S110_1Health sciences in practice Using numbers and handling data About this free course This free course provides a sample of level 1 study in Health & Wellbeing http://www.open.ac.uk/courses/find/health-and-wellbeingThis version of the content may include video, images and interactive content that may not be optimised for your device. You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: http://www.open.edu/openlearn/health-sports-psychology/health/public-health/using-numbers-and-handling-data/content-section-0.There you'll also be able to track your progress via your activity record, which you can use to demonstrate your learning.
The Open University, Walton Hall, Milton Keynes, MK7 6AA
978-1-4730-1191-5 (.epub)
IntroductionThis 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 work-based 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.This OpenLearn course provides a sample of level 1 study in Health & WellbeingAfter studying this course, you should be able to:understand the decimal system of numbering (hundreds, tens, units)explain the best way to write down decimal numbers and associated units of measurement in the healthcare workplace, in a manner that avoids confusionunderstand the concepts of discrete and continuous variables and the best types of graphs used to represent these dataanalyse, construct and extract information from graphs.
1 Decimals
1.1 Introducing the decimal system of numbersMany different systems for writing numbers have been developed over the history of humankind.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?Numbering systems get round this problem by using a system of scale 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.For instance, the number five 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 to represent an individual unit and a star to represent a collection of five individual units? Using this system, the number eleven could be represented as two stars and a line (i.e. five plus five plus one).The decimal system uses a similar concept of scale, but it is arranged such that ten of any particular unit makes one unit of the next size up. This also works the other way round: each larger unit consists of ten units of the next size down.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.These different groups differ by a factor of ten (each group is ten times larger or smaller than the one that precedes or follows it), which is also known as an order of magnitude (see Figure 1).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 positional system 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).The name of the unit of measurement 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.
1.2 Decimal pointsSuppose you have less than one of any particular unit: how would you represent that using the decimal system?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 decimal point is then used to mark the boundary between the whole units and tenths of that unit.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).In Table 1, you can see how the position of each digit in the sequence is used to tell you the overall size of the number. Table 1 Decimal numbers described in terms of their orders of magnitude
Description Tens Units Decimal point Tenths Hundredths
4.5 is 4 units and 5 tenths0450
6.87 is 6 units, 8 tenths and 7 hundredths0687
98.04 is 9 tens, 8 units, 0 tenths and 4 hundredths9804
0.06 is 6 hundredths0006
Now try placing decimal points appropriately and reading the values of decimal numbers for yourself with practice questions 1, 2 and 3.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions.1.2.1 Study Note 1Simple 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.Note: 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.
1.3 Marking decimals on a scale Figure 2 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 tenths 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.In Figure 2 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.Now practise placing decimal values on a similar scale for yourself with practice questions 4 to 8.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions.
1.5 Rounding to decimal placesSometimes 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?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 recurring 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.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 1.6666 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 1.6466 ml then the value corrected to one decimal place would be 1.6 ml (1 dp).Here are some more worked examples for you to practise with: Now try rounding numbers to specific decimal places for yourself with practice question 9.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions. Box 1 General rules for numbers in healthcare Try to avoid the need for a decimal point Use 500 mg not 0.5 g Use 125 mcg not 0.125 mg Never leave a decimal point 'naked' Paracetamol 0.5 mg not Paracetamol .5 mg Avoid using a terminal zero Diazepam 2 mg not Diazepam 2.0 mg Put a space between the drug name and dose Apresoline 55 mg not Apresoline55 mg Study Note 3 'Dos and don'ts' with decimals in the healthcare workplaceAs 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. Look carefully! 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. 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). 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 hand-written 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. Listen to the audio track below. It contains information that reinforces what you have just learned here.Click to listen to the track [3 minutes 30 seconds, 4.02MB]
1.6 Multiplication and division by factors of ten1.6.1 Getting comfortable with factors of tenMoving 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: 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.To multiply a number by 100, move the decimal point 2 places to the right: 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: Look at the animated examples in the second 'Info' box of your practice questions. 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.In Table 2, 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. Table 2 Names, units and symbols for factors of ten found within the decimal system of numberingThe units shown in bold are those that you are likely to encounter on a daily basis in your work.
Name Number Order of magnitude Power Unit Symbol
Million1 000 0006106 mega M
Hundred thousand100 0005105
Ten thousand10 0004104
Thousand1 0003103 kilo k
Hundred1002102 hectoh
Ten101101
One10100
Tenth0.1−110−1 decid
Hundredth0.01−210−2 centic
Thousandth0.001−310−3 milli m
Ten thousandth0.000 1−410−4
Hundred thousandth0.000 01−510−5
Millionth0.000 001−610−6 micro μ
Ten millionth0.000 000 1−710−7
Hundred millionth0.000 000 01−810−8
Billionth0.000 000 001−910−9 nanon
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 power or the exponent, 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, 102 ('ten squared'), means 10 × 10 = 100. Similarly, 104 means 10 × 10 × 10 × 10 = 10 000.Notice also that numbers less than zero can be shown by a negative exponent. For instance, 10−1 means '1 divided by 10', where 1 ÷ 10 = 0.1. Similarly, 10−4 means '1 divided by 104', 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 Table 2 states that 100 = 1.An exponent will often have a number and a multiplication sign written before it, for example 2.4 × 103, 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. If the exponent is positive, then move the decimal point to the right by the same number of places as the exponent. If the exponent is negative, then move the decimal point to the left by the same number of places as the exponent. 1.7 SI units and conversionsThe international system of units (Le Système International d'Unités: 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.There are seven basic units (or base units) to the SI system and these are shown in Table 3. All other units of measurement can be derived from combinations of these base SI units. Table 3 The seven base SI units
Quantity Name Symbol
Lengthmetrem
Masskilogramkg
Timeseconds
TemperaturekelvinK
Electric currentampereA
Luminous intensitycandelacd
Amount of substancemolemol
Many units of measurement arise from a combination of different base SI units, and some of these are given in Table 4. 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−1 where the negative exponent indicates the division. Similarly, the newton is a SI-derived 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−2. Table 4 Commonly used units that are derived from base SI units
Quantity Derived unit Symbol SI units
Speedmetres per secondm s−1 m s−1
ForcenewtonNm kg s−2
EnergyjouleJm2 kg s−2
Volumecubic metrem3 m3
PressurepascalPakg m −1 s−2
In addition, some non-standard SI units are in common usage, and a selection of these is given in Table 5. Table 5 Commonly used units that are not derived from base SI units
Quantity Symbol Non-SI unit Conversion factors
Timeminminute1 min = 60 s
Timehhour1 h = 60 min = 3600 s
Timedday1 d = 24 h = 1440 min = 86400 s
Volume1litre1 l = 0.001 m3
Massttonne1 t = 1000 kg
Energycalcalorie1 cal = 4.18 J
Temperature°CCelsius1 °C = 274.15 K
PressuremmHgmillimetres of mercury1 mmHg = 133.3 Pa
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.For instance, 1000 metres is 1 kilometre and this is abbreviated to 1 km. One hundredth of a metre is one centimetre, abbreviated as 1 cm.Box 2 A note about temperatureAlthough 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.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.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions.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.Activity 1.1 Comparing SI units 0 25 Part 1 Vitamin tabletsA multivitamin tablet contains various vitamins and minerals. Table 6 shows the recommended daily allowance (RDA) for a selection of them, as shown on the packet. Table 6 Label from packet of vitamin tablets
Component %RDA Amount per tablet
Vitamin A100800 μg
Vitamin B61002 mg
Vitamin C10060 mg
Vitamin D1005 μg
Vitamin E10010 mg
Magnesium50150 mg
Note: 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).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?
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.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).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.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.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:150 mg x 2 = 300 mgTherefore, 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).
Activity 1.2 Comparing SI units 0 5 Part 2 TemperatureWhat is the freezing point of water on the Kelvin scale?The freezing point of water is 0 °C, which is +273.15 K.Activity 1.3 Comparing SI units 0 15 Part 3 TemperatureWhat is the normal range of human body temperature in degrees Celsius? Why is there a range of values?Body temperature varies with age, time of the day, menstrual cycle and location in the body: in a 6-year-old 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 36.5-37.2 °C is considered normal for a healthy person. 38 °C or above is a significant fever.
1.9 Addition of decimal numbersIf 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?To compare 109.8 with 6.5, you need to remember that Place the two numbers in a grid on top of each other and make sure that columns representing the same magnitude 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: To add these numbers start from the right, with the smallest sized group and compare the numbers in the Tenths column. 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 one unit and three tenths.Put your addition into the results line at the bottom. Then move up one order of magnitude to the next column to the left (Units) and add these values together. 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. Moving up one order of magnitude and adding the tens together we have 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. Move up to the final order of magnitude and add the hundreds together. So, as you can see from your results line, the sum of 109.8 ml and 6.5 ml = 116.3 ml Now practise some additions for yourself with practice question 12.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions.
1.10 Subtraction of decimal numbersSubtraction 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?' Let's work through an example: how much is 25.18 - 16.87?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 must go below the number you are subtracting it from. Just like addition, you should work through your calculation from right to left.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.Let's see how that works out in practice with our example of 25.18 - 16.87.In the first column from the right (Hundredths), we need to subtract 7 from 8. Eight take away seven leaves us with one hundredth. 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. Now we can subtract 8 from the resulting 11 tenths, to leave a total of 3 tenths. 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. Finally, in the Tens column, we're left with 1 ten to subtract from 1 ten, which leaves 0 tens. So, the result of 25.18 - 16.87 = 8.31.You can double-check you're right by adding 8.31 to 16.87 in order to confirm you get back to the original 25.18.Now practise some subtractions for yourself with practice question 13.Right click and open the practice questions in a separate window, then you can switch easily between the course text and the questions.
1.11 Addition and subtraction in practice - fluid balanceA common healthcare example that uses addition and subtraction involves calculating the fluid balance of a patient.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.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.SAQ 3Calculate the fluid balance for the patient described in the following scenario.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.The fluid balance is the difference between the total fluid input and the total fluid output during a day.To calculate this, first, check whether the units of measurement are in millilitres; convert them to millilitres, if necessary.Then add together all the fluid inputs in millilitres. Then add together all the fluid outputs in millilitres. The fluid balance is the daily total inputs minus the daily total outputs. 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).
2 Accuracy, precision and common errors
2.1 Differences between accuracy and precisionAccuracy 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 Figure 6. 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's-eye 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's-eye. 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.From the example given you can see how it is possible to be very precise, but not at all accurate. This is called a systematic error (sometimes also called bias) and can normally be corrected.
2.2 Checking accuracy and precision2.1.1 AccuracyThe 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.2.2.2 PrecisionMeasuring 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 Section 3.5, how to quantify the precision of a series of measurements.
2.3 Common maths problems and errors in the workplaceIn 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 Figure 7 are sadly typical).However, many of these types of errors can be avoided by pausing and thinking, 'is this right'?
2.5 What is a sensible dose?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.The following dose ranges are the most sensible and practical for adults: Table 7 Typical drug dosesFor each category, the doses for a baby or a child are normally much less than adult doses.
Drug formulationTypical dose at any one time
Liquidoral: 5-20 ml (1-4 teaspoons full)
injection: generally 0.25-2 ml
subcutaneous: 1 ml or less
intramuscular: adults - up to 3 ml in large muscles
children and elderly - up to 2 ml
infants - 1 ml or less
Solid1-4 tablets
Gas0.2-150 litres/min
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 double-check it. If it still doesn't look right, or was written that way on the prescription then check with a senior colleague.
3 Handling data
3.1 GraphsInformation 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.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.
3.2 The anatomy of a graphA graph shows how two different types of data that can take on different values (known as variables) 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. Table 8 shows what these measurements might look like. Table 8 A patient's temperature, measured throughout the dayNotice that a measurement was not made at 17:00.
Time of dayPatient temperature/°C
13:0038.2
14:0037.9
15:0037.9
16:0037.5
18:0037.1
19:0036.9
As we shall see, this information can become easier to interrogate once it has been assembled graphically.3.2.1 AxesA graph is made using two different scales or axes, forming a right angle. The horizontal axis (x-axis) 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 (y-axis) is used to represent a variable that you measure but may not be able to control directly, such as a patient's temperature.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.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 along the corridor (horizontal, x-axis) before you can go up the stairs (vertical, y-axis).3.2.2 Choice of scaleIt'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.
3.3 Types of graphs and their usesMany 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.Here is a bar graph and a line graph plotting the patient's hourly temperature data that we looked at in Table 8.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.Which type of graph is best to use? To help answer this question you can consider the list shown in Table 9. Table 9 Choosing the best graph for your data - advantages and disadvantages of bar and line graphs
Bar and linetrends in data can be seen clearly (how one variable affects the other)
Bar and lineeasy to use the value of one of the variables to determine the value of the other variable
Bar and lineenables predictions to be made about results of data you don't have yet
Barbest for 'discrete' variables (those that change in jumps, with no 'in between' values)
Linebest for 'continuous' variables (those that change smoothly)
You will see from Table 9 that we have identified two different types of variable, and these are defined by the way in which their numerical values change. Discrete variables can only have specific values within any given range (e.g. 1, 2, 3). Continuous variables are not limited in this way, and can have any value within a range.SAQ 4Find some examples of continuous data and discrete data in your workplace.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.
3.4 Bar graphsThe following graph (Figure 9) records how the outside diameter of a hypodermic needle is related to the needle gauge number.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 histogram.) 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 quarter-gauge sizes, only whole numbers).SAQ 5If you wanted a needle with a diameter of 2 mm, what would be the required needle gauge number?Figure 9 shows that no such needle exists. A 2-mm-diameter 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.
3.6 Graphs with unusual scales - graphing exponentials 3.6.1 Radioactivity and bugs!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.3.6.2 Exponential increase: bacteriaBacteria are single-celled 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.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.From your earlier knowledge of exponentsyou might realise that this series of numbers can also be represented as a power series of exponentials: 21, 22, 23, 24, 25, 26, 27. Hence, this type of growth is often referred to as 'exponential growth'. 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.3.6.3 Exponential decrease: radioactive decayThe 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 half-life. For instance, the radioactive isotope 11C, pronounced 'carbon 11', has a half-life of 1224 seconds (a little over 20 minutes). After 1224 seconds, there would only be half of the starting amount of 11C remaining. After another 1224 seconds there would be only half of this amount remaining, i.e. 1/4 of the starting amount, and so on. Thus, the fraction of the starting material that remains after each half-life 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 re-written as a power series of exponentials, except this time they are fractions: 1/21, 1/22, 1/23, 1/24, 1/25, 1/26.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.Box 6 Radioactivity - a brief explanation of atomic instabilityAt 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 isotope number of that element.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: 16O, 17O, and 18O.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 isotope number, so this would be 12C (pronounced, 'carbon 12'). However, other isotopes of carbon exist, with different numbers of neutrons.There is a form of carbon that has 6 protons and 5 neutrons, 11C. However, this arrangement of protons and neutrons is unstable. The nucleus of a 11C atom is poised, like an over-wound spring, to suddenly re-arrange. When this happens, a burst of energy is released that is detected as radioactivity, 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 11B (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.3.6.4 Representing exponential relationships using graphsWhat 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 Figure 12, which shows the number of bacteria present in a millilitre of growth medium (marked on the left side of the graph), counted each hour.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, Td) or if we were dealing with exponential decay, it would give the half-life (often given the symbol, t1/2). 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.Because an exponential increase can also be represented as a power series, if we drew a graph showing the time at which there were 21, 22, 23, 24, 25, 26, 27, 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. 22) or 16 (24) or 128 (27) 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 straight-line graph. To do this we need to convert our bacteria number values into logarithms.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 = 2p). We've already seen that 128 = 27, so the logarithm of 128 with base 2 is 7.A word about bases: you're already familiar with the decimal system of numbering, in which exponents are of the type 'ten to the power something' (e.g. 103, 106 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' (23, 26, etc.).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 = 2.718p. This calculation gives the answer p = 4.852.In Figure 12 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 semi-logarithmic 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 semi-logarithmic 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 semi-logarithmic graphs are used to spot this kind of pattern.The steepness, or gradient, of the line in a semi-logarithmic 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 half-life. You may come across specific names for this gradient: for exponential increases, the gradient is also known as the growth constant, whilst for exponential decreases the gradient is known as the decay constant.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 y-axis (change in y: abbreviated to ΔY) if you moved one unit along on the x-axis (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 y-axis 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 of 5.5 ÷8 = 0.6875, which is the growth constant.The doubling time or half-life is given by dividing the logarithm of 2 by the gradient of the graph, i.e. .Using a calculator we can confirm that ln(2) = 0.6931. Therefore Td = 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.3.6.5 Using the gradient of a semi-logarithmic graph to calculate doubling time or half-lifeKnowing the equation allows you to perform several useful calculations without needing to make a graph, and we'll look at one such example in a moment.First, let's return to the gradient of the exponential increase graph in Figure 12. This gradient is positive: as you move left to right along the x-axis the graph climbs, and the values on the y-axis 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 y-axis values would become smaller as you moved along the x-axis, so the gradient would have a negative value.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.Box 7 Being positive about negative numbersA negative number is any value less than zero. Look at the thermometer scale below.−4 °C −3 °C −2 °C −1 °C 0 °C +1 °C +2 °C +3 °C +4 °CThe 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 difference between 0 °C and 1 °C is 1 °C. Similarly, the temperature difference 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. Addition and subtraction involving negative numbers Using the thermometer scale above, you should be able to verify for yourself that the following sums are correct:3 − 4 = −1−2 + 1 = −1−1 − 1 = −2Now here's the tricky one: subtracting a negative number is the same as adding a positive number, so:−5 − (−3) = −5 + 3 = −2Another helpful way to view the same calculation is as follows:−5 − (−3) = −(5 − 3) = −(2) = −2Multiplication and division involving negative numbersIf 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?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).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.positive × positive = positivepositive × negative = negativenegative × positive = negativenegative × negative = positivepositive ÷ positive = positivepositive ÷ negative = negativenegative ÷ positive = negativenegative ÷ negative = positiveThese rules will come in handy as you work through the following example question.Example 1The radioactive isotope iodine 131 (131I) is used in medicine to diagnose conditions of the thyroid gland. 131I has a half-life of 8 days. If the hospital has just received a delivery of 10 g 131I: how much will remain after 25 days?As an initial check, we can work out a rough answer. Each half-life takes 8 days, therefore, after 8 + 8 + 8 = 24 days the 131I will have gone through three half-lives. Since we lose half of the 131I after each half-life, out of the 10 g starting material there will be 1.25 g left after 3 half-lives. We want to know how much 131I will remain after 25 days, so we would estimate that there should be a little less than 1.25 g remaining.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 Figure 12.On a semi-logarithmic graph we know that, where Td can represent doubling time or half-life; this distinction becomes important in a moment. We are concerned here with the half-life, so let's re-name Td as 'half-life'.If we multiply both sides of the equation by gradient, and divide both sides of the equation by half-life, then this re-arranges to give Using a calculator to find the logarithm of 2 gives, ln(2) = 0.6931The half-life is 8 days, so 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 straight-line graph is defined as Now the distinction between doubling time and half-life 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 131I is decreasing 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.Therefore, = −0.0866Rearranging gives, ΔY = −0.0866 × ΔXThe time interval we are interested in is ΔX = 25 days, so Using the general rules we established in Box 7, 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 = −2.165The y-axis is on a logarithmic scale, so this value that we have deduced for ΔY represents the logarithm of the change in the amount of 131I that has occurred during 25 days.The starting amount of 131I 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.During 25 days, this starting amount of 131I will be reduced by ΔY, therefore: the remaining 131I after 25 days = 2.3026 - 2.165 = 0.1376Once more, this value for the remaining amount of 131I 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 'anti-logarithm' button on your calculator (shown as ex on most calculators). You should find that ex(0.1376) = 1.148, which is how many grams of 131I are remaining after 25 days.Finally, always check that your calculated value fits with your initial estimate from counting half-lives. Our initial rough answer was for there to be a little less than 1.25 g 131I remaining after 25 days. Reassuringly, our accurate answer of 1.148 g is exactly in keeping with the rough calculation.SAQ 11The elimination of a drug from the blood, due to metabolism and excretion, follows an exponential decay with a half-life 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 re-administered?To answer this, first work out a rough answer by counting how many half-lives are needed to get from the initial concentration of drug to the ineffective concentration of drug: Somewhere between the second half-life (4 + 4 = 8 hours) and the third half-life (4 + 4 + 4 = 12 hours), the plasma concentration will reach 0.2 mcg/ml and the drug will become ineffective.Using the equation we can find the gradient, as we have previously seen, by rearranging to give: We are dealing with a decrease, so the slope of the gradient on the graph will be negative, therefore We want to know at what time to re-administer the drug, and time is on the x-axis, so we need to make ΔX the subject of the equation: rearranging gives, ΔX = ΔY ÷ −0.1733.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).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.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 re-administered. This fits with our original estimate of somewhere between 8 and 12 hours.
3.8 Descriptive statistics3.8.1 Standard deviation: finding how reproducible a series of measurements areEven 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?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 59-100 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.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 frequency distribution, because it shows how frequently particular values were recorded.For instance, in the list of my pulse rate measurements from above: 62, 63, and 100 bpm were recorded once 59, 60 and 61 bpm were recorded twice. These data have been plotted on the bar graph shown in Figure 14.With enough measurements, this type of graph eventually resembles a bell-shape, often called a Gaussian curve or a normal distribution, 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 one-off and in fact, most of the measurements were centred around 65 bpm.Where results are very regularly reproduced and don't deviate much from the mean value (high precision), the bell-shaped curve is steep and narrow (like the top graph in Figure 15) and this indicates a small standard deviation from the mean value (as suggested by the condensed spread of values on the x-axis). In contrast, when the results are more variable (low precision), the bell-shaped curve is relatively spread and flat like the bottom graph in Figure 15, and this indicates a large standard deviation from the mean value.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.About 68% of all the results occur within one standard deviation of the mean value on the horizontal, x-axis, and this figure is represented by the red areas on both of the graphs in Figure 15). 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 and blue areas). 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.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.