><p>With experience, you will gain a commonsense knowledge of when particular calculations are correct or not, and by developing a reflective sense of your working practices you will quickly become better and more confident with these routine calculations. In particular, you should get used to the recommended dose ranges of the drugs or reagents you routinely deal with, and check the dose range against your calculations for all new drugs that you come across.</p> <script>
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https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.4
2.4 Sources of errorsS110_1<p>The following is a list of common problems that can lead to medication errors. They fall into three broad categories according to where they occur in the sequence from a drug being prescribed to it being administered to a patient. As you can see, the same types of mistake can occur in each category. Those errors that involve maths are highlighted in <i>italics</i>:
</p><p><b>Prescription errors</b></p><ul class="oucontentbulleted"><li>
<p>Wrong drug prescribed (contraindicated, or allergy, or interferes with existing drug therapy).</p>
</li><li>
<p>Prescription illegible/<i>misread (number, decimal point, units of measurement)</i> or unsigned.</p>
</li><li>
<p>Wrong dosage form selected (tablets vs. liquid).</p>
</li><li>
<p>Wrong administration route selected (oral, intravenous, etc.).</p>
</li><li>
<p>
<i>Wrong prescribed drug concentration, or drug quantity/volume, or drug rate of administration</i>.</p>
</li><li>
<p>Verbal (orally communicated) orders.</p>
</li></ul><p><b>Dispensing errors</b></p><ul class="oucontentbulleted"><li>
<p>Drug not available.</p>
</li><li>
<p>Drug preparation error (wrong drug selected or <i>misreading/miscalculation of drug concentration, or drug quantity/volume</i>).</p>
</li><li>
<p>Equipment failure/malfunction.</p>
</li><li>
<p>Labelling error (e.g. wrongly labelled syringe).</p>
</li></ul><p><b>Administration/omission errors</b></p><ul class="oucontentbulleted"><li>
<p>Misreading/<i>miscalculating drug quantity/volume or drug rate of administration, or dose of radiation</i>, or route of administration.</p>
</li><li>
<p>Wrong drug administration technique/equipment.</p>
</li><li>
<p>Equipment failure/malfunction.</p>
</li><li>
<p>Drug chart not kept up to date/needs rewriting.</p>
</li><li>
<p>Drug not given at correct time, or correct frequency, or not given at all.</p>
</li><li>
<p>Inadequate patient ID or drug given to wrong patient.</p>
</li></ul><p>In addition, the following factors can all contribute to medication errors:</p><ul class="oucontentbulleted"><li>
<p>excessive workload;</p>
</li><li>
<p>lapses in individual performance;</p>
</li><li>
<p>inadequate training;</p>
</li><li>
<p>inadequate communication.</p>
</li></ul><p>(Adapted from ASHP, 1993)</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box003_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 4 How can you help to minimise errors?</h2><div class="oucontentinnerbox"><p>Take a proactive role in minimising errors by keeping yourself up to date with the latest alerts and safe medical practices.</p><p>The <i>Medicines and Healthcare Products Regulatory Agency</i> website (http://www.mhra.gov.uk/) is run by the UK Government (website accessed 13 March 2014). It has uptodate safety information and alerts about problems with medicines, especially problems concerned with the labelling and packaging of medicines that might lead to drug errors.</p><p>The <i>Institute for Safe Medication Practices</i> (in the USA) also has a good website (http://www.ismp.org/) that you can browse for more information (website accessed 13 March 2014). Of particular interest are the newsletters that highlight potential risks to be aware of and good practice to adopt. You may wish to look at some of the articles and extracts that are available from the current issue and past issues of the newsletter, or search for a particular topic of interest (note: for the purposes of this course it is not necessary to subscribe to the newsletters). At present (2007), this website only covers America and Spain, so whilst there's much useful and practical advice here, please be aware that procedural differences may occur in the UK.</p></div></div></div><div class="
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</span></div><div class="oucontentfiguretext"><div class="oucontentmediadownload"><a href="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/f7430b8c/s110_1_005s.mp3?forcedownload=1" title="Download this audio clip">Download</a></div><div class="oucontentcaption oucontentnonumber oucontentcaptionplaceholder"> </div></div></div><div class="oucontentinteractionprint"><div class="oucontentinteractionunavailable">Interactive feature not available in single page view (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.4#mp3005">see it in standard view</a>).</div></div></div></div></div></div><p>As you have just seen from the errors list above, it is not always safe to assume that a prescription has been filled out correctly. The most common numerical mistakes are differences by a factor of 10, 100 or 1000, either because of a mistake with a calculation or because the wrong unit of measurement has been written down. For instance, as you have seen, when handwritten, the symbols for '<i>milli</i>' (m) and '<i>micro</i>' (μ) can often be confused, but are different by a factor of 1000, which is why 'mcg' is routinely used in hospitals to indicate micrograms.</p><p>With experience, you will gain a commonsense knowledge of when particular calculations are correct or not, and by developing a reflective sense of your working practices you will quickly become better and more confident with these routine calculations. In particular, you should get used to the recommended dose ranges of the drugs or reagents you routinely deal with, and check the dose range against your calculations for all new drugs that you come across.</p> <script>
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2.5 What is a sensible dose?
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.5
Tue, 12 Apr 2016 23:00:00 GMT
<p>This will vary from drug to drug and patient to patient, but bear in mind that most drugs need to be swallowed or injected, so the manufacturer has designed the dose sizes to be as easy as possible for a patient to take and for the health worker to administer.</p><p>The following dose ranges are the most sensible and practical for adults:</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl007"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 7 Typical drug doses</h2><div class="oucontenttablewrapper"><table><tr><th scope="col"><b>Drug formulation</b></th><th scope="col"><b>Typical dose at any one time</b></th></tr><tr><td>Liquid</td><td>oral: 520 ml (14 teaspoons full)</td></tr><tr><td></td><td>injection: generally 0.252 ml</td></tr><tr><td></td><td>subcutaneous: 1 ml or less</td></tr><tr><td></td><td>intramuscular: adults  up to 3 ml in large muscles</td></tr><tr><td></td><td> children and elderly  up to 2 ml</td></tr><tr><td></td><td> infants  1 ml or less</td></tr><tr><td>Solid</td><td>14 tablets</td></tr><tr><td>Gas</td><td>0.2150 litres/min</td></tr><tr><td>Radiation dose</td><td>2070 μSv for an Xray, 100200 Sv for radiotherapy</td></tr></table></div><div class="oucontenttablefootnote">For each category, the doses for a baby or a child are normally much less than adult doses.</div></div><p>If you find that the dose you have calculated or the prescription you have been given is outside of this range, especially if it is out by a factor of 10, 100 etc., then it's likely that a mistake has happened somewhere. If it's your own calculation, then doublecheck it. If it still doesn't look right, or was written that way on the prescription then check with a senior colleague.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection2.5
2.5 What is a sensible dose?S110_1<p>This will vary from drug to drug and patient to patient, but bear in mind that most drugs need to be swallowed or injected, so the manufacturer has designed the dose sizes to be as easy as possible for a patient to take and for the health worker to administer.</p><p>The following dose ranges are the most sensible and practical for adults:</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl007"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 7 Typical drug doses</h2><div class="oucontenttablewrapper"><table><tr><th scope="col"><b>Drug formulation</b></th><th scope="col"><b>Typical dose at any one time</b></th></tr><tr><td>Liquid</td><td>oral: 520 ml (14 teaspoons full)</td></tr><tr><td></td><td>injection: generally 0.252 ml</td></tr><tr><td></td><td>subcutaneous: 1 ml or less</td></tr><tr><td></td><td>intramuscular: adults  up to 3 ml in large muscles</td></tr><tr><td></td><td> children and elderly  up to 2 ml</td></tr><tr><td></td><td> infants  1 ml or less</td></tr><tr><td>Solid</td><td>14 tablets</td></tr><tr><td>Gas</td><td>0.2150 litres/min</td></tr><tr><td>Radiation dose</td><td>2070 μSv for an Xray, 100200 Sv for radiotherapy</td></tr></table></div><div class="oucontenttablefootnote">For each category, the doses for a baby or a child are normally much less than adult doses.</div></div><p>If you find that the dose you have calculated or the prescription you have been given is outside of this range, especially if it is out by a factor of 10, 100 etc., then it's likely that a mistake has happened somewhere. If it's your own calculation, then doublecheck it. If it still doesn't look right, or was written that way on the prescription then check with a senior colleague.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.1 Graphs
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Information is everywhere these days  in the form of images, written records, tables and graphs. In this part of the course we want you to realise how useful graphs can be to analyse numerical information, and to show you some techniques that can help you decide how reliable this numerical information is.</p><p>It's often difficult to spot a trend or a relationship in a long list of numbers. Because the human mind is highly adapted to recognising visual patterns, it is often much easier to understand a series of numbers or measurements by representing them visually as a graph.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.1
3.1 GraphsS110_1<p>Information is everywhere these days  in the form of images, written records, tables and graphs. In this part of the course we want you to realise how useful graphs can be to analyse numerical information, and to show you some techniques that can help you decide how reliable this numerical information is.</p><p>It's often difficult to spot a trend or a relationship in a long list of numbers. Because the human mind is highly adapted to recognising visual patterns, it is often much easier to understand a series of numbers or measurements by representing them visually as a graph.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.2 The anatomy of a graph
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>A graph shows how two different types of data that can take on different values (known as <b>variables</b>) are related, or change in relation to each other; for instance, how a patient's temperature changes over time. Each measurement consists of two variable values: the patient's temperature and the time at which the temperature was taken. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2#tbl008">Table 8</a> shows what these measurements might look like.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl008"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 8 A patient's temperature, measured throughout the day</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">Time of day</th><th scope="col">Patient temperature/°C</th></tr><tr><td>13:00</td><td class="oucontenttablemiddle ">38.2</td></tr><tr><td>14:00</td><td class="oucontenttablemiddle ">37.9</td></tr><tr><td>15:00</td><td class="oucontenttablemiddle ">37.9</td></tr><tr><td>16:00</td><td class="oucontenttablemiddle ">37.5</td></tr><tr><td>18:00</td><td class="oucontenttablemiddle ">37.1</td></tr><tr><td>19:00</td><td class="oucontenttablemiddle ">36.9</td></tr></table></div><div class="oucontenttablefootnote">Notice that a measurement was not made at 17:00.</div></div><p>As we shall see, this information can become easier to interrogate once it has been assembled graphically.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2
3.2 The anatomy of a graphS110_1<p>A graph shows how two different types of data that can take on different values (known as <b>variables</b>) are related, or change in relation to each other; for instance, how a patient's temperature changes over time. Each measurement consists of two variable values: the patient's temperature and the time at which the temperature was taken. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2#tbl008">Table 8</a> shows what these measurements might look like.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl008"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 8 A patient's temperature, measured throughout the day</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">Time of day</th><th scope="col">Patient temperature/°C</th></tr><tr><td>13:00</td><td class="oucontenttablemiddle ">38.2</td></tr><tr><td>14:00</td><td class="oucontenttablemiddle ">37.9</td></tr><tr><td>15:00</td><td class="oucontenttablemiddle ">37.9</td></tr><tr><td>16:00</td><td class="oucontenttablemiddle ">37.5</td></tr><tr><td>18:00</td><td class="oucontenttablemiddle ">37.1</td></tr><tr><td>19:00</td><td class="oucontenttablemiddle ">36.9</td></tr></table></div><div class="oucontenttablefootnote">Notice that a measurement was not made at 17:00.</div></div><p>As we shall see, this information can become easier to interrogate once it has been assembled graphically.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.2.1 Axes
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>A graph is made using two different scales or <i>axes</i>, forming a right angle. The horizontal axis (xaxis) is used to represent the variable that changes in a consistent way, such as time, or in a way that you can control. The vertical axis (yaxis) is used to represent a variable that you measure but may not be able to control directly, such as a patient's temperature.</p><p>Each axis should be carefully labelled to indicate what it represents. To plot a graph, you put a mark at the point where the two variables in each measurement meet.</p><p>One way to remember which way round the axes go is to remember that X comes before Y in the alphabet and then picture entering someone's house; you go <i>along</i> the corridor (horizontal, xaxis) before you can go <i>up</i> the stairs (vertical, yaxis).</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2.1
3.2.1 AxesS110_1<p>A graph is made using two different scales or <i>axes</i>, forming a right angle. The horizontal axis (xaxis) is used to represent the variable that changes in a consistent way, such as time, or in a way that you can control. The vertical axis (yaxis) is used to represent a variable that you measure but may not be able to control directly, such as a patient's temperature.</p><p>Each axis should be carefully labelled to indicate what it represents. To plot a graph, you put a mark at the point where the two variables in each measurement meet.</p><p>One way to remember which way round the axes go is to remember that X comes before Y in the alphabet and then picture entering someone's house; you go <i>along</i> the corridor (horizontal, xaxis) before you can go <i>up</i> the stairs (vertical, yaxis).</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.2.2 Choice of scale
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>It's important to choose a scale that covers the range of values you have recorded for that particular axis. If the scale is too big, then all of your measurements will be bunched up at one end of the graph, making it difficult to read. It is also very important to keep the scale consistent all along the axis, i.e. don't suddenly change the spacing between the units of measurement on an axis.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2.2
3.2.2 Choice of scaleS110_1<p>It's important to choose a scale that covers the range of values you have recorded for that particular axis. If the scale is too big, then all of your measurements will be bunched up at one end of the graph, making it difficult to read. It is also very important to keep the scale consistent all along the axis, i.e. don't suddenly change the spacing between the units of measurement on an axis.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.3 Types of graphs and their uses
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3
Tue, 12 Apr 2016 23:00:00 GMT
<p>Many different types of graphs exist, and each has something different about it that makes it useful in a unique way. Here we will be looking at just two types of graph: bar graphs and line graphs.</p><p>Here is a bar graph and a line graph plotting the patient's hourly temperature data that we looked at in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2#tbl008">Table 8</a>.</p><div class="oucontentfigure" style="width:511px;" id="fig005_001"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6370816" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/2e78ad1f/s110_1_008i.small.jpg" alt="Figure 8" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6370816">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 Bar graph and line graph of the same data (a patient's temperature measured over a period of time)</span></div></div><a id="back_thumbnailfigure_idm6370816"></a></div><p>From either of these graphs, you can quickly see that the patient's temperature is gradually declining through the day, and by the seventh hour (19:00) it is at a normal level of 36.9 °C. Representing the data as a graph also allows you to estimate what the patient's temperature probably was at the fifth hour of the measurement period (17:00), when someone forgot to take a reading. If you imagine a straight line connection between the temperature values at the fourth and sixth hours (16:00 and 18:00 respectively) this line would intersect the fifth time point at about 37.3 °C.</p><p>Which type of graph is best to use? To help answer this question you can consider the list shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3#tbl009">Table 9</a>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl009"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 9 Choosing the best graph for your data  advantages and disadvantages of bar and line graphs</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">Graph type</th><th scope="col">Advantages</th></tr><tr><td>Bar and line</td><td>trends in data can be seen clearly (how one variable affects the other)</td></tr><tr><td>Bar and line</td><td>easy to use the value of one of the variables to determine the value of the other variable</td></tr><tr><td>Bar and line</td><td>enables predictions to be made about results of data you don't have yet</td></tr><tr><td>Bar</td><td>best for 'discrete' variables (those that change in jumps, with no 'in between' values)</td></tr><tr><td>Line</td><td>best for 'continuous' variables (those that change smoothly)</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>You will see from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3#tbl009">Table 9</a> that we have identified two different types of variable, and these are defined by the way in which their numerical values change. <b>Discrete variables</b> can only have specific values within any given range (e.g. 1, 2, 3). <b>Continuous variables</b> are not limited in this way, and can have any value within a range.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq003_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Find some examples of continuous data and discrete data in your workplace.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Examples of continuous variables could include: temperature, blood pressure and pH. Examples of discrete variables might be: blood type, numbers of patients and needle size.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3
3.3 Types of graphs and their usesS110_1<p>Many different types of graphs exist, and each has something different about it that makes it useful in a unique way. Here we will be looking at just two types of graph: bar graphs and line graphs.</p><p>Here is a bar graph and a line graph plotting the patient's hourly temperature data that we looked at in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.2#tbl008">Table 8</a>.</p><div class="oucontentfigure" style="width:511px;" id="fig005_001"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6370816" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/2e78ad1f/s110_1_008i.small.jpg" alt="Figure 8" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6370816">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 Bar graph and line graph of the same data (a patient's temperature measured over a period of time)</span></div></div><a id="back_thumbnailfigure_idm6370816"></a></div><p>From either of these graphs, you can quickly see that the patient's temperature is gradually declining through the day, and by the seventh hour (19:00) it is at a normal level of 36.9 °C. Representing the data as a graph also allows you to estimate what the patient's temperature probably was at the fifth hour of the measurement period (17:00), when someone forgot to take a reading. If you imagine a straight line connection between the temperature values at the fourth and sixth hours (16:00 and 18:00 respectively) this line would intersect the fifth time point at about 37.3 °C.</p><p>Which type of graph is best to use? To help answer this question you can consider the list shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3#tbl009">Table 9</a>.</p><div class="oucontenttable oucontentsnormal noborder oucontentsbox" id="tbl009"><h2 class="oucontenth3 oucontentheading oucontentnonumber">
Table 9 Choosing the best graph for your data  advantages and disadvantages of bar and line graphs</h2><div class="oucontenttablewrapper"><table><tr><th scope="col">Graph type</th><th scope="col">Advantages</th></tr><tr><td>Bar and line</td><td>trends in data can be seen clearly (how one variable affects the other)</td></tr><tr><td>Bar and line</td><td>easy to use the value of one of the variables to determine the value of the other variable</td></tr><tr><td>Bar and line</td><td>enables predictions to be made about results of data you don't have yet</td></tr><tr><td>Bar</td><td>best for 'discrete' variables (those that change in jumps, with no 'in between' values)</td></tr><tr><td>Line</td><td>best for 'continuous' variables (those that change smoothly)</td></tr></table></div><div class="oucontentsourcereference"></div></div><p>You will see from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.3#tbl009">Table 9</a> that we have identified two different types of variable, and these are defined by the way in which their numerical values change. <b>Discrete variables</b> can only have specific values within any given range (e.g. 1, 2, 3). <b>Continuous variables</b> are not limited in this way, and can have any value within a range.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq003_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Find some examples of continuous data and discrete data in your workplace.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Examples of continuous variables could include: temperature, blood pressure and pH. Examples of discrete variables might be: blood type, numbers of patients and needle size.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.4 Bar graphs
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4
Tue, 12 Apr 2016 23:00:00 GMT
<p>The following graph (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4#fig005002">Figure 9</a>) records how the outside diameter of a hypodermic needle is related to the needle gauge number.</p><div class="oucontentfigure" style="width:511px;" id="fig005_002"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6350736" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5e92f5fd/s110_1_009i.small.jpg" alt="Figure 9" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6350736">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 Relationship between needle diameter and gauge size</span></div></div><a id="back_thumbnailfigure_idm6350736"></a></div><p>This is an example of data that do not vary continuously, but instead change in discrete jumps, and discrete data are often best represented using a bar chart. (An alternative name for a bar chart is a <i>histogram</i>.) To express discrete data as a line chart would be misleading, since it would give the impression that gauge number and needle diameter are changing smoothly all of the time, which is not the case (there are no half or quartergauge sizes, only whole numbers).</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq003_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>If you wanted a needle with a diameter of 2 mm, what would be the required needle gauge number?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4#fig005002">Figure 9</a> shows that no such needle exists. A 2mmdiameter needle falls between gauge 14 (2.1 mm) and gauge 15 (1.8 mm). However, because 2.1 mm is closer to 2 mm than 1.8 mm is, you'd probably choose the gauge 14 needle.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4
3.4 Bar graphsS110_1<p>The following graph (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4#fig005002">Figure 9</a>) records how the outside diameter of a hypodermic needle is related to the needle gauge number.</p><div class="oucontentfigure" style="width:511px;" id="fig005_002"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6350736" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5e92f5fd/s110_1_009i.small.jpg" alt="Figure 9" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6350736">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 Relationship between needle diameter and gauge size</span></div></div><a id="back_thumbnailfigure_idm6350736"></a></div><p>This is an example of data that do not vary continuously, but instead change in discrete jumps, and discrete data are often best represented using a bar chart. (An alternative name for a bar chart is a <i>histogram</i>.) To express discrete data as a line chart would be misleading, since it would give the impression that gauge number and needle diameter are changing smoothly all of the time, which is not the case (there are no half or quartergauge sizes, only whole numbers).</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq003_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>If you wanted a needle with a diameter of 2 mm, what would be the required needle gauge number?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.4#fig005002">Figure 9</a> shows that no such needle exists. A 2mmdiameter needle falls between gauge 14 (2.1 mm) and gauge 15 (1.8 mm). However, because 2.1 mm is closer to 2 mm than 1.8 mm is, you'd probably choose the gauge 14 needle.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.5 Line graphs
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5
Tue, 12 Apr 2016 23:00:00 GMT
<p>To illustrate how to create and use line graphs, we will use the example of a <b>calibration curve</b>.
</p><p>A calibration curve is a type of line graph in which the response of a measuring device to a series of known concentrations of a substance is plotted. You can then make a measurement of an unknown sample  in the case we're about to examine, blood serum samples from newborn infants  and use the calibration curve to work out what concentration of substance is present.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq003_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 6</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Think of a type of machine that can perform measurements to give readings that can then be used to make a standard curve.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>A spectrophotometer provides these types of measurement data, which can be used to produce standard curves.</p></div></div></div></div><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 5 Spectrophotometers explained</h2><div class="oucontentinnerbox"><p>A spectrophotometer is an instrument that determines how much light of a particular colour is absorbed by a liquid sample. The more there is of a coloured substance in the solution, the more light will be absorbed (i.e. the less light passes through the solution). After measuring how much light is absorbed by a series of solutions containing known concentrations of the coloured substance, you can draw a graph of this data and use it to calculate the concentration of that substance in an unknown sample from a measurement of how much light it absorbs.</p><p>
<b>Here's how a spectrophotometer works:</b>
</p><ol class="oucontentnumbered"><li>
<p>White light from a bulb (source) is focused into a narrow beam by passing it through a thin slit.</p>
</li><li>
<p>A prism is used to split the beam of white light into its component colours, in the same way that water droplets can split sunlight into its component colours to make a rainbow. Different colours of light have a different wavelength: the distance between the peaks of the light waves, measured in nanometres (where 1 nanometre is 10<sup>9</sup> metres). For an idea of scale, individual virus particles range in size from about 20300 nm).</p>
</li><li>
<p>A second thin slit, just after the prism, can be moved from side to side to select just one colour of light to pass through to the sample.</p>
</li><li>
<p>The light passes through a container with the liquid sample inside (usually the light passes through 1 cm thickness of the liquid).</p>
</li><li>
<p>A light detector measures how much light is transmitted through the sample, and compares this with how much light was emitted by the source. The difference between these values gives a measure of how much light was absorbed by the sample: i.e. the absorbance (A), often also called the <i>optical density</i> (OD). The absorbance varies with wavelength, so measurements of this type always specify the wavelength of light that was shone through the sample. In the following example with Creactive protein, the wavelength was 450 nm, so this would be quoted as A<sub>450</sub> or OD<sub>450</sub>.</p>
</li></ol><p>This process is summarised below in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005003">Figure 10</a>, which also gives an indication of the wavelengths of different parts of the visible light spectrum.</p></div></div></div><div class="oucontentfigure" style="width:511px;" id="fig005_003"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6326288" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/87d59bfe/s110_1_010i.small.jpg" alt="Figure 10" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6326288">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 A spectrophotometer measures how much light of a certain wavelength is absorbed by a liquid</span></div></div><a id="back_thumbnailfigure_idm6326288"></a></div><p>In this particular example, we are looking at how the concentration of Creactive protein (CRP: a blood component produced in response to infection) changes the intensity of a bluecoloured test solution: the more CRP present, the more intense the blue colour becomes. The intensity of the blue colour is determined using a spectrophotometer to measure the Optical Density at 450 nm (O.D. 450 nm). More specifically, this is a measure of the amount of a blue light with a wavelength of 450 nanometres (450 nm) that is absorbed by a known thickness of the solution before the light reaches a detector.</p><p>When making a calibration curve, the following points need to be considered:</p><ol class="oucontentnumbered"><li>
<p>What is the expected range of concentrations?</p>
<ul class="oucontentunnumbered"><li>
<p>The range of blood serum CRP concentrations that might be encountered in infants, and the level of infection that this correlates with, are as follows:</p>
</li><li>
<p>normal: <2 μg/ml</p>
</li><li>
<p>sepsis: >22 μg/ml</p>
</li><li>
<p>dying: >120 μg/ml</p>
</li><li>
<p>Where < means 'less than' and > means 'more than'.</p>
</li><li>
<p>(Romagnoli et al., 2001)</p>
</li><li>
<p>From these values, a range of known standards containing between 0100 μg/ml CRP should cover the most likely range of infant serum CRP concentrations.</p>
</li></ul>
</li><li>
<p>Am I within the working range of the instrument?</p>
<ul class="oucontentunnumbered"><li>
<p>Many instruments are only accurate over a specific range of values. As we continue to increase the CRP concentration, the blue dye will become more and more intense. However, this can't continue forever and after a point the solution will be saturated with blue dye. Beyond that the solution can't get any more intense, no matter how much CRP we add. As we begin to reach this saturation point, the measurements will plateau out to the maximum value as CRP is increased. The most reliable part of the calibration curve covers the middle range of concentrations, where it is closest to being a straight line. For this reason, it is called the <b>linear</b> part of the graph.</p>
</li></ul>
</li><li>
<p>How random or 'noisy' is the assay and the measuring device?</p>
<ul class="oucontentunnumbered"><li>
<p>Depending on the precision of the measuring device you may not get the same reading every time from the same sample. To compensate for this it is a good idea to repeat the sample measurement two or three times and then calculate the <b>average</b> or <b>mean</b> value. For example, if you repeated the measurement of the same sample once, then you'd add both results together and then divide the resulting figure by 2 to find the average value.</p>
</li></ul>
</li></ol><p>With these points in mind, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005004">Figure 11</a> shows a table and graph of OD<sub>450</sub> values measured using a series of known concentrations of CRP.</p><div class="oucontentfigure" style="width:511px;" id="fig005_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/621103b1/s110_1_011i.jpg" alt="Figure 11" width="511" height="305" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 Calibration curve for CRP</span></div></div></div><p>In general, the xaxis is used to represent a variable that changes in a consistent way, or in a way that you can control (here, it's the known concentrations of CRP that you have measured out). The yaxis is normally used to represent a variable that you measure but may not be able to affect directly (in this case the optical density of the CRP test solutions that you read from the measuring device).</p><p>The small circular symbols on the graph mark the measured data points and they are joined by a curved line.</p><p>Notice that the graph shows a <b>positive correlation</b> between the two variables: the more CRP that is present, the larger the optical density reading. Other types of data may show a <b>negative correlation</b> between the variables, whereby one of the measured entities decreases as the other increases, e.g. the further you drive your car, the less petrol you have in the petrol tank.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5
3.5 Line graphsS110_1<p>To illustrate how to create and use line graphs, we will use the example of a <b>calibration curve</b>.
</p><p>A calibration curve is a type of line graph in which the response of a measuring device to a series of known concentrations of a substance is plotted. You can then make a measurement of an unknown sample  in the case we're about to examine, blood serum samples from newborn infants  and use the calibration curve to work out what concentration of substance is present.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq003_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 6</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Think of a type of machine that can perform measurements to give readings that can then be used to make a standard curve.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>A spectrophotometer provides these types of measurement data, which can be used to produce standard curves.</p></div></div></div></div><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 5 Spectrophotometers explained</h2><div class="oucontentinnerbox"><p>A spectrophotometer is an instrument that determines how much light of a particular colour is absorbed by a liquid sample. The more there is of a coloured substance in the solution, the more light will be absorbed (i.e. the less light passes through the solution). After measuring how much light is absorbed by a series of solutions containing known concentrations of the coloured substance, you can draw a graph of this data and use it to calculate the concentration of that substance in an unknown sample from a measurement of how much light it absorbs.</p><p>
<b>Here's how a spectrophotometer works:</b>
</p><ol class="oucontentnumbered"><li>
<p>White light from a bulb (source) is focused into a narrow beam by passing it through a thin slit.</p>
</li><li>
<p>A prism is used to split the beam of white light into its component colours, in the same way that water droplets can split sunlight into its component colours to make a rainbow. Different colours of light have a different wavelength: the distance between the peaks of the light waves, measured in nanometres (where 1 nanometre is 10<sup>9</sup> metres). For an idea of scale, individual virus particles range in size from about 20300 nm).</p>
</li><li>
<p>A second thin slit, just after the prism, can be moved from side to side to select just one colour of light to pass through to the sample.</p>
</li><li>
<p>The light passes through a container with the liquid sample inside (usually the light passes through 1 cm thickness of the liquid).</p>
</li><li>
<p>A light detector measures how much light is transmitted through the sample, and compares this with how much light was emitted by the source. The difference between these values gives a measure of how much light was absorbed by the sample: i.e. the absorbance (A), often also called the <i>optical density</i> (OD). The absorbance varies with wavelength, so measurements of this type always specify the wavelength of light that was shone through the sample. In the following example with Creactive protein, the wavelength was 450 nm, so this would be quoted as A<sub>450</sub> or OD<sub>450</sub>.</p>
</li></ol><p>This process is summarised below in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005003">Figure 10</a>, which also gives an indication of the wavelengths of different parts of the visible light spectrum.</p></div></div></div><div class="oucontentfigure" style="width:511px;" id="fig005_003"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6326288" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/87d59bfe/s110_1_010i.small.jpg" alt="Figure 10" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6326288">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 A spectrophotometer measures how much light of a certain wavelength is absorbed by a liquid</span></div></div><a id="back_thumbnailfigure_idm6326288"></a></div><p>In this particular example, we are looking at how the concentration of Creactive protein (CRP: a blood component produced in response to infection) changes the intensity of a bluecoloured test solution: the more CRP present, the more intense the blue colour becomes. The intensity of the blue colour is determined using a spectrophotometer to measure the Optical Density at 450 nm (O.D. 450 nm). More specifically, this is a measure of the amount of a blue light with a wavelength of 450 nanometres (450 nm) that is absorbed by a known thickness of the solution before the light reaches a detector.</p><p>When making a calibration curve, the following points need to be considered:</p><ol class="oucontentnumbered"><li>
<p>What is the expected range of concentrations?</p>
<ul class="oucontentunnumbered"><li>
<p>The range of blood serum CRP concentrations that might be encountered in infants, and the level of infection that this correlates with, are as follows:</p>
</li><li>
<p>normal: <2 μg/ml</p>
</li><li>
<p>sepsis: >22 μg/ml</p>
</li><li>
<p>dying: >120 μg/ml</p>
</li><li>
<p>Where < means 'less than' and > means 'more than'.</p>
</li><li>
<p>(Romagnoli et al., 2001)</p>
</li><li>
<p>From these values, a range of known standards containing between 0100 μg/ml CRP should cover the most likely range of infant serum CRP concentrations.</p>
</li></ul>
</li><li>
<p>Am I within the working range of the instrument?</p>
<ul class="oucontentunnumbered"><li>
<p>Many instruments are only accurate over a specific range of values. As we continue to increase the CRP concentration, the blue dye will become more and more intense. However, this can't continue forever and after a point the solution will be saturated with blue dye. Beyond that the solution can't get any more intense, no matter how much CRP we add. As we begin to reach this saturation point, the measurements will plateau out to the maximum value as CRP is increased. The most reliable part of the calibration curve covers the middle range of concentrations, where it is closest to being a straight line. For this reason, it is called the <b>linear</b> part of the graph.</p>
</li></ul>
</li><li>
<p>How random or 'noisy' is the assay and the measuring device?</p>
<ul class="oucontentunnumbered"><li>
<p>Depending on the precision of the measuring device you may not get the same reading every time from the same sample. To compensate for this it is a good idea to repeat the sample measurement two or three times and then calculate the <b>average</b> or <b>mean</b> value. For example, if you repeated the measurement of the same sample once, then you'd add both results together and then divide the resulting figure by 2 to find the average value.</p>
</li></ul>
</li></ol><p>With these points in mind, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005004">Figure 11</a> shows a table and graph of OD<sub>450</sub> values measured using a series of known concentrations of CRP.</p><div class="oucontentfigure" style="width:511px;" id="fig005_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/621103b1/s110_1_011i.jpg" alt="Figure 11" width="511" height="305" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 Calibration curve for CRP</span></div></div></div><p>In general, the xaxis is used to represent a variable that changes in a consistent way, or in a way that you can control (here, it's the known concentrations of CRP that you have measured out). The yaxis is normally used to represent a variable that you measure but may not be able to affect directly (in this case the optical density of the CRP test solutions that you read from the measuring device).</p><p>The small circular symbols on the graph mark the measured data points and they are joined by a curved line.</p><p>Notice that the graph shows a <b>positive correlation</b> between the two variables: the more CRP that is present, the larger the optical density reading. Other types of data may show a <b>negative correlation</b> between the variables, whereby one of the measured entities decreases as the other increases, e.g. the further you drive your car, the less petrol you have in the petrol tank.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Questions relating to Figure 11
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>The following OD<sub>450</sub> values were measured from serum samples taken from three babies: 0.12, 0.40, and 1.74.</p><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 7</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Use the calibration curve shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005004">Figure 11</a> to estimate the serum concentration of CRP for each sample.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>From the curve you can deduce the values are about 2 g/ml, 7 g/ml, and 90 g/ml, respectively.</p></div></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 8</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Does the existing scale allow you to confidently estimate these values?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>You probably found it difficult to make accurate CRP readings for the OD<sub>450</sub> values of 0.12 and 1.74. This is because the scale of the yaxis (the OD<sub>450</sub> values) is too compressed. If the graph were redrawn, with the yaxis maybe twice its current height (but still running from 0 to 2) then this would provide a better resolution.</p></div></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 9</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>One of your estimated concentrations might be particularly inaccurate; which one do you think this is likely to be and why?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Apart from the problem already mentioned concerning the size of the yaxis, the OD<sub>450</sub> measurement of 1.74 is likely to be inaccurate. This is because the measuring assay is becoming saturated, as you can see by the calibration curve flattening off towards a constant high value in this region. The colour of the test solution is almost at its maximum intensity and changes in CRP concentration make almost no difference to the OD<sub>450</sub> reading.</p></div></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq007"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 10</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What do you infer about the health of each baby from these results?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Earlier on in this section you were told what level of infection corresponded with a particular range of serum CRP concentrations:</p><p>normal: <2 μg/ml</p><p>sepsis: >22 μg/ml</p><p>dying: >120 μg/ml</p><p>You have calculated the CRP values of three babies to be about 2 μg/ml, 7 μg/ml, and 90 μg/ml. Using these figures for reference, you can see that the baby with a serum CRP concentration of 2 μg/ml is healthy, although at the high end of the range for a healthy baby. The baby with a CRP concentration of 7 μg/ml is beyond the normal level and would be expected to have a mild fever, but is not yet dangerously unwell. The baby with a CRP concentration of 90 μg/ml is very seriously ill and will require intensive care.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5.1
Questions relating to Figure 11S110_1<p>The following OD<sub>450</sub> values were measured from serum samples taken from three babies: 0.12, 0.40, and 1.74.</p><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 7</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Use the calibration curve shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.5#fig005004">Figure 11</a> to estimate the serum concentration of CRP for each sample.</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>From the curve you can deduce the values are about 2 g/ml, 7 g/ml, and 90 g/ml, respectively.</p></div></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 8</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Does the existing scale allow you to confidently estimate these values?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>You probably found it difficult to make accurate CRP readings for the OD<sub>450</sub> values of 0.12 and 1.74. This is because the scale of the yaxis (the OD<sub>450</sub> values) is too compressed. If the graph were redrawn, with the yaxis maybe twice its current height (but still running from 0 to 2) then this would provide a better resolution.</p></div></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 9</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>One of your estimated concentrations might be particularly inaccurate; which one do you think this is likely to be and why?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Apart from the problem already mentioned concerning the size of the yaxis, the OD<sub>450</sub> measurement of 1.74 is likely to be inaccurate. This is because the measuring assay is becoming saturated, as you can see by the calibration curve flattening off towards a constant high value in this region. The colour of the test solution is almost at its maximum intensity and changes in CRP concentration make almost no difference to the OD<sub>450</sub> reading.</p></div></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq007"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 10</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>What do you infer about the health of each baby from these results?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>Earlier on in this section you were told what level of infection corresponded with a particular range of serum CRP concentrations:</p><p>normal: <2 μg/ml</p><p>sepsis: >22 μg/ml</p><p>dying: >120 μg/ml</p><p>You have calculated the CRP values of three babies to be about 2 μg/ml, 7 μg/ml, and 90 μg/ml. Using these figures for reference, you can see that the baby with a serum CRP concentration of 2 μg/ml is healthy, although at the high end of the range for a healthy baby. The baby with a CRP concentration of 7 μg/ml is beyond the normal level and would be expected to have a mild fever, but is not yet dangerously unwell. The baby with a CRP concentration of 90 μg/ml is very seriously ill and will require intensive care.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.6.1 Radioactivity and bugs!
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Many natural processes involve repeated doublings or halving at regular intervals. You may have come across this already in your work, in the context of bacterial growth or radioactivity. In this section, we are going to look in more detail at bacterial growth and radioactivity and we will be using graphs to examine how the numbers of bacteria or numbers of radioactive atoms change over time.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.1
3.6.1 Radioactivity and bugs!S110_1<p>Many natural processes involve repeated doublings or halving at regular intervals. You may have come across this already in your work, in the context of bacterial growth or radioactivity. In this section, we are going to look in more detail at bacterial growth and radioactivity and we will be using graphs to examine how the numbers of bacteria or numbers of radioactive atoms change over time.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.6.2 Exponential increase: bacteria
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.2
Tue, 12 Apr 2016 23:00:00 GMT
<p>Bacteria are singlecelled organisms. Many different types of bacteria exist and they populate almost every environment on earth, from deep oceans to soil to human intestines. Several bacteria are beneficial to us: for instance, our gut bacteria can help to break down foodstuffs that we would otherwise find difficult to digest. However, some bacteria produce harmful toxins and if they grow in an uncontrolled way in our bodies this can have serious health consequences.</p><p>If a bacterium is growing under perfect conditions, then it will divide in two at a constant rate, which is called the doubling time (generally every 12 minutes to 24 hours, depending on the type of bacterium). Each of the newly produced bacteria will then themselves divide in two, and so on. Thus, from one original bacterium, the following number of offspring will be generated during each round of division: 2, 4, 8, 16, 32, 64, 128, etc. In other words, a doubling at each division: 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32, 32 × 2 = 64, 64 × 2 = 128.</p><p>From your earlier knowledge of exponentsyou might realise that this series of numbers can also be represented as a power series of exponentials: 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>, 2<sup>6</sup>, 2<sup>7</sup>. Hence, this type of growth is often referred to as '<i>exponential growth</i>'. In reality, it's rare to find organisms undergoing exponential growth, except at the beginning of an infection where growth conditions are the closest to perfect. Later on, factors such as cell death, nutrient availability, waste product production and overcrowding can all restrict the growth rate.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.2
3.6.2 Exponential increase: bacteriaS110_1<p>Bacteria are singlecelled organisms. Many different types of bacteria exist and they populate almost every environment on earth, from deep oceans to soil to human intestines. Several bacteria are beneficial to us: for instance, our gut bacteria can help to break down foodstuffs that we would otherwise find difficult to digest. However, some bacteria produce harmful toxins and if they grow in an uncontrolled way in our bodies this can have serious health consequences.</p><p>If a bacterium is growing under perfect conditions, then it will divide in two at a constant rate, which is called the doubling time (generally every 12 minutes to 24 hours, depending on the type of bacterium). Each of the newly produced bacteria will then themselves divide in two, and so on. Thus, from one original bacterium, the following number of offspring will be generated during each round of division: 2, 4, 8, 16, 32, 64, 128, etc. In other words, a doubling at each division: 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32, 32 × 2 = 64, 64 × 2 = 128.</p><p>From your earlier knowledge of exponentsyou might realise that this series of numbers can also be represented as a power series of exponentials: 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>, 2<sup>6</sup>, 2<sup>7</sup>. Hence, this type of growth is often referred to as '<i>exponential growth</i>'. In reality, it's rare to find organisms undergoing exponential growth, except at the beginning of an infection where growth conditions are the closest to perfect. Later on, factors such as cell death, nutrient availability, waste product production and overcrowding can all restrict the growth rate.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.6.3 Exponential decrease: radioactive decay
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.3
Tue, 12 Apr 2016 23:00:00 GMT
<p>The most familiar example of exponential decrease is provided by radioactive decay. Radioactivity is a natural phenomenon that is used routinely in many medical applications, from imaging (radioactive tracers in PET scanning) to therapy (radiotherapy to destroy tumours). During radioactive decay, the number of radioactive atoms halves at a constant rate, called the halflife. For instance, the radioactive isotope <sup>11</sup>C, pronounced 'carbon 11', has a halflife of 1224 seconds (a little over 20 minutes). After 1224 seconds, there would only be half of the starting amount of <sup>11</sup>C remaining. After another 1224 seconds there would be only half of <i>this</i> amount remaining, i.e. 1/4 of the starting amount, and so on. Thus, the fraction of the starting material that remains after each halflife follows this series: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, etc. Just as we saw with exponential increase, this sequence can be rewritten as a power series of exponentials, except this time they are fractions: 1/2<sup>1</sup>, 1/2<sup>2</sup>, 1/2<sup>3</sup>, 1/2<sup>4</sup>, 1/2<sup>5</sup>, 1/2<sup>6</sup>.</p><p>Exponential decay occurs in many more situations than just radioactivity. For instance, most drugs become metabolised in the body according to an exponential decay pattern. The clearance of most substances from the blood by the kidneys (or their clearance from the blood using a dialysis machine) also follows an exponential decay pattern.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 6 Radioactivity  a brief explanation of atomic instability</h2><div class="oucontentinnerbox"><p>At the core of every atom is a nucleus, made up of a fixed number of protons and neutrons, and orbiting the nucleus are electrons. The number of protons present defines what the element is (hydrogen, oxygen, gold, etc.) whilst the number of protons plus the number of neutrons defines the <b>isotope</b> number of that element.</p><p>Although each isotope of the same element has a different configuration of protons and neutrons in its nucleus, most interactions between atoms involve the electrons that orbit at a relatively large distance around the nucleus. Because the behaviour of the electrons isn't greatly altered by changing the number of neutrons, isotopes of the same element generally look and behave the same as each other, it's just that their weights are very slightly different because of the different numbers of neutrons present. Almost every element has many different isotopes and these often exist together naturally as a mixture. The air you're breathing right now contains a mixture of three different oxygen isotopes: <sup>16</sup>O, <sup>17</sup>O, and <sup>18</sup>O.</p><p>Let's look at one element, carbon, in a little more detail. The nucleus of a carbon atom normally has 6 protons and 6 neutrons. These two numbers added together give the <i>isotope number</i>, so this would be <sup>12</sup>C (pronounced, 'carbon 12'). However, other isotopes of carbon exist, with different numbers of neutrons.</p><p>There is a form of carbon that has 6 protons and 5 neutrons, <sup>11</sup>C. However, this arrangement of protons and neutrons is unstable. The nucleus of a <sup>11</sup>C atom is poised, like an overwound spring, to suddenly rearrange. When this happens, a burst of energy is released that is detected as <b>radioactivity</b>, and in the process, one of the protons is converted into a neutron. The resulting atom has a much more stable nucleus, containing 5 protons and 6 neutrons, but is now a different element (remember, the number of protons defines the element) with its own unique properties; it has become an atom of <sup>11</sup>B (boron 11). Some nuclear arrangements are inherently more unstable than others are, and this explains why different radioactive isotopes undergo these rearrangements (often called 'radioactive decay') at different rates.</p></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.3
3.6.3 Exponential decrease: radioactive decayS110_1<p>The most familiar example of exponential decrease is provided by radioactive decay. Radioactivity is a natural phenomenon that is used routinely in many medical applications, from imaging (radioactive tracers in PET scanning) to therapy (radiotherapy to destroy tumours). During radioactive decay, the number of radioactive atoms halves at a constant rate, called the halflife. For instance, the radioactive isotope <sup>11</sup>C, pronounced 'carbon 11', has a halflife of 1224 seconds (a little over 20 minutes). After 1224 seconds, there would only be half of the starting amount of <sup>11</sup>C remaining. After another 1224 seconds there would be only half of <i>this</i> amount remaining, i.e. 1/4 of the starting amount, and so on. Thus, the fraction of the starting material that remains after each halflife follows this series: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, etc. Just as we saw with exponential increase, this sequence can be rewritten as a power series of exponentials, except this time they are fractions: 1/2<sup>1</sup>, 1/2<sup>2</sup>, 1/2<sup>3</sup>, 1/2<sup>4</sup>, 1/2<sup>5</sup>, 1/2<sup>6</sup>.</p><p>Exponential decay occurs in many more situations than just radioactivity. For instance, most drugs become metabolised in the body according to an exponential decay pattern. The clearance of most substances from the blood by the kidneys (or their clearance from the blood using a dialysis machine) also follows an exponential decay pattern.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 6 Radioactivity  a brief explanation of atomic instability</h2><div class="oucontentinnerbox"><p>At the core of every atom is a nucleus, made up of a fixed number of protons and neutrons, and orbiting the nucleus are electrons. The number of protons present defines what the element is (hydrogen, oxygen, gold, etc.) whilst the number of protons plus the number of neutrons defines the <b>isotope</b> number of that element.</p><p>Although each isotope of the same element has a different configuration of protons and neutrons in its nucleus, most interactions between atoms involve the electrons that orbit at a relatively large distance around the nucleus. Because the behaviour of the electrons isn't greatly altered by changing the number of neutrons, isotopes of the same element generally look and behave the same as each other, it's just that their weights are very slightly different because of the different numbers of neutrons present. Almost every element has many different isotopes and these often exist together naturally as a mixture. The air you're breathing right now contains a mixture of three different oxygen isotopes: <sup>16</sup>O, <sup>17</sup>O, and <sup>18</sup>O.</p><p>Let's look at one element, carbon, in a little more detail. The nucleus of a carbon atom normally has 6 protons and 6 neutrons. These two numbers added together give the <i>isotope number</i>, so this would be <sup>12</sup>C (pronounced, 'carbon 12'). However, other isotopes of carbon exist, with different numbers of neutrons.</p><p>There is a form of carbon that has 6 protons and 5 neutrons, <sup>11</sup>C. However, this arrangement of protons and neutrons is unstable. The nucleus of a <sup>11</sup>C atom is poised, like an overwound spring, to suddenly rearrange. When this happens, a burst of energy is released that is detected as <b>radioactivity</b>, and in the process, one of the protons is converted into a neutron. The resulting atom has a much more stable nucleus, containing 5 protons and 6 neutrons, but is now a different element (remember, the number of protons defines the element) with its own unique properties; it has become an atom of <sup>11</sup>B (boron 11). Some nuclear arrangements are inherently more unstable than others are, and this explains why different radioactive isotopes undergo these rearrangements (often called 'radioactive decay') at different rates.</p></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.6.4 Representing exponential relationships using graphs
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4
Tue, 12 Apr 2016 23:00:00 GMT
<p>What do exponential increase and decrease look like when plotted as a graph? Although exponentials describe anything that continually doubles or halves, the specific assumption of 'exponential increase' and 'exponential decay' are that these happen during a constant time interval. If the time taken for doubling or halving remains constant, then an exponential increase looks like the thick blue line in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>, which shows the number of bacteria present in a millilitre of growth medium (marked on the left side of the graph), counted each hour.</p><div class="oucontentfigure" style="width:511px;" id="fig005_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/888b8a69/s110_1_014i.jpg" alt="Figure 12" width="511" height="368" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 12 Graph showing an exponential increase of bacteria numbers over time</span></div></div></div><p>You can see that at 3 hours there were about 10 bacteria/ml, at 4 hours there were about 20 bacteria/ml and at 5 hours there were about 40 bacteria/ml, so, in this case the bacteria are dividing with a doubling time of around 1 hour. In practice, it's unlikely that any cell type would be doubling, rather conveniently, at each time point you had chosen to make a measurement. Instead, it's best to estimate the doubling time by reading off the time at which there were a known number of cells e.g. 50 bacteria/ml, and then reading off the time when that number of bacteria had doubled to 100 bacteria/ml. The time difference between these values gives the doubling time (often given the symbol, T<sub>d</sub>) or if we were dealing with exponential decay, it would give the halflife (often given the symbol, t<sub>1/2</sub>). In this case, there were 50 bacteria/ml at about 5.5 hours and 100 bacteria/ml at about 6.5 hours, so the bacteria doubled in about 6.5  5.5 = 1 hour. Try this for yourself to estimate how long it took to go from 100 bacteria/ml to 200 bacteria/ml. If the bacteria are still growing exponentially then you should get the same value for the doubling time.</p><p></p><p>Because an exponential increase can also be represented as a power series, if we drew a graph showing the time at which there were 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>, 2<sup>6</sup>, 2<sup>7</sup>, etc. bacteria present then this would produce a straight line if the bacteria were doubling exponentially. In practice, it's extremely unlikely that we would be lucky enough to make a measurement when there were exactly 4 (i.e. 2<sup>2</sup>) or 16 (2<sup>4</sup>) or 128 (2<sup>7</sup>) bacteria present. More likely, we might find that we had 130 bacteria/ml instead of 128 bacteria/ml. Nevertheless, it is possible to convert our existing data of bacteria number measured at fixed time points into a straightline graph. To do this we need to convert our bacteria number values into logarithms.</p><p>A logarithm isn't a lumberjack pop group, but is a mathematical method to calculate the values of exponents. In essence, a logarithm asks the question 'what power do I need to raise something to in order to get the answer I want?' For instance, 2 raised to what power, 'p', gives the value 128? (i.e. 128 = 2<sup>p</sup>). We've already seen that 128 = 2<sup>7</sup>, so the logarithm of 128 with base 2 is 7.</p><p>A word about <i>bases</i>: you're already familiar with the decimal system of numbering, in which exponents are of the type 'ten to the power something' (e.g. 10<sup>3</sup>, 10<sup>6</sup> etc.). In the decimal system, 10 is the base number. However, we could actually choose any base we like. Things that change by halving or doubling are best expressed using base 2, i.e. you'd express exponents in a base 2 system of numbering as 'two to the power something' (2<sup>3</sup>, 2<sup>6</sup>, etc.).</p><p>For many complex historical and mathematical reasons, most logarithms use a base value of 2.718, a special mathematical constant. For instance, if you use the logarithm button on a calculator (marked 'ln', or 'natural logarithm') to find the logarithm of 128 you are asking it to calculate the value of 'p' in the equation 128 = 2.718<sup>p</sup>. This calculation gives the answer p = 4.852.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a> we have converted the measurements of the numbers of bacteria/ml into their logarithm values (marked on the right side of the graph) and have plotted these against time, using a red colour. Because one axis is logarithmic and the other is normal, this is called a <i>semilogarithmic</i> graph. Notice that plotting the logarithmic data gives a straight line, as we predicted it would, and this confirms that the growth of the bacteria is exponential. If their growth had started to slow down then the semilogarithmic plot would deviate from a straight line at that point and begin to flatten off. It is for this reason  that it's easier to tell by eye if a line is straight than to tell if it's an exponential curve  that semilogarithmic graphs are used to spot this kind of pattern.</p><p>The steepness, or gradient, of the line in a semilogarithmic graph can be used to determine how fast the bacteria are dividing, i.e. to find the doubling time, or in the case of exponential decay to find the halflife. You may come across specific names for this gradient: for exponential increases, the gradient is also known as the <i>growth constant</i>, whilst for exponential decreases the gradient is known as the <i>decay constant</i>.</p><p>Just like the gradient of a road, the gradient of a line graph tells you how far you would go up (or down) on the yaxis (change in y: abbreviated to ΔY) if you moved one unit along on the xaxis (change in x: abbreviated to ΔX). (The symbol 'Δ' is the Greek letter 'delta', and is used to denote a difference or change.) On this graph, it's difficult to read accurately how much the yaxis logarithm value changes per hour, but it looks to be somewhere between 0.6 and 0.7. A more accurate estimate can be made if we look over a wider range. Over the entire 8 hours of measurement, the logarithm of the cell density changes by about 5.5, so that gives a gradient <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie004i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> of 5.5 ÷8 = 0.6875, which is the growth constant.</p><p>The doubling time or halflife is given by dividing the logarithm of 2 by the gradient of the graph, i.e. <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie005i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span>.</p><p>Using a calculator we can confirm that ln(2) = 0.6931. Therefore T<sub>d</sub> = 0.6931 ÷ 0.6875 = 1.008 hours. This agrees well with our estimate from the exponential curve that the bacteria are doubling every (one) hour.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4
3.6.4 Representing exponential relationships using graphsS110_1<p>What do exponential increase and decrease look like when plotted as a graph? Although exponentials describe anything that continually doubles or halves, the specific assumption of 'exponential increase' and 'exponential decay' are that these happen during a constant time interval. If the time taken for doubling or halving remains constant, then an exponential increase looks like the thick blue line in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>, which shows the number of bacteria present in a millilitre of growth medium (marked on the left side of the graph), counted each hour.</p><div class="oucontentfigure" style="width:511px;" id="fig005_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/888b8a69/s110_1_014i.jpg" alt="Figure 12" width="511" height="368" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 12 Graph showing an exponential increase of bacteria numbers over time</span></div></div></div><p>You can see that at 3 hours there were about 10 bacteria/ml, at 4 hours there were about 20 bacteria/ml and at 5 hours there were about 40 bacteria/ml, so, in this case the bacteria are dividing with a doubling time of around 1 hour. In practice, it's unlikely that any cell type would be doubling, rather conveniently, at each time point you had chosen to make a measurement. Instead, it's best to estimate the doubling time by reading off the time at which there were a known number of cells e.g. 50 bacteria/ml, and then reading off the time when that number of bacteria had doubled to 100 bacteria/ml. The time difference between these values gives the doubling time (often given the symbol, T<sub>d</sub>) or if we were dealing with exponential decay, it would give the halflife (often given the symbol, t<sub>1/2</sub>). In this case, there were 50 bacteria/ml at about 5.5 hours and 100 bacteria/ml at about 6.5 hours, so the bacteria doubled in about 6.5  5.5 = 1 hour. Try this for yourself to estimate how long it took to go from 100 bacteria/ml to 200 bacteria/ml. If the bacteria are still growing exponentially then you should get the same value for the doubling time.</p><p></p><p>Because an exponential increase can also be represented as a power series, if we drew a graph showing the time at which there were 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>, 2<sup>5</sup>, 2<sup>6</sup>, 2<sup>7</sup>, etc. bacteria present then this would produce a straight line if the bacteria were doubling exponentially. In practice, it's extremely unlikely that we would be lucky enough to make a measurement when there were exactly 4 (i.e. 2<sup>2</sup>) or 16 (2<sup>4</sup>) or 128 (2<sup>7</sup>) bacteria present. More likely, we might find that we had 130 bacteria/ml instead of 128 bacteria/ml. Nevertheless, it is possible to convert our existing data of bacteria number measured at fixed time points into a straightline graph. To do this we need to convert our bacteria number values into logarithms.</p><p>A logarithm isn't a lumberjack pop group, but is a mathematical method to calculate the values of exponents. In essence, a logarithm asks the question 'what power do I need to raise something to in order to get the answer I want?' For instance, 2 raised to what power, 'p', gives the value 128? (i.e. 128 = 2<sup>p</sup>). We've already seen that 128 = 2<sup>7</sup>, so the logarithm of 128 with base 2 is 7.</p><p>A word about <i>bases</i>: you're already familiar with the decimal system of numbering, in which exponents are of the type 'ten to the power something' (e.g. 10<sup>3</sup>, 10<sup>6</sup> etc.). In the decimal system, 10 is the base number. However, we could actually choose any base we like. Things that change by halving or doubling are best expressed using base 2, i.e. you'd express exponents in a base 2 system of numbering as 'two to the power something' (2<sup>3</sup>, 2<sup>6</sup>, etc.).</p><p>For many complex historical and mathematical reasons, most logarithms use a base value of 2.718, a special mathematical constant. For instance, if you use the logarithm button on a calculator (marked 'ln', or 'natural logarithm') to find the logarithm of 128 you are asking it to calculate the value of 'p' in the equation 128 = 2.718<sup>p</sup>. This calculation gives the answer p = 4.852.</p><p>In <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a> we have converted the measurements of the numbers of bacteria/ml into their logarithm values (marked on the right side of the graph) and have plotted these against time, using a red colour. Because one axis is logarithmic and the other is normal, this is called a <i>semilogarithmic</i> graph. Notice that plotting the logarithmic data gives a straight line, as we predicted it would, and this confirms that the growth of the bacteria is exponential. If their growth had started to slow down then the semilogarithmic plot would deviate from a straight line at that point and begin to flatten off. It is for this reason  that it's easier to tell by eye if a line is straight than to tell if it's an exponential curve  that semilogarithmic graphs are used to spot this kind of pattern.</p><p>The steepness, or gradient, of the line in a semilogarithmic graph can be used to determine how fast the bacteria are dividing, i.e. to find the doubling time, or in the case of exponential decay to find the halflife. You may come across specific names for this gradient: for exponential increases, the gradient is also known as the <i>growth constant</i>, whilst for exponential decreases the gradient is known as the <i>decay constant</i>.</p><p>Just like the gradient of a road, the gradient of a line graph tells you how far you would go up (or down) on the yaxis (change in y: abbreviated to ΔY) if you moved one unit along on the xaxis (change in x: abbreviated to ΔX). (The symbol 'Δ' is the Greek letter 'delta', and is used to denote a difference or change.) On this graph, it's difficult to read accurately how much the yaxis logarithm value changes per hour, but it looks to be somewhere between 0.6 and 0.7. A more accurate estimate can be made if we look over a wider range. Over the entire 8 hours of measurement, the logarithm of the cell density changes by about 5.5, so that gives a gradient <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie004i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> of 5.5 ÷8 = 0.6875, which is the growth constant.</p><p>The doubling time or halflife is given by dividing the logarithm of 2 by the gradient of the graph, i.e. <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie005i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span>.</p><p>Using a calculator we can confirm that ln(2) = 0.6931. Therefore T<sub>d</sub> = 0.6931 ÷ 0.6875 = 1.008 hours. This agrees well with our estimate from the exponential curve that the bacteria are doubling every (one) hour.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.6.5 Using the gradient of a semilogarithmic graph to calculate doubling time or halflife
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.5
Tue, 12 Apr 2016 23:00:00 GMT
<p>Knowing the equation <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie006i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span> allows you to perform several useful calculations without needing to make a graph, and we'll look at one such example in a moment.</p><p>First, let's return to the gradient of the exponential increase graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>. This gradient is <i>positive</i>: as you move left to right along the xaxis the graph climbs, and the values on the yaxis increase. What would be the gradient if this graph were reversed, to show an exponential decrease? In that situation, the graph would fall from left to right and the yaxis values would become smaller as you moved along the xaxis, so the gradient would have a <i>negative</i> value.</p><p>There are some basic rules for dealing with negative numbers that you need to be aware of before moving onto the next example, which deals with exponential decay.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 7 Being positive about negative numbers</h2><div class="oucontentinnerbox"><p>A negative number is any value less than zero. Look at the thermometer scale below.</p><p>−4 °C −3 °C −2 °C −1 °C 0 °C +1 °C +2 °C +3 °C +4 °C</p><p>The freezing point of water is marked as 0 °C. Temperatures above freezing go up incrementally in positive numbers. As temperatures get colder and colder below 0 °C their values also increase, but in negative numbers. The temperature <i>difference</i> between 0 °C and 1 °C is 1 °C. Similarly, the temperature <i>difference</i> between 0 °C and −1 °C is also 1 °C. The temperature difference between −1 °C and +1 °C is 2 °C. Using a scale like this we can come up with some general rules for addition and subtraction involving negative numbers.</p><p>
<b>Addition and subtraction involving negative numbers</b>
</p><p>Using the thermometer scale above, you should be able to verify for yourself that the following sums are correct:</p><ul class="oucontentunnumbered"><li>3 − 4 = −1</li><li>−2 + 1 = −1</li><li>−1 − 1 = −2</li></ul><p>Now here's the tricky one: subtracting a negative number is the same as adding a positive number, so:</p><ul class="oucontentunnumbered"><li>−5 − (−3) = −5 + 3 = −2</li></ul><p>Another helpful way to view the same calculation is as follows:</p><ul class="oucontentunnumbered"><li>−5 − (−3) = −(5 − 3) = −(2) = −2</li></ul><p><b>Multiplication and division involving negative numbers</b></p><p>If you multiply 2 × 2, you get 4, so, a positive number multiplied by a positive number, gives a positive number. However, what do you get if you multiply 2 × −2?</p><p>The rule is that a positive number multiplied by a negative number (or the other way round), gives a negative number, so 2 × −2 = −4 (and −2 × 2 = −4).</p><p>The general idea is that if you multiply (or divide) a positive number by a negative number then that changes the sign of your eventual answer; or, put another way: multiplying or dividing the same signs together results in a positive number, whilst doing the same with opposite signs give a negative number.</p><ul class="oucontentunnumbered"><li>positive × positive = positive</li><li>positive × negative = negative</li><li>negative × positive = negative</li><li>negative × negative = positive</li><li>positive ÷ positive = positive</li><li>positive ÷ negative = negative</li><li>negative ÷ positive = negative</li><li>negative ÷ negative = positive</li></ul></div></div></div><p>These rules will come in handy as you work through the following example question.</p><div class="oucontentexample oucontentsheavybox1 oucontentsbox " id="exa001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Example 1</h2><div class="oucontentinnerbox"><p>The radioactive isotope iodine 131 (<sup>131</sup>I) is used in medicine to diagnose conditions of the thyroid gland. <sup>131</sup>I has a halflife of 8 days. If the hospital has just received a delivery of 10 g <sup>131</sup>I: how much will remain after 25 days?</p><p>As an initial check, we can work out a rough answer. Each halflife takes 8 days, therefore, after 8 + 8 + 8 = 24 days the <sup>131</sup>I will have gone through three halflives. Since we lose half of the <sup>131</sup>I after each halflife, out of the 10 g starting material there will be 1.25 g left after 3 halflives.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/650d2d86/s110_1_ue015i.gif" alt=""/></div><p>We want to know how much <sup>131</sup>I will remain after 25 days, so we would estimate that there should be a little less than 1.25 g remaining.</p><p>Now let's work out the answer accurately using some of the principles we would apply if we were plotting data of this radioactive decay on a graph like that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>.</p><p>On a semilogarithmic graph we know that, <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie007i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span>where T<sub>d</sub> can represent doubling time or halflife; this distinction becomes important in a moment. We are concerned here with the halflife, so let's rename T<sub>d</sub> as 'halflife'.</p><p>If we multiply both sides of the equation by gradient, and divide both sides of the equation by halflife, then this rearranges to give</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/fa967ae0/s110_1_ue016i.gif" alt=""/></div><p>Using a calculator to find the logarithm of 2 gives, ln(2) = 0.6931</p><p>The halflife is 8 days, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c1934146/s110_1_ue017i.gif" alt=""/></div><p>We know that the gradient of the graph can also be calculated in a different way, directly from the graph itself, since the gradient of any straightline graph is defined as <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie008i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> </p><p>Now the distinction between doubling time and halflife becomes important. The initial equation we used doesn't distinguish between exponential increase or decrease. However, because we are dealing with an exponential decay (the amount of <sup>131</sup>I is <i>decreasing</i> with time) the gradient will be negative. The actual numerical value, the steepness, of the gradient is unchanged; it's just that we are moving 'downhill' (exponential decrease) rather than uphill (exponential increase), so this value must have a negative sign.</p><p>Therefore, <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie009i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> = −0.0866</p><p>Rearranging gives, ΔY = −0.0866 × ΔX</p><p>The time interval we are interested in is ΔX = 25 days, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4d5a34a5/s110_1_ue018i.gif" alt=""/></div><p>Using the general rules we established in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.5#box005003">Box 7</a>, we know that because we are multiplying a positive number by a negative number the answer will be a negative number, thus, ΔY = −0.0866 × 25 = −2.165</p><p>The yaxis is on a logarithmic scale, so this value that we have deduced for ΔY represents the logarithm of the change in the amount of <sup>131</sup>I that has occurred during 25 days.</p><p>The starting amount of <sup>131</sup>I is 10 g, so in order to be able to compare this with our value for ΔY (which is in logarithms) it too needs to be converted into a logarithm, and the logarithm of 10 is ln(10) = 2.3026.</p><p>During 25 days, this starting amount of <sup>131</sup>I will be reduced by ΔY, therefore: the remaining <sup>131</sup>I after 25 days = 2.3026  2.165 = 0.1376</p><p>Once more, this value for the remaining amount of <sup>131</sup>I is a logarithm because the values that we have used to calculate it have been in logarithmic form. In order to convert it to a conventional number you need to use the 'antilogarithm' button on your calculator (shown as e<sup>x</sup> on most calculators). You should find that e<sup>x</sup>(0.1376) = 1.148, which is how many grams of <sup>131</sup>I are remaining after 25 days.</p><p>Finally, always check that your calculated value fits with your initial estimate from counting halflives. Our initial rough answer was for there to be a little less than 1.25 g <sup>131</sup>I remaining after 25 days. Reassuringly, our accurate answer of 1.148 g is exactly in keeping with the rough calculation.</p></div></div></div><div class=" oucontentsaq oucontentsheavybox1 oucontentsbox " id="saq008"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 11</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>The elimination of a drug from the blood, due to metabolism and excretion, follows an exponential decay with a halflife of 4 hours. Below a blood plasma concentration of 0.2 mcg/ml the drug is not effective. (Blood consists of cells floating in a liquid. This liquid component of the blood is called plasma and can be produced by centrifuging a blood sample to sediment out all of the cells.) If a patient is dosed intravenously to a blood plasma concentration of 1.5 mcg/ml at 13:00 one day, at what time will the blood plasma concentration of drug have become reduced to a level where the drug is no longer effective and needs to be readministered?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>To answer this, first work out a rough answer by counting how many halflives are needed to get from the initial concentration of drug to the ineffective concentration of drug:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn002"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6158496" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ed0c2d95/s110_1_ue020.gif" alt=""/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6158496">View larger image</a></div><div class="oucontentcaption oucontentnonumber oucontentcaptionplaceholder"> </div></div><a id="back_thumbnailfigure_idm6158496"></a></div><p>Somewhere between the second halflife (4 + 4 = 8 hours) and the third halflife (4 + 4 + 4 = 12 hours), the plasma concentration will reach 0.2 mcg/ml and the drug will become ineffective.</p><p>Using the equation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn002a"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/bcf0194e/s110_1_ue021.gif" alt=""/></div><p>we can find the gradient, as we have previously seen, by rearranging to give:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5793f431/s110_1_ue022.gif" alt=""/></div><p>We are dealing with a decrease, so the slope of the gradient on the graph will be negative, therefore <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/fb153269/s110_1_ue023.gif" alt="" width="92" height="36" style="maxwidth:92px;" class="oucontentinlinefigureimage"/></span>
</p><p>We want to know at what time to readminister the drug, and time is on the xaxis, so we need to make ΔX the subject of the equation: rearranging gives, ΔX = ΔY ÷ −0.1733.</p><p>We can calculate ΔY because we already know the start and end concentrations of the drug. They are 1.5 mcg/ml at the start and 0.2 mcg/ml at the end. As such, ΔY = ln(1.5) − ln(0.2) = (0.4055) − (−1.6094) = 0.4055 + 1.6094 = 2.0149 (remembering that subtracting a negative number is like adding a positive number).</p><p>Because ΔY describes a decrease, this number should be represented as a negative number, i.e. −2.0149, then ΔX = −2.0149 ÷ −0.1733 = 11.627 hours.</p><p>So, after 11.627 hours the drug concentration will have decreased from 1.5 mcg/ml to 0.2 mcg/ml, and the drug will have to be readministered. This fits with our original estimate of somewhere between 8 and 12 hours.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.5
3.6.5 Using the gradient of a semilogarithmic graph to calculate doubling time or halflifeS110_1<p>Knowing the equation <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie006i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span> allows you to perform several useful calculations without needing to make a graph, and we'll look at one such example in a moment.</p><p>First, let's return to the gradient of the exponential increase graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>. This gradient is <i>positive</i>: as you move left to right along the xaxis the graph climbs, and the values on the yaxis increase. What would be the gradient if this graph were reversed, to show an exponential decrease? In that situation, the graph would fall from left to right and the yaxis values would become smaller as you moved along the xaxis, so the gradient would have a <i>negative</i> value.</p><p>There are some basic rules for dealing with negative numbers that you need to be aware of before moving onto the next example, which deals with exponential decay.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box005_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Box 7 Being positive about negative numbers</h2><div class="oucontentinnerbox"><p>A negative number is any value less than zero. Look at the thermometer scale below.</p><p>−4 °C −3 °C −2 °C −1 °C 0 °C +1 °C +2 °C +3 °C +4 °C</p><p>The freezing point of water is marked as 0 °C. Temperatures above freezing go up incrementally in positive numbers. As temperatures get colder and colder below 0 °C their values also increase, but in negative numbers. The temperature <i>difference</i> between 0 °C and 1 °C is 1 °C. Similarly, the temperature <i>difference</i> between 0 °C and −1 °C is also 1 °C. The temperature difference between −1 °C and +1 °C is 2 °C. Using a scale like this we can come up with some general rules for addition and subtraction involving negative numbers.</p><p>
<b>Addition and subtraction involving negative numbers</b>
</p><p>Using the thermometer scale above, you should be able to verify for yourself that the following sums are correct:</p><ul class="oucontentunnumbered"><li>3 − 4 = −1</li><li>−2 + 1 = −1</li><li>−1 − 1 = −2</li></ul><p>Now here's the tricky one: subtracting a negative number is the same as adding a positive number, so:</p><ul class="oucontentunnumbered"><li>−5 − (−3) = −5 + 3 = −2</li></ul><p>Another helpful way to view the same calculation is as follows:</p><ul class="oucontentunnumbered"><li>−5 − (−3) = −(5 − 3) = −(2) = −2</li></ul><p><b>Multiplication and division involving negative numbers</b></p><p>If you multiply 2 × 2, you get 4, so, a positive number multiplied by a positive number, gives a positive number. However, what do you get if you multiply 2 × −2?</p><p>The rule is that a positive number multiplied by a negative number (or the other way round), gives a negative number, so 2 × −2 = −4 (and −2 × 2 = −4).</p><p>The general idea is that if you multiply (or divide) a positive number by a negative number then that changes the sign of your eventual answer; or, put another way: multiplying or dividing the same signs together results in a positive number, whilst doing the same with opposite signs give a negative number.</p><ul class="oucontentunnumbered"><li>positive × positive = positive</li><li>positive × negative = negative</li><li>negative × positive = negative</li><li>negative × negative = positive</li><li>positive ÷ positive = positive</li><li>positive ÷ negative = negative</li><li>negative ÷ positive = negative</li><li>negative ÷ negative = positive</li></ul></div></div></div><p>These rules will come in handy as you work through the following example question.</p><div class="oucontentexample oucontentsheavybox1 oucontentsbox " id="exa001_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Example 1</h2><div class="oucontentinnerbox"><p>The radioactive isotope iodine 131 (<sup>131</sup>I) is used in medicine to diagnose conditions of the thyroid gland. <sup>131</sup>I has a halflife of 8 days. If the hospital has just received a delivery of 10 g <sup>131</sup>I: how much will remain after 25 days?</p><p>As an initial check, we can work out a rough answer. Each halflife takes 8 days, therefore, after 8 + 8 + 8 = 24 days the <sup>131</sup>I will have gone through three halflives. Since we lose half of the <sup>131</sup>I after each halflife, out of the 10 g starting material there will be 1.25 g left after 3 halflives.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/650d2d86/s110_1_ue015i.gif" alt=""/></div><p>We want to know how much <sup>131</sup>I will remain after 25 days, so we would estimate that there should be a little less than 1.25 g remaining.</p><p>Now let's work out the answer accurately using some of the principles we would apply if we were plotting data of this radioactive decay on a graph like that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.4#fig005007">Figure 12</a>.</p><p>On a semilogarithmic graph we know that, <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/aa90aedb/s110_1_ie007i.gif" alt="" width="87" height="35" style="maxwidth:87px;" class="oucontentinlinefigureimage"/></span>where T<sub>d</sub> can represent doubling time or halflife; this distinction becomes important in a moment. We are concerned here with the halflife, so let's rename T<sub>d</sub> as 'halflife'.</p><p>If we multiply both sides of the equation by gradient, and divide both sides of the equation by halflife, then this rearranges to give</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/fa967ae0/s110_1_ue016i.gif" alt=""/></div><p>Using a calculator to find the logarithm of 2 gives, ln(2) = 0.6931</p><p>The halflife is 8 days, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c1934146/s110_1_ue017i.gif" alt=""/></div><p>We know that the gradient of the graph can also be calculated in a different way, directly from the graph itself, since the gradient of any straightline graph is defined as <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie008i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> </p><p>Now the distinction between doubling time and halflife becomes important. The initial equation we used doesn't distinguish between exponential increase or decrease. However, because we are dealing with an exponential decay (the amount of <sup>131</sup>I is <i>decreasing</i> with time) the gradient will be negative. The actual numerical value, the steepness, of the gradient is unchanged; it's just that we are moving 'downhill' (exponential decrease) rather than uphill (exponential increase), so this value must have a negative sign.</p><p>Therefore, <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5ccc1a72/s110_1_ie009i.gif" alt="" width="18" height="29" style="maxwidth:18px;" class="oucontentinlinefigureimage"/></span> = −0.0866</p><p>Rearranging gives, ΔY = −0.0866 × ΔX</p><p>The time interval we are interested in is ΔX = 25 days, so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/4d5a34a5/s110_1_ue018i.gif" alt=""/></div><p>Using the general rules we established in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.6.5#box005003">Box 7</a>, we know that because we are multiplying a positive number by a negative number the answer will be a negative number, thus, ΔY = −0.0866 × 25 = −2.165</p><p>The yaxis is on a logarithmic scale, so this value that we have deduced for ΔY represents the logarithm of the change in the amount of <sup>131</sup>I that has occurred during 25 days.</p><p>The starting amount of <sup>131</sup>I is 10 g, so in order to be able to compare this with our value for ΔY (which is in logarithms) it too needs to be converted into a logarithm, and the logarithm of 10 is ln(10) = 2.3026.</p><p>During 25 days, this starting amount of <sup>131</sup>I will be reduced by ΔY, therefore: the remaining <sup>131</sup>I after 25 days = 2.3026  2.165 = 0.1376</p><p>Once more, this value for the remaining amount of <sup>131</sup>I is a logarithm because the values that we have used to calculate it have been in logarithmic form. In order to convert it to a conventional number you need to use the 'antilogarithm' button on your calculator (shown as e<sup>x</sup> on most calculators). You should find that e<sup>x</sup>(0.1376) = 1.148, which is how many grams of <sup>131</sup>I are remaining after 25 days.</p><p>Finally, always check that your calculated value fits with your initial estimate from counting halflives. Our initial rough answer was for there to be a little less than 1.25 g <sup>131</sup>I remaining after 25 days. Reassuringly, our accurate answer of 1.148 g is exactly in keeping with the rough calculation.</p></div></div></div><div class="
oucontentsaq
oucontentsheavybox1 oucontentsbox " id="saq008"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">SAQ 11</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>The elimination of a drug from the blood, due to metabolism and excretion, follows an exponential decay with a halflife of 4 hours. Below a blood plasma concentration of 0.2 mcg/ml the drug is not effective. (Blood consists of cells floating in a liquid. This liquid component of the blood is called plasma and can be produced by centrifuging a blood sample to sediment out all of the cells.) If a patient is dosed intravenously to a blood plasma concentration of 1.5 mcg/ml at 13:00 one day, at what time will the blood plasma concentration of drug have become reduced to a level where the drug is no longer effective and needs to be readministered?</p></div>
<div class="oucontentsaqanswer" datashowtext="Reveal answer" datahidetext="Hide answer"><h3 class="oucontenth4">Answer</h3><p>To answer this, first work out a rough answer by counting how many halflives are needed to get from the initial concentration of drug to the ineffective concentration of drug:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn002"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6158496" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ed0c2d95/s110_1_ue020.gif" alt=""/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2148&extra=thumbnailfigure_idm6158496">View larger image</a></div><div class="oucontentcaption oucontentnonumber oucontentcaptionplaceholder"> </div></div><a id="back_thumbnailfigure_idm6158496"></a></div><p>Somewhere between the second halflife (4 + 4 = 8 hours) and the third halflife (4 + 4 + 4 = 12 hours), the plasma concentration will reach 0.2 mcg/ml and the drug will become ineffective.</p><p>Using the equation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn002a"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/bcf0194e/s110_1_ue021.gif" alt=""/></div><p>we can find the gradient, as we have previously seen, by rearranging to give:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/5793f431/s110_1_ue022.gif" alt=""/></div><p>We are dealing with a decrease, so the slope of the gradient on the graph will be negative, therefore <span class="oucontentinlinefigure" style="verticalalign:4px;"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/fb153269/s110_1_ue023.gif" alt="" width="92" height="36" style="maxwidth:92px;" class="oucontentinlinefigureimage"/></span>
</p><p>We want to know at what time to readminister the drug, and time is on the xaxis, so we need to make ΔX the subject of the equation: rearranging gives, ΔX = ΔY ÷ −0.1733.</p><p>We can calculate ΔY because we already know the start and end concentrations of the drug. They are 1.5 mcg/ml at the start and 0.2 mcg/ml at the end. As such, ΔY = ln(1.5) − ln(0.2) = (0.4055) − (−1.6094) = 0.4055 + 1.6094 = 2.0149 (remembering that subtracting a negative number is like adding a positive number).</p><p>Because ΔY describes a decrease, this number should be represented as a negative number, i.e. −2.0149, then ΔX = −2.0149 ÷ −0.1733 = 11.627 hours.</p><p>So, after 11.627 hours the drug concentration will have decreased from 1.5 mcg/ml to 0.2 mcg/ml, and the drug will have to be readministered. This fits with our original estimate of somewhere between 8 and 12 hours.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.7.1 Averages: finding the middle of a group of numbers
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.7.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>The average of a group of numbers (which is sometimes called the 'mean') represents the <i>balancing point</i> or middle of the data. It is found by adding together all of the individual data values and dividing by the sample size.</p><p>For example, I ask five friends how many children they have, and I get the answers: 0, 1, 2, 4, and 1. The total number of children is 0 + 1 + 2 + 4 + 1 = 8 and there are 5 friends, so the average number of children per friend is 8 ÷ 5 = 1.6. This has been represented graphically in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.7.1#fig005008">Figure 13</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig005_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c328e3f1/s110_1_015i.jpg" alt="Figure 13" width="264" height="117" style="maxwidth:264px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 The average is the <i>balancing point</i> of the data</span></div></div></div><p>Notice that the average doesn't tell you anything about the <i>range</i> of values that were measured (the maximum and minimum values within the given group of numbers). For instance, someone with their head in the oven and their feet in the freezer might have an average temperature of 37 °C, even though their hair is on fire and their feet are frozen!</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.7.1
3.7.1 Averages: finding the middle of a group of numbersS110_1<p>The average of a group of numbers (which is sometimes called the 'mean') represents the <i>balancing point</i> or middle of the data. It is found by adding together all of the individual data values and dividing by the sample size.</p><p>For example, I ask five friends how many children they have, and I get the answers: 0, 1, 2, 4, and 1. The total number of children is 0 + 1 + 2 + 4 + 1 = 8 and there are 5 friends, so the average number of children per friend is 8 ÷ 5 = 1.6. This has been represented graphically in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.7.1#fig005008">Figure 13</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig005_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/c328e3f1/s110_1_015i.jpg" alt="Figure 13" width="264" height="117" style="maxwidth:264px;" class="oucontentfigureimage"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 The average is the <i>balancing point</i> of the data</span></div></div></div><p>Notice that the average doesn't tell you anything about the <i>range</i> of values that were measured (the maximum and minimum values within the given group of numbers). For instance, someone with their head in the oven and their feet in the freezer might have an average temperature of 37 °C, even though their hair is on fire and their feet are frozen!</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

3.8.1 Standard deviation: finding how reproducible a series of measurements are
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1
Tue, 12 Apr 2016 23:00:00 GMT
<p>Even if we know the maximum and minimum and middle values in a group of numbers, we still don't have a clear idea about the distribution of values within that range: are most of the values all bunched up at one end or spread evenly across the results?</p><p>For instance, if I count my pulse rate on the hour every hour, nine times over the course of a day, I might get the following values for the number of beats per minute (bpm): 61, 59, 60, 62, 60, 100, 59, 63, 61. The average result is 65 bpm and range of values is 59100 bpm. From looking solely at the range you might get the impression that my heart rate fluctuates wildly throughout the day. In fact, my heart rate is remarkably constant, and the value of 100 bpm was a reading, taken after running up the stairs just before 14:00.</p><p>The way to find out whether a series of measurements are all tightly grouped together or are spread out more evenly is to make a graph that shows how often a particular value was recorded. This type of graph is called a <i>frequency distribution</i>, because it shows how frequently particular values were recorded.</p><p>For instance, in the list of my pulse rate measurements from above:</p><ul class="oucontentunnumbered"><li>
<p>62, 63, and 100 bpm were recorded once</p>
</li><li>
<p>59, 60 and 61 bpm were recorded twice.</p>
</li></ul><p>These data have been plotted on the bar graph shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005009">Figure 14</a>.</p><div class="oucontentfigure" style="width:511px;" id="fig005_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/bf1ad553/s110_1_016i.jpg" alt="Figure 14" width="511" height="317" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A frequency distribution histogram showing how often my pulse rate was measured to be a particular value</span></div></div></div><p>With enough measurements, this type of graph eventually resembles a bellshape, often called a Gaussian curve or a <i>normal distribution</i>, where the most common value is at the top of the curve and there's a spread of less and less common results (some larger and some smaller) on either side. For example if I'd continued to make pulse rate measurements, I would soon have found that my measurement of 100 bpm was a oneoff and in fact, most of the measurements were centred around 65 bpm.</p><p>Where results are very regularly reproduced and don't deviate much from the mean value (high precision), the bellshaped curve is steep and narrow (like the top graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>) and this indicates a small <b>standard deviation</b> from the mean value (as suggested by the condensed spread of values on the xaxis). In contrast, when the results are more variable (low precision), the bellshaped curve is relatively spread and flat like the bottom graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>, and this indicates a large standard deviation from the mean value.</p><div class="oucontentfigure" style="width:428px;" id="fig005_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/6c2aad9c/s110_1_017i.jpg" alt="Figure 15" width="428" height="401" style="maxwidth:428px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 A normal frequency distribution looks like a bellshaped curve. Top: data showing a small standard deviation from the mean value. Bottom: data showing a large standard deviation from the mean value</span></div></div></div><p>The exact value of the standard deviation for a group of numbers is calculated using a complex equation that you are not required to know. Suffice to say that in my pulse rate data above, the mean value is 65 bpm and the standard deviation is 13.2 bpm. Because the standard deviation indicates the spread of data both greater and less than the mean value, it is shown with a 'plus or minus' symbol. Thus, the mean value with the standard deviation is 65 ± 13.2 bpm.</p><p>About 68% of all the results occur within one standard deviation of the mean value on the horizontal, xaxis, and this figure is represented by the red areas on both of the graphs in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>). About 95% of the results lie within two standard deviations from the mean (the red plus the green areas on these two graphs), and about 99% of the results lie within three standard deviations of the mean value (the red, green <i>and</i> blue areas).</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ac1c916c/s110_1_i016i.jpg" alt="Source: www.CartoonStock.com" width="342" height="309" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>This information can be used to find out if measurements are unusual or not. For instance, we know that 95% of the measurements should be within 2 standard deviations of the mean value, meaning that only 5% of the results will fall outside of two standard deviations. Because the graph is symmetrical, this 5% includes results that are both larger and smaller than the mean value. If we are only interested in results larger than the mean value, then we can see that only 5 ÷ 2 = 2.5% of results occur outside of the green area to the right of the graph. i.e. only 2.5% of the results would be expected to be more than 2 standard deviations greater than the mean value.</p><p>In my pulse rate data, one standard deviation was 13.2 bpm and the mean value was 65 bpm. Therefore a pulse rate two standard deviations larger than the mean would be (13.2 × 2) + 65 = 91.4 bpm. As such, I would expect any pulse rate of 91.4 bpm or above to occur less than 2.5% of the time. If today I measured my pulse rate on 5 occasions and it was above 91 bpm on one occasion then that could happen by chance, but if it was this high on subsequent measurements then I should become increasingly worried, since 2 out of the 5 measurements made (i.e. 40%) were above 91 bpm.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1
3.8.1 Standard deviation: finding how reproducible a series of measurements areS110_1<p>Even if we know the maximum and minimum and middle values in a group of numbers, we still don't have a clear idea about the distribution of values within that range: are most of the values all bunched up at one end or spread evenly across the results?</p><p>For instance, if I count my pulse rate on the hour every hour, nine times over the course of a day, I might get the following values for the number of beats per minute (bpm): 61, 59, 60, 62, 60, 100, 59, 63, 61. The average result is 65 bpm and range of values is 59100 bpm. From looking solely at the range you might get the impression that my heart rate fluctuates wildly throughout the day. In fact, my heart rate is remarkably constant, and the value of 100 bpm was a reading, taken after running up the stairs just before 14:00.</p><p>The way to find out whether a series of measurements are all tightly grouped together or are spread out more evenly is to make a graph that shows how often a particular value was recorded. This type of graph is called a <i>frequency distribution</i>, because it shows how frequently particular values were recorded.</p><p>For instance, in the list of my pulse rate measurements from above:</p><ul class="oucontentunnumbered"><li>
<p>62, 63, and 100 bpm were recorded once</p>
</li><li>
<p>59, 60 and 61 bpm were recorded twice.</p>
</li></ul><p>These data have been plotted on the bar graph shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005009">Figure 14</a>.</p><div class="oucontentfigure" style="width:511px;" id="fig005_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/bf1ad553/s110_1_016i.jpg" alt="Figure 14" width="511" height="317" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A frequency distribution histogram showing how often my pulse rate was measured to be a particular value</span></div></div></div><p>With enough measurements, this type of graph eventually resembles a bellshape, often called a Gaussian curve or a <i>normal distribution</i>, where the most common value is at the top of the curve and there's a spread of less and less common results (some larger and some smaller) on either side. For example if I'd continued to make pulse rate measurements, I would soon have found that my measurement of 100 bpm was a oneoff and in fact, most of the measurements were centred around 65 bpm.</p><p>Where results are very regularly reproduced and don't deviate much from the mean value (high precision), the bellshaped curve is steep and narrow (like the top graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>) and this indicates a small <b>standard deviation</b> from the mean value (as suggested by the condensed spread of values on the xaxis). In contrast, when the results are more variable (low precision), the bellshaped curve is relatively spread and flat like the bottom graph in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>, and this indicates a large standard deviation from the mean value.</p><div class="oucontentfigure" style="width:428px;" id="fig005_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/6c2aad9c/s110_1_017i.jpg" alt="Figure 15" width="428" height="401" style="maxwidth:428px;" class="oucontentfigureimage oucontentmediawide"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 A normal frequency distribution looks like a bellshaped curve. Top: data showing a small standard deviation from the mean value. Bottom: data showing a large standard deviation from the mean value</span></div></div></div><p>The exact value of the standard deviation for a group of numbers is calculated using a complex equation that you are not required to know. Suffice to say that in my pulse rate data above, the mean value is 65 bpm and the standard deviation is 13.2 bpm. Because the standard deviation indicates the spread of data both greater and less than the mean value, it is shown with a 'plus or minus' symbol. Thus, the mean value with the standard deviation is 65 ± 13.2 bpm.</p><p>About 68% of all the results occur within one standard deviation of the mean value on the horizontal, xaxis, and this figure is represented by the red areas on both of the graphs in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection3.8.1#fig005010">Figure 15</a>). About 95% of the results lie within two standard deviations from the mean (the red plus the green areas on these two graphs), and about 99% of the results lie within three standard deviations of the mean value (the red, green <i>and</i> blue areas).</p><p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/4a2ff72b/ac1c916c/s110_1_i016i.jpg" alt="Source: www.CartoonStock.com" width="342" height="309" style="maxwidth:342px;" class="oucontentinlinefigureimage"/></span>
</p><p>This information can be used to find out if measurements are unusual or not. For instance, we know that 95% of the measurements should be within 2 standard deviations of the mean value, meaning that only 5% of the results will fall outside of two standard deviations. Because the graph is symmetrical, this 5% includes results that are both larger and smaller than the mean value. If we are only interested in results larger than the mean value, then we can see that only 5 ÷ 2 = 2.5% of results occur outside of the green area to the right of the graph. i.e. only 2.5% of the results would be expected to be more than 2 standard deviations greater than the mean value.</p><p>In my pulse rate data, one standard deviation was 13.2 bpm and the mean value was 65 bpm. Therefore a pulse rate two standard deviations larger than the mean would be (13.2 × 2) + 65 = 91.4 bpm. As such, I would expect any pulse rate of 91.4 bpm or above to occur less than 2.5% of the time. If today I measured my pulse rate on 5 occasions and it was above 91 bpm on one occasion then that could happen by chance, but if it was this high on subsequent measurements then I should become increasingly worried, since 2 out of the 5 measurements made (i.e. 40%) were above 91 bpm.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Conclusion
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection4
Tue, 12 Apr 2016 23:00:00 GMT
<p>This free course provided an introduction to studying Health & Wellbeing. It took you through a series of exercises designed to develop your approach to study and learning at a distance, and helped to improve your confidence as an independent learner.</p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection4
ConclusionS110_1<p>This free course provided an introduction to studying Health & Wellbeing. It took you through a series of exercises designed to develop your approach to study and learning at a distance, and helped to improve your confidence as an independent learner.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University

Keep on learning
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsection5
Tue, 12 Apr 2016 23:00:00 GMT
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Keep on learningS110_1<div class="oucontentfigure oucontentmediamini"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/61471/mod_oucontent/oucontent/409/8ff4c822/d3c986e6/ol_skeleton_keeponlearning_image.jpg" alt="" width="300" height="200" style="maxwidth:300px;" class="oucontentfigureimage"/></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Study another free course</h2><p>There are more than <b>800 courses on OpenLearn</b> for you to choose from on a range of subjects. </p><p>Find out more about all our <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">free courses</a></span>.</p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Take your studies further</h2><p>Find out more about studying with The Open University by <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">visiting our online prospectus</a>. </p><p>If you are new to university study, you may be interested in our <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Access Courses</a> or <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Certificates</a>.</p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">What's new from OpenLearn?</h2><p>
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Acknowledgements
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsectionacknowledgements
Tue, 12 Apr 2016 23:00:00 GMT
<p>Except for third party materials and otherwise stated (see <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/conditions">terms and conditions</a></span>), this content is made available under a <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/4.0/">Creative Commons AttributionNonCommercialShareAlike 4.0 Licence</a></p><p>Course image: <a class="oucontenthyperlink" href="https://www.flickr.com/photos/tz1_1zt/">David J Morgan</a> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/2.0/legalcode">Creative Commons AttributionNonCommercialShareAlike 2.0 Licence</a>.</p><p>Grateful acknowledgement is made to the following sources for permission:</p><p><b><i>Sections 1.10 and 3.8</i></b>: www.CartoonStock.com
</p><p>This extract is taken from S110 © 2007 The Open University.</p><p>All other materials included in this course are derived from content originated at the Open University.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>
https://www.open.edu/openlearn/sciencemathstechnology/mathematicsstatistics/usingnumbersandhandlingdata/contentsectionacknowledgements
AcknowledgementsS110_1<p>Except for third party materials and otherwise stated (see <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/conditions">terms and conditions</a></span>), this content is made available under a <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/4.0/">Creative Commons AttributionNonCommercialShareAlike 4.0 Licence</a></p><p>Course image: <a class="oucontenthyperlink" href="https://www.flickr.com/photos/tz1_1zt/">David J Morgan</a> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/2.0/legalcode">Creative Commons AttributionNonCommercialShareAlike 2.0 Licence</a>.</p><p>Grateful acknowledgement is made to the following sources for permission:</p><p><b><i>Sections 1.10 and 3.8</i></b>: www.CartoonStock.com
</p><p>This extract is taken from S110 © 2007 The Open University.</p><p>All other materials included in this course are derived from content originated at the Open University.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBUsing numbers and handling data  S110_1Copyright © 2016 The Open University