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Introduction

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

This free course, Prices, location and spread, examines aspects of the question:

Are people getting better or worse off?

The course concentrates on the statistical aspects of the question, focusing on statistics about prices. However, it is not the case that statistics can provide all the answers – or even the best answer – to the question of whether people are getting better or worse off. There are many non-statistical issues which are relevant and it is important to put the statistical approach in its correct perspective.

In the question, people does not refer specifically to you, the readers of the course, but to the whole of society in the UK. That is quite a big batch (more than 62 million in 2010, according to an estimate from the UK’s Office for National Statistics), consisting of men, women and children, living alone, in large or small households, or in institutions; some of them working, others unemployed, some retired and others still at school.

It is not possible, using statistical techniques, to provide a complete answer to this one question covering such a big theme, particularly an answer which is valid for all these people and their varied economic and social circumstances; data and techniques both have to be used with common sense. Instead, the aim of this text is more modest: to explore small batches of data relevant to the question (and relating to some individuals and groups in society), using basic analytical and graphical techniques. The sections in this course cover the following:

Section 1 looks at ways to measure the overall ‘location’ (the typical value) of a batch of data. Two very important measures looked at are the ‘median’ and the ‘arithmetic mean’. The section also looks at patterns in data using diagrammatic methods.
Section 2 shows how to calculate the ‘weighted mean’, which is a quantity related to the arithmetic mean. You will learn about some circumstances where it makes sense to calculate a weighted mean.
Section 3 shows how to calculate one particular measure of spread for a batch: the ‘interquartile range’. It also shows some diagrammatic methods for representing the spread and shape of the distribution of values in a batch.
Section 4 introduces the notion of a ‘price index’ for indicating changes in the price of a single item and for two or more different items.
Section 5 looks at the UK’s Retail Prices Index (RPI) and Consumer Prices Index (CPI), which measure changes in prices over time. (This section is longer than all the other sections, so you should plan your study time accordingly.)

This OpenLearn course is an adapted extract from the Open University course M140 Introducing statistics.

Learning outcomes

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

After studying this course, you should be able to:

find the mean and median of a batch of data
find the weighted mean of two or more numbers
find the lower and upper quartiles, interquartile range and five-figure summary of a batch of data
calculate a simple chained price index
use the Retail Prices Index and the Consumer Prices Index to measure price changes.

1 Measuring location

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Measuring location has two components:

gathering data about the quantity of interest
determining a value to represent the location of the data.

The task of gathering appropriate data is somewhat problem-specific – general strategies are available, but exact details usually need to be decided for each problem. To determine the price of an electric kettle, for example, we would have to decide the size and type of kettle we’re interested in, where and when its purchased, and so forth. In contrast, choosing a value to summarise the location of a set of data is more straightforward. In this section, we will focus on the two most common measures of location: the median and the mean. The data gathered about the quantity of interest does not affect the way we calculate these location measures.

1.1 Data on prices

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

In order to measure how prices change, we need data on prices and some way of measuring their overall location. Price data take many forms.

In examining the overall location, prices of all goods are relevant, but some are more important than others. Ballpoint pens are relatively unimportant in most people’s shopping baskets, coffee prices are unimportant for tea drinkers, and chicken prices are of little concern to vegetarians. The first batch of price data we will look at is coffee prices.

Example 1 Price data for jars of coffee

Table 1 shows prices of a 100 g jar of a well-known brand of instant coffee obtained in 15 different shops in Milton Keynes on the same day in February 2012.

Table 1 Coffee prices (in pence)
299	315	268	269	295
295	369	275	268	295
279	268	268	295	305

There are several points to note concerning these prices.

They relate to a particular brand of coffee. You might expect the price to vary between brands.
They relate to a standard 100 g jar. You might expect the price per gram of this brand of coffee to vary depending upon the size of the jar – larger jars are often cheaper (per gram).
They relate to a particular locality. You might expect the price to vary depending upon where you buy the coffee (e.g. central London, a suburb, a provincial town, a country village or a Hebridean island).
They relate to a particular day. You might expect the price to vary from time to time depending upon changes in the cost of raw coffee beans, costs of production and distribution, and the availability of special offers.

Nevertheless, although we have data for a fixed brand of coffee, size of jar, locality and date of purchase, this batch of prices still varies from the lower extreme of 268p to the upper extreme of 369p. (In symbols: and .) One of the most likely reasons for this is that the prices were collected from different kinds of shops (e.g. supermarket, petrol station, ethnic grocery and corner shop).

For all these reasons, it is impossible to state exactly what the price of this brand of instant coffee is. Yet its price is, in its own small way, relevant to the question: Are people getting better or worse off? That is, if you drink this particular coffee, then changes in its price in your locality will affect your cost of living. Similarly, your costs and economic well-being will also be affected by what happens to the prices of all the other things you need or like to consume.

On the other hand, someone who never buys instant coffee will be unaffected by any change in its price; they will be much more interested in what happens to the prices of alternative products such as ground coffee, tea, milk or fruit juice. The problem of measuring the effect of price changes on individuals with different consumption patterns will be considered in Section 5.

1.2 The median

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Despite the variability in the data, Table 1 does provide some idea of the price you would expect to pay for a 100 g jar of that particular instant coffee in the Milton Keynes area on that particular day. The information provided by the batch can be seen more clearly when drawn as a stemplot, shown in Figure 1 of Example 2.

Example 2 Picturing the coffee data

Figure 1 Stemplot of coffee prices from Table 1

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An ordered stemplot of coffee prices from Table 1. The first column contains the numbers in the stem while the second column contains the relevant leaf values. There are 11 levels. The numbers in the stem start at 26, increase in steps of one and end at 36. The leaf values correspond to the relevant numbers in the stem, though sometimes there is more than one leaf and sometimes none at all. In this stemplot they are in numerical order.

At level 26 there are five leaves, 8, 8, 8, 8, 9. Level 27 has two leaves, 5, 9. Level 28 has no leaves while level 29 has five leaves, 5, 5, 5, 5, 9. Level 30 has a single leaf, 5, and level 31 has a single leaf, 5. Levels 32, 33, 34 and 35 have no leaves. Level 36 has a single leaf, 9. Beneath the stemplot is written n = 15, followed by 26 vertical line 8 represents 268 pence. The vertical line sits horizontally between the 26 and the 8.

Stemplot of coffee prices from Table 1

This stemplot shows at a glance that if you shop around, you might well find this brand of coffee on sale at less than 270p. (Indeed some stores seem to have been ‘price matching’ at the lowest price of 268p.) On the other hand, if you are not too careful about making price comparisons then you might pay considerably more than 300p (£3). However, you are most likely to find a shop with the coffee priced between about 270p and 300p. Although there is no one price for this coffee, it seems reasonable to say that the overall location of the price is a bit less than 300p.

The median of the batch is a useful measure of the overall location of the values in a batch. It is defined as the middle value of a batch of figures when the values are placed in order. Let us examine in more detail what that means.

The stemplot in Figure 1 shows the prices arranged in order of size. We can label each of these 15 prices with a symbol indicating where it comes in the ordered batch. A convenient way of showing this is to write each value as the symbol plus a subscript number in brackets, where the subscript number shows the position of that value within the ordered batch. Figure 2 shows the 15 prices written out in ascending order using this subscript notation.

Figure 2 Subscript notation for ordered data

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Subscript notation for ordered data. This shows a table with 2 rows and 15 columns. The first row contains x subscript 1, x subscript 2 and so on, ending at x subscript 15. The subscript numbers are in round brackets and positioned to the right of and slightly below each x. Above this row is a cloud with an arrow pointing to the term x subscript 3. In the cloud is written ‘The subscript is (3), so this is the third value in the ordered batch.’

The second row contains the corresponding values. These are 268 (shown below x subscript 1), 268, 268, 268, 269, 275, 279, 295, 295, 295, 295, 299, 305, 315 and 369 (shown below x subscript 15). Below the table are three clouds with arrows pointing upwards. The first cloud contains ‘capital E subscript capital L’ and the arrow points to the first number in the second row, 268. The middle cloud contains the word ‘Median’ and the arrow points to the first 295 in the second row, which is under x subscript 8. The last cloud contains ‘capital E subscript capital U’ and the arrow points to the last number in the second row, 369.

Subscript notation for ordered data

The lower extreme, , is labelled and the upper extreme, , is labelled . The middle value is the value labelled since there are as many values, namely 7, above the value of as there are below it. (This is not strictly true here, since the values of , and happen also to be actually equal to the median.)

This is illustrated in Figure 3 by a V-shaped formation. The median is the middle value, so it lies at the bottom of the V. (This way of picturing a batch will be developed further in Subsection 3.2.)

Figure 3 Median of 15 values

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Median of 15 values. The letters x subscript 1, x subscript 2 and so on, each with the subscripts in round brackets, are arranged in order, but in a V-shaped formation. At the top left-hand end of the V is x subscript 1. The letters then descend in order down the slope until they reach x subscript 8. Then they begin to rise up a slope to form the right-hand side of the V, starting with x subscript 9, which is level with x subscript 7. They continue upwards to end at x subscript 15, which is level with and to the far right of x subscript 1. Below the point of the V is a cloud containing the word ‘Median’. From the cloud an arrow points to x subscript 8.

Median of 15 values

If you wanted to make a more explicit statement, then you could write: The median price of this batch of 15 prices is 295p.

If we picture any batch of data as a V-shape like Figure 3, the median of the batch will always lie at the bottom of the V. In the ordered batch, it is more places away from the extremes than any other value.

In general, the median is the value of the middle item when all the items of the batch are arranged in order. For a batch size , the position of the middle value is . For example, when , this gives a position of , indicating that is the median value. When is an even number, the middle position is not a whole number and the median is the average of the two numbers either side of it. For example, when , the median position is , indicating that the median value is taken as halfway between and .

Example 3 uses prices of a digital camera to illustrate how the median is found for an even number of values.

Example 3 Digital cameras

Table 2 shows prices for a particular model of digital camera as given on a price comparison website in March 2012.

Table 2 Prices for a digital camera (to the nearest £)
60	70	53	81	74
85	90	79	65	70

If we put these prices in order and arrange them in a V-shape, they look like Figure 4.

Figure 4 Prices of 10 digital cameras

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V-shaped formation for prices of ten digital cameras. Reading down the slope from the top left, the numbers are 53, 60, 65, 70 and 70, but this V shape has no number at its point. The number 74 is to the right of but level with the preceding number 70. The remaining numbers 79, 81, 85, 90 then rise up the slope, with 90 being level with and to the far right of 53.

Prices of 10 digital cameras

Because 10 is an even number, there is no single middle value in this batch: the position of the middle item is . The two values closest to the middle are those shown at the bottom of the V: and . Their average is 72, so we say that the median price of this batch of camera prices is £72.

The following activity asks you to find the median for an even number of values, using a stemplot of prices for small flat-screen televisions.

Activity 1 Small flat-screen televisions

Figure 5 is a stemplot of data on the prices of small flat-screen televisions. (The prices have been rounded to the nearest £10. Originally all but one ended in 9.99, so in this case it makes reasonable sense to ignore the rounding and treat the data as if the prices were exact multiples of £10.) Find the median of these data.

Figure 5 Prices of all flat-screen televisions with a screen size of 24 inches or less on a major UK retailer’s website on a day in February 2012

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Stemplot with 10 levels, though there are repeated numbers in the stem. The numbers in the stem are 0, 1, 1, 1, 1, 1, 2, 2, 2, 2. At level 0 there is one leaf 9. At the first level 1 there is one leaf, 0. At the second level 1 there are four leaves 2, 3, 3, 3. The third level 1 has five leaves, 4, 5, 5, 5, 5. The fourth level 1 has three leaves, 6, 6, 7. The fifth and last level 1 has three leaves, 8, 8, 9. The first and second level 2s have no leaves. The third level 2 has two leaves, 4, 5. The last level 2 has one leaf, 7. Beneath the stemplot is written n = 20, followed by 0 vertical line 9 represents 90 pounds sterling. The vertical line sits horizontally between the 0 and the 9.

Prices of all flat-screen televisions with a screen size of 24 inches or less on a major UK retailer’s website on a day in February 2012...

Discussion

For a batch size of 20, the median position is . So, the median will be halfway between and . These are both 150, so the median is £150.

This subsection can now be finished by using some of the methods we have met to examine a batch of data consisting of two parts, or sub-batches.

Activity 2 The price of gas in UK cities

Table 3 presents the average price of gas, in pence per kilowatt hour (kWh), in 2010, for typical consumers on credit tariffs in 14 cities in the UK. These cities have been divided into two sub-batches: as seven northern cities and seven southern cities. (Legally, at the time of writing, Ipswich is a town, not a city, but we shall ignore that distinction here.)

Table 3 Average gas prices in 14 cities
Northern cities	Average gas price (pence per kWh)	Southern cities	Average gas price (pence per kWh)
Aberdeen	3.740	Birmingham	3.805
Edinburgh	3.740	Canterbury	3.796
Leeds	3.776	Cardiff	3.743
Liverpool	3.801	Ipswich	3.760
Manchester	3.801	London	3.818
Newcastle-upon-Tyne	3.804	Plymouth	3.784
Nottingham	3.767	Southampton	3.795

(a) Draw a stemplot of all 14 prices shown in the table.

Discussion

A stemplot of all 14 prices in the table is shown below.

Figure 6 Stemplot of 14 gas prices

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A stemplot with 8 levels. The numbers in the stem start at 374, increase in steps of one and end at 381. At level 374 there are 3 leaves, 0, 0, 3. Level 375 has no leaves. Level 376 has two leaves, 0, 7 while level 377 has one leaf, 6. Level 378 has one leaf, 4. Level 379 has two leaves, 5, 6. Level 380 has four leaves, 1, 1, 4, 5. Level 381 has one leaf, 8. Beneath the stemplot is written n = 14, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0.

Stemplot of 14 gas prices

(b) Draw separate stemplots for the seven prices for northern cities and the seven prices for southern cities.

Discussion

Stemplots for the prices for northern and southern cities are shown below.

Figure 7 Stemplots for northern and southern cities separately.

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There are two stemplots, one for northern cities and one for southern cities. The one for the northern cities has 7 levels. The numbers in the stem start at 374, increase in steps of one and end at 380.

At level 374 there are 2 leaves, 0, 0. Level 375 has no leaves. Level 376 has one leaf, 7 and level 377 also has one leaf, 6. Levels 378 and level 379 have no leaves. Level 380 has three leaves, 1, 1, 4. Beneath the stemplot is written n = 7, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0. The one for the southern cities has 8 levels. The numbers in the stem start at 374, increase in steps of one and end at 381. At level 374 there is one leaf, 3. Level 375 has no leaves. Level 376 has one leaf, 0 but level 377 has no leaves. Level 378 has one leaf, 4, and level 379 has two leaves, 5, 6. Level 380 has one leaf, 5 and level 381 has one leaf, 8.

Beneath the stemplot is written n = 7, followed by 374 vertical line 3 represents 3.743 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 3.

Stemplots for northern and southern cities separately.

(c) For each of these three batches (northern cities, southern cities and all cities) find the median and the range. Then use these figures to find the general level and the range of gas prices for typical consumers in the country as a whole, and to compare the north and south of the country.

Discussion

For a batch size of 14, the median position is . So, the all-cities median will be halfway between and . These are 3.784 and 3.795, so the median is 3.7895, which is 3.790 when rounded to three decimal places. (The rounded median should be written as 3.790 and not 3.79, to show it is accurate to three decimal places and not just two.)

For the northern and southern batches, both of size 7, the median for each is the value of (that is, ). This is 3.776 for the northern batch and 3.795 for the southern batch.

The range is the difference between the upper extreme, , and the lower extreme, (range ). So the all-cities range is

the range for the northern batch is

and the range for the southern batch is

The medians and ranges are summarised below.

Batch	Median	Range
All cities	3.790	0.078
Northern cities	3.776	0.064
Southern cities	3.795	0.075

Thus the general level of gas prices in the country as a whole was about 3.790p per kWh. The average price differed by only 0.078p per kWh across the 14 cities.

The difference between the median prices for the northern and southern cities is 0.019p per kWh , with the south having the higher median.

The analysis does not clearly reveal whether the general level of gas prices for typical consumers in 2010 was higher in the south or in the north, though there is an indication that prices were a little higher in the south. The range of prices was also rather greater in the south. It is worth noting that the differences in gas prices between the cities in Table 3 were generally small, when measured in pence per kWh – although, with a typical annual gas usage of 18 000 kWh, the price difference between the most expensive city and the cheapest would amount to an annual difference in bills of about £14 on a typical bill of somewhere around £700.

Activity 2 illustrates two general properties of sub-batches:

The range of the complete batch is greater than or equal to the ranges of all the sub-batches.
The median of the complete batch is greater than or equal to the smallest median of a sub-batch and less than or equal to the largest median of a sub-batch.

1.3 The arithmetic mean

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Another important measure of location is the arithmetic mean. (Pronounced arithmetic.)

Arithmetic mean

The arithmetic mean is the sum of all the values in the batch divided by the size of the batch. More briefly,

There are other kinds of mean, such as the geometric mean and the harmonic mean, but in this course we shall be using only the arithmetic mean; the word mean will therefore normally be used for arithmetic mean.

Example 4 An arithmetic mean

Suppose we have a batch consisting of five values: 4, 8, 4, 2, 9. In this simple example, the mean is

Note that in calculating the mean, the order in which the values are summed is irrelevant.

For a larger batch size, you may find it helpful to set out your calculations systematically in a table. However, in practice the raw data are usually fed directly into a computer or calculator. In general, it is a good idea to check your calculations by reworking them. If possible, use a different method in the reworking; for example, you could sum the numbers in the opposite order.

The formula ‘’ can be expressed more concisely as follows. Referring to the values in the batch by , the ‘sum’ can be written as . Here is the Greek (capital) letter Sigma, the Greek version of S, and is used in statistics to denote ‘the sum of’. Also, the symbol is often used to denote the mean – and as you have already seen in stemplots, can be used to denote the batch size. (Some calculators use keys marked and to produce the sum and the mean of a batch directly.)

Using this notation,

can be written as

In this course we shall normally round the mean to one more figure than the original data.

Activity 3 Small televisions: the mean

The prices of 20 small televisions were given in Activity 1 (Subsection 1.2). Find the mean of these prices. Round your answer appropriately (if necessary), given that the original data were rounded to the nearest £10.

Discussion

Using the data for the prices from Activity 1:

Or using the notation, and , so

The prices were rounded to the nearest £10, so it is appropriate to keep one more significant figure for the mean, that is, to show it accurate to the nearest £1. So since the exact value is £162, it needs no further rounding.

1.4 The mean and median compared

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Both the mean and median of a batch are useful indicators of the location of the values in the batch. They are, however, calculated in very different ways. To find the median you must first order the batch of data, and if you are not using a computer, you will often do the sorting by means of a stemplot. On the other hand, the major step in finding the mean consists of summing the values in the batch, and for this they do not need to be ordered.

For large batches, at least when you are not using a computer, it is often much quicker to sum the values in the batch than it is to order them. However, for small batches, like some of those you will be analysing in this course without a computer, it can be just as fast to calculate the median as it is to calculate the mean. Moreover, placing the batch values in order is not done solely to help calculate the median – there are many other uses. Drawing a stemplot to order the values also enables us to examine the general shape of the batch. In Section 3 you will read about some other uses of the stemplot.

Comparisons based on the method of calculation can be of great practical interest, but the rest of this subsection will consider more fundamental differences between the mean and the median – differences which should influence you when you are deciding which measure to use in summarising the general location of the values in a batch.

Many of the problems with the mean, as well as some advantages, lie in the fact that the precise value of every item in the batch enters into its calculation. In calculating the median, most of the data values come into the calculation only in terms of whether they are in the 50% above the median value or the 50% below it. If one of them changes slightly, but without moving into the other half of the batch, the median will not change. In particular, if the extreme values in the batch are made smaller or larger, this will have no effect on the value of the median – the median is resistant to outliers. In contrast, changes to the extremes could have an appreciable effect on the value of the mean, as the following examples show.

Example 5 Changing the extreme coffee prices

For the batch of coffee prices in Figure 1 (Subsection 1.2), the sum of the values is 4363p, so the mean is

Suppose the highest and lowest coffee prices are reduced so that

The median of this altered batch is the same as before, 295p. However, the sum of the values is now 4306p and so the mean is

Example 6 Changing the small television prices

Suppose the highest two television prices in Activity 1 (Subsection 1.2) are altered to £350 and £400. The median, at £150, remains the same as that of the original batch, whereas the new mean is

compared with the original mean of £162.

Now, even with the very high prices of £350 and £400 for two televisions, the overall location of the main body of the data is still much the same as for the original batch of data. For the original batch the mean, £162, was a reasonably good measure of this. However, for the new batch the mean, £174, is much too high to be a representative measure since, as we can see from the stemplot in Activity 1, most of the values are below £174.

Example 6 is the subject of the following screencast. [Note that the reference to ‘Unit 2’ should be ‘this course’. Unit 2 is a reference to the Open University course from which this material is adapted.]

Download this video clip.Video player: Effects on the median and mean when data points change

Screencast 1 Effects on the median and mean when data points change

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Transcript: Screencast 1 Effects on the median and mean when data points change

INSTRUCTOR

In this screencast, I’m going to talk about calculating the mean and the median from the stemplot and showing how the mean and the median change when some of the data changes. So I’m going to start off with a stemplot. And the stemplot I’ve got here is a stemplot of the prices of the small flat screen televisions that are shown in Activity 1 in Subsection 1.2 of Unit 2.

And the first thing I’m going to do is calculate the median. And we note first that we’ve got 20 data points in our batch. And so the median is the average of the 10th and 11th largest values.

So it’s just a question of finding out from the stemplot what the 10th and the 11th largest values are. And we can do that by counting down from the top value in our stemplot. That’s 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

But we could also have counted from the bottom of the stemplot. Again, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. And notice, it’s come at the same two numbers, as it should. So for these data, the median is 150 plus 150 over 2, which is £150, taking into account this key for this stemplot.

Now the mean is just the sum of all the numbers divided by the number of numbers. And going through stemplot, we can write down what all the numbers are. So the lowest number is 90. The next one is 100. Next one is 120. And so on, so forth, until we get to 240 and 250 and 270.

And that’s all divided by the number of numbers, which is 20. And the sum on top of the fraction happens to be 3240. That’s divided by 20, and that comes to £162. So the mean of the television prices is £162, and the median is £150. So the mean is £12 bigger than the median.

Now, what happens if some of the data changes? For example, what happens if a couple of the prices for the televisions goes up? And in particular, what happens if the prices of the most expensive televisions go up? So instead of having a television that costs £250 and a television that costs £270, we actually had a couple of televisions that cost £350 and £400? What difference does this make to the median and the mean?

Well, notice that the two values we work the median out from haven’t changed. So we can just write that down immediately. The median is £150, just as it was before.

So what about the mean? Well again, we’ve got to work out the sum of all the data points and divide by the number of numbers. For most of the numbers, they haven’t changed. So most the numbers in the sum don’t change. But the last two have – so instead of 250, we’ve now got 350, and instead of 270, we’ve got 400. Again, that’s divided by 20. That’s equal to 3470 over 20, which equals 173.5, or we can say that’s £174 to the nearest pound.

So what we notice here that when the data is changed, the median has stayed the same. We say that the median is a resistant measure. It’s been resistant to a change in the data. On the other hand, the mean is bigger. We say that the mean is a sensitive measure. It has been sensitive to changes in the data.

End transcript: Screencast 1 Effects on the median and mean when data points change

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Screencast 1 Effects on the median and mean when data points change

Interactive feature not available in single page view (see it in standard view).

A measure which is insensitive to changes in the values near the extremes is called a resistant measure.

The median is a resistant measure whereas the mean is sensitive.

In the following activities, you can investigate some other ways in which the median is more resistant than the mean.

Activity 4 Changing the gas prices

In Activity 2 (Subsection 1.2) you may have noticed that Cardiff and Ipswich had rather low gas prices compared to the other southern cities. Here you are going to examine the effect of deleting them from the batch of southern cities. Complete the following table and comment on your results.

Batch	Mean	Median
Seven southern cities
Five southern cities (excluding Cardiff and Ipswich)

Discussion

The completed table is:

Batch	Mean	Median
Seven southern cities	3.7859	3.795
Five southern cities (excluding Cardiff and Ipswich)	3.7996	3.796

Whereas deletion of Cardiff and Ipswich has the effect of increasing the mean price by 0.0137p per kWh, the median price increases by only 0.001p per kWh. This is what we would expect as, in general, the more resistant a measure is, the less it changes when a few extreme values are deleted.

Activity 5 A misprint in the gas prices

Suppose the value for London had been misprinted as 8.318 instead of 3.818 (quite an easy mistake to make!). How would this affect your results for the batch of five southern cities (again omitting Cardiff and Ipswich)?

Batch	Mean	Median
Five cities (correct data)
Five cities (with misprint)

Discussion

The completed table is:

Batch	Mean	Median
Five cities (correct data)	3.7996	3.796
Five cities (with misprint)	4.6996	3.796

Here the median is completely unaffected by the misprint, although the mean changes considerably.

Suppose you wanted to use these values – the correct ones, of course – to estimate the average price of gas over the whole country. The simple arithmetic mean of the 14 values given in Table 3 (Subsection 1.2) would not allow for the fact that much more gas is consumed in London, at a relatively high price, than in other cities. To take account of this you would need to calculate what is known as a weighted arithmetic mean. Weighted means are the subject of Section 2.

Exercises on Section 1

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

The following exercises provide extra practice on the topics covered in Section 1.

Exercise 1 Finding medians

For each of the following batches of data, find the median of the batch. (We shall also use these batches of data in some of the exercises in Section 3.)

(a) Percentage scores in arithmetic obtained by 33 school students.

Figure 8

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A stemplot with 11 levels. The numbers in the stem start at 0, increase in steps of one and end at 10. At level 0 there is one leaf, 7. Level 1 has one leaf, 5. Level 2 has no leaves, while level 3 has two leaves, 3, 5. Level 4 has three leaves, 2, 2, 3 and level 5 has two leaves, 5, 8. Level 6 has three leaves, 4, 6, 8. Level 7 has five leaves, 1, 1, 6, 8, 9. Level 8 has nine leaves, 0, 1, 1, 3, 4, 5, 5, 6, 9. Level 9 has five leaves, 1, 1, 3, 5, 9. Level 10 has two leaves which are both 0. Beneath the stemplot is written n = 33, followed by 0 vertical line 7 represents a score of 7 per cent. The vertical lines sits horizontally between the 0 and the 7.

Figure 8

Discussion

For the arithmetic scores, the position of the median is , so the median is 79%.

(b) Prices of 26 digital televisions with 22- to 26-inch LED screens, quoted online by a large department store in February 2012. The prices have been rounded to the nearest pound (£).

170	180	190	200	220	229	230	230	230
230	250	269	269	270	279	299	300	300
315	320	349	350	400	429	649	699

Discussion

For the television prices, the position of the median is , so the median is halfway between and . Thus, the median is

Exercise 2 Finding means

Calculate the mean for each of the batches in Exercise 1.

Discussion

For the batch of arithmetic scores in part (a) of Exercise 1, the sum of the 33 values is 2326 and

Therefore, the mean is 70.5%. (The original data are given to the nearest whole number, so the mean is rounded to one decimal place.)

For the batch of television prices in part (b) of Exercise 1, the sum of the 26 values is 7856 and

Therefore, the mean is £302.2.

Exercise 3 The effect of removing values on the median and mean

In the data on prices for small televisions in Activity 1 (Subsection 1.2), the three highest-priced televisions were considerably more expensive than all the others (which all cost under £200). Suppose that in fact these prices had been for a different, larger type of television that should not have been in the batch. (In fact that is not the case – but this is only an exercise!) Leave these three prices out of the batch and calculate the median and the mean of the remaining prices.

How do these values compare with the original median (150) and mean (162)? What does this comparison demonstrate about how resistant the median and mean are?

Discussion

For the median, there are now 17 prices left in the batch, so the median is at position . It is therefore 150.

The sum of the remaining 17 values is 2480, so the mean is

In this case, removing the three highest prices has not changed the median at all, but it has reduced the mean considerably. This illustrates that the median is a more resistant measure than the mean.

2 Weighted means

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

For goods and services, price changes vary considerably from one to another. Central to the theme question of this course, Are people getting better or worse off?, there is a need to find a fair method of calculating the average price change over a wide range of goods and services. Clearly a 10% rise in the price of bread is of greater significance to most people than a similar rise in the price of clothes pegs, say. What we need to take account of, then, are the relative weightings attached to the various price changes under consideration.

2.1 The mean of a combined batch

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

This first subsection looks at how a mean can be calculated when two unequally weighted batches are combined.

Example 7 Alan’s and Beena’s biscuits

Suppose we are conducting a survey to investigate the general level of prices in some locality. Two colleagues, Alan and Beena, have each visited several shops and collected information on the price of a standard packet of a particular brand of biscuits. They report as follows (Figure 9).

Alan visited five shops, and calculated that the mean price of the standard packet at these shops was 81.6p.
Beena visited eight shops, and calculated that the mean price of the standard packet at these shops was 74.0p.

Figure 9 Means of biscuit prices

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The figure shows a horizontal arrow pointing to the right and labelled pence. Part way in from the left is a marker above which is written 74.0. Farther to the right, but before the end of the arrow, is another marker, above which is written 81.6.

Means of biscuit prices

If we had all the individual prices, five from Alan and eight from Beena, then they could be amalgamated into a single batch of 13 prices, and from this combined batch we could calculate the mean price of the standard packet at all 13 shops. However, our two investigators have unfortunately not written down, nor can they fully remember, the prices from individual shops. Is there anything we can do to calculate the mean of the combined batch?

Fortunately there is, as long as we are interested in arithmetic means. (If they had recorded the medians instead, then there would have been very little we could do.)

The mean of the combined batch of all 13 prices will be calculated as

We already know that the size of the combined batch is the sum of the sizes of the two original batches; that is, . The problem here is how to find the sum of the combined batch of Alan’s and Beena’s prices. The solution is to rearrange the familiar formula

so that it reads

This will allow us to find the sums of Alan’s five prices and Beena’s eight prices separately. Adding the results will produce the sum of the combined batch prices. Finally, dividing by 13 completes the calculation of finding the combined batch mean.

Let us call the sum of Alan’s prices ‘sum(A)’ and the sum of Beena’s prices ‘sum(B)’.

For Alan: and , so .

For Beena: and , so .

For the combined batch:

Here, the result has been rounded to give the same number of digits as in the two original means.

The process that we have used above is an important one. It will be used several times in the rest of this course. The box below summarises the method, using symbols.

Mean of a combined batch

The formula for the mean of a combined batch is

where batch consists of batch combined with batch , and

For our survey in Example 7,

The formula summarises the calculations we did as

This expression is an example of a weighted mean. The numbers 5 and 8 are the weights. We call this expression the weighted mean of 81.6 and 74.0 with weights 5 and 8, respectively.

To see why the term weighted mean is used for such an expression, imagine that Figure 10 shows a horizontal bar with two weights, of sizes 5 and 8, hanging on it at the points 81.6 and 74.0, and that you need to find the point at which the bar will balance. This point is at the weighted mean: approximately 76.9.

Figure 10 Point of balance at the weighted mean

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Point of balance at the weighted mean. The figure shows a horizontal arrow pointing to the right and labelled pence. Three appropriately spaced points are marked on the line: 74.0, 76.9 and 81.6. The distance between the 74.0 and 76.9 is less than the distance between 76.9 and 81.6. Immediately below 76.9 there is a solid arrow head pointing upwards, which is the fulcrum or balancing point of the balance. Suspended from the point marked 74.0 is a pile of 8 discs and suspended from the point marked 81.6 is a pile of 5 discs.

Point of balance at the weighted mean

This physical analogy illustrates several important facts about weighted means.

It does not matter whether the weights are 5 kg and 8 kg or 5 tonnes and 8 tonnes; the point of balance will be in the same place. It will also remain in the same place if we use weights of 10 kg and 16 kg or 40 kg and 64 kg – it is only the relative sizes (i.e. the ratio) of the weights that matter.
The point of balance must be between the points where we hang the weights, and it is nearer to the point with the larger weight.
If the weights are equal, then the point of balance is halfway between the points.

This gives the following rules.

Rules for weighted means

Rule 1 The weighted mean depends on the relative sizes (i.e. the ratio) of the weights.

Rule 2 The weighted mean of two numbers always lies between the numbers and it is nearer the number that has the larger weight.

Rule 3 If the weights are equal, then the weighted mean of two numbers is the number halfway between them.

Example 8 Two batches of small televisions

Suppose that we have two batches of prices (in pounds) for small televisions:

To find the mean of the combined batch we use the formula above, with

This gives

Note that this is the weighted mean of 119 and 185 with weights 7 and 13 respectively. It lies between 119 and 185 but it is nearer to 185 because this has the greater weight: 13 compared with 7.

Example 8 is the subject of the following screencast. [Note that references to ‘the unit’ and ‘the units’ should be interpreted as ‘this course’. The original wording refers to the Open University course from which this material is adapted.]

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Screencast 2 Calculating a weighted mean

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Transcript: Screencast 2 Calculating a weighted mean

INSTRUCTOR

OK, here’s an example that you’ve seen in the units about the price of two batches of small TVs. The first batch, that’s Batch A, that’s got a mean of 119, that’s £119 actually, and size 7, there are 7 TVs in that batch. Batch B, the second batch, has got a mean of £185, good bit more expensive, and the size of that batch is 13. And what we’re asked to do here is to find the mean of the combined batch. OK, so how are we going to do that?

Well, there’s a formula in the unit which I’ll write down in a minute. But I want to build it up a bit at a time to show you where the formula comes from, and the formula looks like this. We’ve got x bar, that means mean. And c, that’s because it’s the mean of the combined batch.

Now, we usually calculate a mean by calculating the total of the values in the batch, and dividing it by the size of the batch. And essentially, that’s all that’s happening here – we just have to do it in a bit of a complicated way. So how does that work?

Well, first of all we need the sum of the values in Batch A. Now, we’ve got the mean, we haven’t got the sum. But that mean, £119, is the sum divided by the size, which is 7. So you can get back to the sum by multiplying the mean by the size. So that is it’s the mean of the values in Batch A, x bar A, times the size of Batch A.

Anyway, that deals with the TV prices in Batch A, and now we’ve got to add on the TV prices in Batch B. So you do the same trick. You add on x bar B, the mean of the values in Batch B, times the size, nB, of Batch B, and that gives you the sum part of this mean calculation.

Then you’ve got to divide that by the size, so there’s the division sign. And the size of the combined batches is the total number of TVs in this combined batch. So it’s the number you’ve got in Batch A, plus the number you’ve got in Batch B – nA plus nB. So that’s the formula, and that’s the same one as in the unit.

The next step is to convert the numbers we have to make clear how the notation we got in this formula fits with the numbers we’ve got. So let’s just translate the mean and the size of the batches into the notation we need for the formula. So for Batch A, the mean’s 119 – that is, in this notation we got x bar A is 119, and nA the size of Batch A is 7 because there’s 7 TVs. And then we do the same with Batch B, x bar B is 185 to match what it says over the left there, and nB is 13.

So we’ve got the formula here, and we’ve got the values we need to put in the formula. So now we’ve just got to go ahead and put the values in and do the arithmetic. So the first step, let’s work out what we’ve got to put on the top. Well, on the top of this expression we’ve got x bar A times nA, and that is 119, that’s x bar A, times nA, which is 7, and I’ve put the multiplication sign in there to make it clear what’s going on.

And then the other aspect of this is we’re going to add something on in a minute, but we need to do the multiplication first. That is, we need to do this multiplication before we get to this plus. So to make that absolutely clear, we put this in some brackets to indicate that’s what we’re going to do first.

So we do that, and then we add on what we get from the B batch, that is x bar B times nB. And again, we’re going to put that in brackets, so that’s 185 times 13. Close the brackets, and then divide the whole thing by nA plus nB, which is 7 plus 13. So that’s put in the values we got into the formula we had. And now it really is just a case of bashing through the arithmetic.

So I’ve got my trusty calculator here – 119 times 7, that comes to 833. And then we’ve got to add on what we get from this bit here – 185 times 13, and that comes to 2,405. So that’s the top of the thing we’ve got to calculate. We’ve got to divide by the bottom, 7 plus 13, and doing that all in my head it’s 20 and we’re nearly there.

What we’ve got now is we added 833 to 2405, that comes to 3238, and we’ve got to divide that by 20. And calculating that, that comes to 161.9. Well, we’re nearly there now. It looks as if it’s calculated the mean of the combined batch but there’s a final step we got to do, and that’s to relate it back to what’s really going on here.

If you look at these means up here, or they were here originally, they’re given in whole pounds. That is, these means are rounded to the whole pound. Well, it might have been coincidence that they actually come to whole numbers, but what’s really happened is that it’s not appropriate to express the mean to the full accuracy you get from your calculator, because that’s just too much accuracy. The data don’t support it.

So what we got to do is we got to round the means to an appropriate value. And in this case, the original means were rounded to the nearest pound, so our combined mean also needs to be rounded to the nearest pound. So let’s have look at what it is – it’s 161.9. And to round that to the nearest pound, is it going to go up or down?

Well, look at the last figure here, the one we’re going to take away in the rounding – it’s 9, that’s bigger than 5. So to round it to the nearest pound, we have to round it up to 162. That is, we finish off by saying this is £162 rounded to the nearest pound, and that really is the finish of the calculation.

But before we leave this, I just want to point out one thing to you. Look at this value here of 162 and compare it with the means of the two batches we had – Batch A: 119, Batch B: 185. The 162 we’ve got here is a good bit nearer the mean of Batch B than it is to the mean of Batch A. And that’s because Batch B is bigger. It’s got more TVs in it – it’s got 13, and this has only got 7.

So what’s happened is that the Batch B has dominated in the calculation because there’s just more TVs in Batch B. There are 13 of them compared to 7. So Batch B has had more influence on this number than Batch A has.

Now in the unit, it explains that this formula here that we had for a mean of a combined batch is a kind of weighted mean. And the unit also describes that there are various rules that weighted means follow, various properties they have. And one of the properties they have is that the value of the weighted mean of two numbers is closest to the value that had the largest weight.

Now what’s playing the role of weights here is the batch sizes, nA and nB. nB is bigger than nA, so this rule tells us that the combined value, this weighted mean, is going to be nearer to this, the mean of Batch B, than it is to this. And indeed that’s what we find – 162 is nearer to 185 than it is to 119.

So I’ll just record the fact that we found that. This is closer to the mean x bar B, the mean of Batch B, as Batch B is bigger. And that’s that.

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2.2 Further uses of weighted means

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

We shall now look at another similar problem about mean prices – one which is perhaps closer to your everyday experience.

Example 9 Buying petrol

Suppose that, in a particular week in 2012, a motorist purchased petrol on two occasions. On the first she went to her usual, relatively low-priced filling station where the price of unleaded petrol was 136.9p per litre and she filled the tank; the quantity she purchased was 41.2 litres. The second occasion saw her obliged to purchase petrol at an expensive service station where the price of unleaded petrol was 148.0p per litre; she therefore purchased only 10 litres. What was the mean price, in pence per litre, of the petrol she purchased during that week?

To calculate this mean price we need to work out the total expenditure on petrol, in pence, and divide it by the total quantity of petrol purchased, in litres.

The total quantity purchased is straightforward as it is just the sum of the two quantities, so .

To find the expenditure on each occasion, we need to apply the formula:

This gives and , respectively.

So the total expenditure, in pence, is . The mean price, in pence per litre, for which we were asked, is this total expenditure divided by the total number of litres bought:

We have left the answer in this form, rather than working out the individual products and sums as we went along, to show that it has the same form as the calculation of the combined batch mean. (The answer is 139.07p per litre, rounded from 139.067 97p per litre.)

The phrase ‘goods and services’ is an awkward way of referring to the things that are relevant to the cost of living; that is, physical things you might buy, such as bread or gas, and services that you might pay someone else to do for you, such as window-cleaning. Economists sometimes use the word commodity to cover both goods and services that people pay for, and we shall use that word from time to time in this course. (Note that there are other, different, technical meanings of commodity that you might meet in different contexts.)

The mean price of a quantity bought on two different occasions

In general, if you purchase units of some commodity at pence per unit and units of the same commodity at pence per unit, then the mean price of this commodity, pence per unit, can be calculated from the following formula:

Example 10 Buying potatoes

Suppose that, in one month, a family purchased potatoes on two occasions. On one occasion they bought 10 kg at 40p per kg, and on another they bought 6 kg at 45p per kg. We can use this formula to calculate the mean price (in pence per kg) that they paid for potatoes in that month. We have

and

This gives

So the mean price for that month is 41.9p per kg.

The two formulas we have been using,

are basically the same; they are both examples of weighted means.

The first formula is the weighted mean of the numbers and , using the batch sizes, and , as weights.

The second formula is the weighted mean of the unit prices and , using the quantities bought, and , as weights.

The general form of a weighted mean of two numbers having associated weights is as follows.

Weighted mean of two numbers

The weighted mean of the two numbers and with corresponding weights and is

Weighted means have many uses, two of which you have already met. The type of weights depends on the particular use. In our uses, the weights were the following.

The sizes of the batches, when we were calculating the combined batch mean from two batch means.
The quantities bought, when we were calculating the mean price of a commodity bought on two separate occasions.

Another very important use is in the construction of an index, such as the Retail Prices Index; we shall therefore be making much use of weighted means in the final sections of this course.

In the next example, we do not have all the information required to calculate the mean, but we can still get a reasonable answer by using weights.

Example 11 Weighted means of two gas prices

Let us return to the gas prices in Table 3 (Subsection 1.2). This has information about the price of gas for typical consumers in individual cities, but no national figure. Suppose that you want to combine these figures to get an average figure for the whole country; how could you do it? At the end of Section 1, it was suggested that weighted means could provide a solution. The complete answer to this question, using weighted means, is in Example 13 towards the end of this section. To introduce the method used there, let us now consider a similar, but simpler, question.

Here we use just two cities, London and Edinburgh, where the prices were 3.818p per kWh and 3.740p per kWh respectively. How can we combine these two values into one sensible average figure?

One possibility would be to take the simple mean of the two numbers. This gives

However, this gives both cities equal weight. Because London is a lot larger than Edinburgh, we should expect the average to be nearer the London price than the Edinburgh price.

This suggests that we use a weighted mean of the form

where and are suitably chosen weights, with the weight of the London price larger than the weight of the Edinburgh price.

The best weights would be the total quantities of gas consumed in 2010 in each city. However, even if this information is not available to us, we can still find a reasonable average figure by using as weights a readily available measure of the sizes of the two cities: their populations.

The populations of the urban areas of these cities are approximately 8 300 000 and 400 000 respectively. So we could put and .

However, we know that the weighted mean depends only on the ratio of the weights. Therefore, the weights 83 and will give the same answer.

These weights give

Activity 6 Using the rules for weighted means

Using the rules for weighted means, would you expect the weighted mean price to be nearer the London price or the Edinburgh price? To check, calculate the weighted mean price.

Discussion

You should expect the weighted mean price to be nearer the London price, because of Rule 2 for weighted means (Subsection 2.1) and given that London has a much larger weight then Edinburgh.

The weighted mean price given by the formula in Example 11 is (after rounding) 3.814p per kWh, which is indeed much closer to the London price than to the Edinburgh price.

Although we cannot think of the weighted mean price in Activity 6 as a calculation of the total cost divided by the total consumption, the answer is an estimate of the average price, in pence per kWh, for typical consumers in the two cities, and it is the best estimate we can calculate with the available information.

Sometimes the weights in a weighted mean do not have any significance in themselves: they are neither quantities, nor sizes, etc., but simply weights. This is illustrated in the following activity.

Activity 7 Weighted means of Open University marks

Open University students become familiar with the combination of interactive computer-marked assignment (iCMA) and tutor-marked assignment (TMA) scores to provide an overall continuous assessment score (OCAS) for a course.

Suppose that a student obtains a score of 80 for their iCMAs and a score of 60 for their TMAs. Calculate what this student’s overall continuous assessment score will be if the weights for the two components are as follows.

(a) iCMA 50, TMA 50

Discussion

This is the same as a simple (unweighted) mean of the two scores, because the two component scores have equal weight. It lies exactly halfway between the two scores ().

(b) iCMA 40, TMA 60

Discussion

This is slightly less than the simple mean in (a) because the component with the lower score (TMA) has the greater weight.

Discussion

This is slightly higher than the simple mean in (a) because the component with the higher score (iCMA) has the greater weight.

(Note that the weights need not necessarily sum to 100, even when dealing with percentages.)

(d) iCMA 25, TMA 75

Discussion

This is even lower than (b), so even nearer the lower score (TMA), because the TMA score has even greater weight.

(e) iCMA 30, TMA 90

Discussion

This is the same as (d) because the ratios of the weights are the same; they are both in the ratio 1 to 3. That is, ().

(We say this as follows: ‘the ratio 25 to 75 equals the ratio 30 to 90’.)

We have seen, in Activity 7 and in Example 11, that only the ratio of the weights affects the answer, not the individual weights. So weights are often chosen to add up to a convenient number like 100 or 1000. (This is Rule 1 for weighted means (see Subsection 2.1).)

Activity 7 should also have reminded you of another important property of a weighted mean of two numbers: the weighted mean lies nearer to the number having the larger weight. (This is part of Rule 2 for weighted means.)

2.3 More than two numbers

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

The idea of a weighted mean can be extended to more than two numbers. To see how the calculation is done in general, remind yourself first how we calculated the weighted mean of two numbers and with corresponding weights and .

Multiply each number by its weight to get the products and .
Sum these products to get .
Sum the weights to get .
Divide the sum of the products by the sum of the weights.

This leads to the following formula.

Weighted mean of two or more numbers

The weighted mean of two or more numbers is

This is the formula which is used to find the weighted mean of any set of numbers, each with a corresponding weight.

Example 12 A weighted mean of wine prices

Suppose we have the following three batches of wine prices (in pence per bottle).

We want to calculate the weighted mean of these three batch means using, as corresponding weights, the three batch sizes. Rather than applying the formula directly, the calculations can be set out in columns.

Table 4 Data on wine purchases
Batch	Number (batch mean)	Weight (batch size)	Number weight ( = product)
Batch 1	525.5	6	3 153.0
Batch 2	468.0	2	936.0
Batch 3	504.2	12	6 050.4
Sum		20	10 139.4

The weighted mean is

We round this to the same accuracy as the original means, to get a weighted mean of 507.0. (Note that this lies between 468.0 and 525.5. This is a useful check, as a weighted mean always lies within the range of the original means.)

The physical analogy in Example 12 can be extended to any set of numbers and weights. Suppose that you calculate the weighted mean for:

This is given by

This is pictured in Figure 11, with the point of balance for these three weights shown at 1.6.

Figure 11 Point of balance for three means

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Point of balance for three means. The figure shows a horizontal arrow pointing to the right. Four suitably spaced points are marked on it, 1.3, 1.6, 1.7 and 1.9. Immediately below 1.6 there is a solid arrow head pointing upwards, which is the fulcrum or balancing point of the balance. Suspended from the point marked 1.3 is a pile of 2 discs, suspended from the point marked 1.7 is a pile of 3 discs and suspended from the point marked 1.9 is a single disc.

Point of balance for three means

You will meet many examples of weighted means of larger sets of numbers in Subsection 5.2, but we shall end this section with one more example.

Example 13 Weighted means of many gas prices

Example 11 showed the calculation of a weighted mean of gas prices using, for simplicity, just the two cities London and Edinburgh. We can extend Example 11 to calculate a weighted mean of all 14 gas prices from Table 3, using as weights the populations of the 14 cities. The calculations are set out in Table 5.

Table 5 Product of gas price and weight by city
City	Price (p/kWh):	Weight:	Price weight:
Aberdeen	3.740	19	71.060
Edinburgh	3.740	42	157.080
Leeds	3.776	150	566.400
Liverpool	3.801	82	311.682
Manchester	3.801	224	851.424
Newcastle-upon-Tyne	3.804	88	334.752
Nottingham	3.767	67	252.389
Birmingham	3.805	228	867.540
Canterbury	3.796	5	18.980
Cardiff	3.743	33	123.519
Ipswich	3.760	14	52.640
London	3.818	828	3161.304
Plymouth	3.784	24	90.816
Southampton	3.795	30	113.850
Sum		1834	6973.436

The entries in the weight column, , are the approximate populations, in 10 000s, of the urban areas that include each city (as measured in the 2001 Census). For each city, we multiply the price, , by the weight, , to get the entry in the last column, .

The weighted mean of the gas prices using these weights is then

or, in symbols,

As and , the weighted mean is

So the weighted mean of these gas prices, using approximate population figures as weights, is 3.802p per kWh.

Note that this weighted mean is larger than all but three of the gas prices for individual cities. That is because the cities with the two highest populations, London and Birmingham, also have the highest gas prices, and the weighted mean gas price is pulled towards these high prices.

Although the details of the calculation above are written out in full in Table 5, in practice, using even a simple calculator, this is not necessary. It is usually possible to keep a running sum of both the weights and the products as the data are being entered. One way of doing this is to accumulate the sum of the weights into the calculator’s memory while the sum of the products is cumulated on the display. If you are using a specialist statistics calculator, the task is generally very straightforward. Simply enter each price and its corresponding weight using the method described in your calculator instructions for finding a weighted mean.

Activity 8 Weighted means on your calculator

Use your calculator to check that the sum of weights and sum of products of the data in Table 5 are, respectively, 1834 and 6973.436, and that the weighted mean is 3.802. (No solution is given to this activity.)

Activity 9 Weighted mean electricity price

Table 6 is similar to Table 5, but this time it presents the average price of electricity, in pence per kilowatt hour (kWh). These data are again for the year 2010 for typical consumers on credit tariffs in the same 14 cities we have been considering for gas prices, with the addition of Belfast. Again, the weights are the approximate populations of the relevant urban areas, in 10 000s.

Table 6 Populations and electricity prices in 15 cities
City	Price (p/kWh):	Weight:
Aberdeen	13.76	19
Belfast	15.03	58
Edinburgh	13.86	42
Leeds	12.70	150
Liverpool	13.89	82
Manchester	12.65	224
Newcastle-upon-Tyne	12.97	88
Nottingham	12.64	67
Birmingham	12.89	228
Canterbury	12.92	5
Cardiff	13.83	33
Ipswich	12.84	14
London	13.17	828
Plymouth	13.61	24
Southampton	13.41	30
Sum

Use these data to calculate the weighted mean electricity price. (Your calculator will almost certainly allow you to do this without writing out all the values in the column.)

Discussion

The table showing the required sums (and the values in the column, that you may not have had to write down), is as follows.

City	Price (p/kWh):	Weight:	Price weight:
Aberdeen	13.76	19	261.44
Belfast	15.03	58	871.74
Edinburgh	13.86	42	582.12
Leeds	12.70	150	1 905.00
Liverpool	13.89	82	1 138.98
Manchester	12.65	224	2 833.60
Newcastle-upon-Tyne	12.97	88	1 141.36
Nottingham	12.64	67	846.88
Birmingham	12.89	228	2 938.92
Canterbury	12.92	5	64.60
Cardiff	13.83	33	456.39
Ipswich	12.84	14	179.76
London	13.17	828	10 904.76
Plymouth	13.61	24	326.64
Southampton	13.41	30	402.30
Sum		1892	24 854.49

Thus , and

So the weighted mean of electricity prices is 13.14p per kWh.

Exercises on Section 2

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

The following exercises provide extra practice on the topics covered in Section 2.

Exercise 4 A combined batch of camera prices

Find the mean price of the batch formed by combining the following two batches, and , of camera prices.

Discussion

Mean price of all the cameras is

which is £79.3 (rounded to the same accuracy as the original means).

Exercise 5 The mean price of fabric

Suppose you buy 8.5 metres of fabric in a sale, at £10.95 per metre, to make some bedroom curtains. The following year you decide to make a matching bedspread and so you buy 6 metres of the same material, but the price is now £12.70 per metre. Calculate the mean price of all the material, in £ per metre.

Discussion

Mean price of all the material is

which is £11.67 (rounded to the nearest penny).

3 Measuring spread

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

As you have already seen, it is difficult to measure price changes when they so often vary from shop to shop and region to region. Taking some average value, such as the median or the mean, helps to simplify the problem. However, it would be a mistake to ignore the notion of spread, as averages on their own can be misleading.

Information about spread can be very important in statistical analysis, where you are often interested in comparing two or more batches. In this section we shall look first at measures of spread, and then at some methods of summarising the shape of a batch of data.

But how can spread be measured? Just as there are several ways of measuring location (mean, median, etc.), there are also several ways of measuring spread. Here, we shall examine two such measures: the range and the interquartile range. (A further, even more important, measure of spread is the standard deviation. It is, however, beyond the scope of this course.)

3.1 The range

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

The range is defined below.

The range

The range is the distance between the lower and the upper extremes. It can be calculated from the formula:

where is the upper extreme and is the lower extreme.

Given an ordered batch of data, for example in a stemplot, the range can easily be calculated. However, the range tells us very little about how the values in the main body of the data are spread. It is also very sensitive to changes in the extreme values, like those considered in Subsection 1.4. It would be better to have a measure of spread that conveys more information about the spread of values in the main body of the data. One such measure is based upon the difference between two particular values in the batch, known as the quartiles. As the name suggests, the two quartiles lie one quarter of the way into the batch from either end. The major part of the next subsection describes how to find them.

3.2 Quartiles and the interquartile range

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Finding the quartiles of a batch is very similar to finding the median.

In Subsection 1.2, we represented a batch as a V-shaped formation, with the median at the ‘hinge’ where the two arms of the V meet. The median splits the batch into two equal parts. Similarly, we can put another hinge in each side of the V and get four roughly equal parts, shaped like this: . For a batch of size 15, it looks like Figure 12.

Figure 12 Median and quartiles

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An extended V-shaped diagram showing the median and quartiles. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 15 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2. Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, so the next entry x subscript 5 is slightly lower down and to the right and level with x subscript 3. The same line is followed by x subscript 6, x subscript 7, which is level with x subscript 1, and x subscript 8, which is lower than x subscript 1. At x subscript 8 the direction changes again and the letters move slightly to the right and begin to rise, so x subscript 9 is level with x subscript 7. They continue to rise until x subscript 12 is reached. This is level with x subscript 4. The letters then begin to fall again, moving to the right each time, ending with x subscript 15. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the label x subscript 4, which is at the top of the first line. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the label x subscript 12. The third cloud contains the word ‘Median’. From it an arrow points to the lowest label, x subscript 8, in the middle of the diagram.

Median and quartiles

The points at the side hinges, in this case and , are the quartiles. There are two quartiles which, as with the extremes, we call the lower quartile and the upper quartile. The lower quartile separates off the bottom quarter, or lowest 25%. The upper quartile separates off the top quarter, or highest 25%. They are denoted and respectively. (Sometimes they are referred to as the first quartile and the third quartile.)

You might be wondering, if these are and , what happened to ? Well, have a think about that for a moment.

separates the bottom quarter of the data (from the top three quarters), and separates the bottom three quarters (from the top quarter). So it would make sense to say that separates the bottom two quarters (from the top two quarters). But two quarters make a half, so would denote the median, and since there is already a separate word for that, it’s not usual to call it the second quartile.

Usually we cannot divide the batch exactly into quarters. Indeed, this is illustrated in Figure 12 where the two central parts of the are larger than the outer ones. As with calculating the median for an even-sized batch, some rule is needed to tell us how many places we need to count along from the smallest value to find the quartiles. However, there are several alternatives that we could adopt and the particular rule described below is somewhat arbitrary. Different authors and different software may use slightly different rules. If your calculator can find quartiles, note that it may use a different rule.

As you might have expected, the rule involves dividing by 4, where is the batch size (as opposed to dividing by 2 to find the median). However, the rule is slightly more complicated for the quartiles and it depends on whether is exactly divisible by 4.

The quartiles

The lower quartile is at position in the ordered batch.

The upper quartile is at position in the ordered batch.

If is exactly divisible by 4, these positions correspond to a single value in the batch.

If is not exactly divisible by 4, then the positions are to be interpreted as follows.

A position which is a whole number followed by means ‘halfway between the two positions either side’ (as was the case for finding the median).
A position which is a whole number followed by means ‘one quarter of the way from the position below to the position above’. So for instance if a position is , the quartile is the number one quarter of the way from to .
A position which is a whole number followed by means ‘three quarters of the way from the position below to the position above’. So for instance if a position is , the quartile is the number three quarters of the way from to .

Before we actually use these rules to find quartiles, let us look at some more examples of -shaped diagrams for different batch sizes . The case where is exactly divisible by 4, so that is a whole number, was shown in Figure 12. The following three figures show the three other possible scenarios, where is not exactly divisible by 4.

For , and . So is halfway between and , and is halfway between and .

Figure 13 Quartiles for sample size

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Diagram showing the quartiles for sample size n = 17. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 17 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 5 is level with but to the right of x subscript 4. The same line is followed by x subscript 6, x subscript 7 and x subscript 8, which is level with x subscript 1. Continuing in the same direction leads to x subscript 9, which is at a lower level than x subscript 1. At x subscript 9 the direction changes again and the letters begin to rise, moving slightly to the right again until x subscript 13 is reached. This is level with x subscript 5. X subscript 14 is to the right of and level with x subscript 13 and forms the start of the last slope. The letters then begin to fall again, moving to the right each time, ending with x subscript 17. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 4 and x subscript 5. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 13 and x subscript 14. The third cloud contains the word ‘Median’. From it an arrow points to the lowest label, x subscript 9, in the middle of the diagram.

Quartiles for sample size MathJax failure: MathML - Unexpected text node: '

For , and . So is three quarters of the way from to , and is one quarter of the way from to .

Figure 14 Quartiles for sample size

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Diagram showing the quartiles for sample size n = 18. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 18 are shown.

The shape starts on the left with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3 and x subscript 4. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 5 is level with but to the right of x subscript 4. The same line is followed by x subscript 6, x subscript 7 and x subscript 8, which is level with x subscript 1. Continuing in the same direction leads to x subscript 9, which is at a lower level than x subscript 1. At x subscript 9 there is a gap. The direction changes again and the letters begin to rise. Slightly to the right but level with x subscript 9 is x subscript 10. They continue to rise and move to the right until x subscript 14 is reached. This is level with x subscript 5. There is then a gap and the letters then begin to fall again, starting with x subscript 15 which is level with x subscript 14. They continue down, moving to the right each time, ending with x subscript 18. This is on the same horizontal level as x subscript 1.

There are three clouds, each containing words. The first contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 4 and x subscript 5 with the arrowhead closer to x subscript 5 than x subscript 4. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 14 and x subscript 15 with the arrowhead closer to x subscript 14 than x subscript 15. Below the diagram a third cloud contains the word ‘Median’. From it an arrow points to the gap between x subscript 9 and x subscript 10, in the middle of the diagram.

Quartiles for sample size MathJax failure: MathML - Unexpected text node: '

For , and . So is one quarter of the way from to , and is three quarters of the way from to .

Figure 15 Quartiles for sample size

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Diagram showing the quartiles for sample size n = 20. The V-shape has two additional arms, one to the left and one to the right, so the shape now resembles a capital letter M, but with the sides sloping. The numbers represented by x subscript 1, x subscript 2 and so on, ending with x subscript 20 are shown.

The shape starts with x subscript 1, followed, slightly higher and to the right by x subscript 2, Following the same line are x subscript 3, x subscript 4 and x subscript 5. At that point the V formation begins, but there is a gap at the top. The next entry x subscript 6 is level with but to the right of x subscript 5. The same line is followed by x subscript 7, x subscript 8, x subscript 9 and x subscript 10, which is level with x subscript 1. At x subscript 10 there is a gap. The direction changes again and the letters begin to rise. Slightly to the right but level with x subscript 10 is x subscript 11. The letters continue to rise and move to the right until x subscript 15 is reached. This is level with x subscript 6. There is then a gap and the letters move to the right and begin to fall again, starting with x subscript 16 which is level with x subscript 15. They continue down, moving to the right each time, ending with x subscript 20. This is on the same horizontal level as x subscript 11.

There are three clouds, each containing words. Above the diagram the first cloud contains the words ‘Lower quartile’ and has an arrow pointing to the gap between x subscript 5 and x subscript 6 with the arrowhead closer to x subscript 5 than x subscript 6. To the right and at the same level is another cloud, containing the words ‘Upper quartile’, from which an arrow points to the gap between x subscript 15 and x subscript 16 with the arrowhead closer to x subscript 16 than x subscript 15. The third cloud is below the diagram and contains the word ‘Median’. From it an arrow points to the gap between x subscript 10 and x subscript 11, in the middle of the diagram.

Quartiles for sample size MathJax failure: MathML - Unexpected text node: '

Example 14 Quartiles for the prices of small televisions

Figure 15 showed you where the quartiles are for a batch of size 20. Let us now use the stemplot of the 20 television prices in Figure 16, which you first met in Figure 5 (Subsection 1.2), to find the lower and upper quartiles, and , of this batch.

Figure 16 Prices of flat-screen televisions with a screen size of 24 inches or less

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Stemplot with 10 levels, though there are repeated numbers in the stem. The numbers in the stem are 0, 1, 1, 1, 1, 1, 2, 2, 2, 2. At level 0 there is one leaf 9. At the first level 1 there is one leaf, 0. At the second level 1 there are four leaves 2, 3, 3, 3. The third level 1 has five leaves, 4, 5, 5, 5, 5. The fourth level 1 has three leaves, 6, 6, 7. The fifth and last level 1 has three leaves, 8, 8, 9. The first level 2 and the second level 2 have no leaves. The third level 2 has two leaves, 4, 5. The last level 2 has one leaf, 7. Beneath the stemplot is written n = 20, followed by 0 vertical line 9 represents 90 pounds sterling. The vertical line sits horizontally between the 0 and the 9.

Prices of flat-screen televisions with a screen size of 24 inches or less

To calculate the lower quartile you need to find the number that is one quarter of the way from to . These values are both 130, so is 130. To calculate the upper quartile you need to find the number three quarters of the way from to . These values are both 180, so is 180.

That example was easier than it might have been, because for each quartile the two numbers we had to consider turned out to be equal!

Example 15 Quartiles for the camera prices

Table 2 (Subsection 1.2) gave ten prices for a particular model of digital camera (in pounds). In order, the prices are as follows.

To find the lower and upper quartiles, and , of this batch, first find and .

The lower quartile is the number three quarters of the way from to . These values are 60 and 65. The difference between them is , and three quarters of that difference is . Therefore is 3.75 larger than 60, so it is 63.75. As with the median, in this course we will generally round the quartiles to the accuracy of the original data, so in this case we round to the nearest whole number, 64. In symbols, .

The upper quartile is the number one quarter of the way from to . These values are 81 and 85. The difference between them is , and one quarter of that difference is . Therefore is 1 larger than 81, so it is 82. (No rounding necessary this time.) In symbols, .

Example 15 is the subject of the following screencast. [Note that references to ‘the unit’ should be interpreted as ‘this course’. The original wording refers to the Open University course from which this material is adapted.]

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Screencast 3 Calculating quartiles

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Transcript: Screencast 3 Calculating quartiles

INSTRUCTOR

Here’s another batch of data from the unit. This time, it’s the prices of a particular model of digital camera. And there are the prices along the top in pounds. And what we’re asked to do is to find the lower and upper quartiles of this batch of data.

So usually the first thing you have to do when you’re finding the median or the lower and upper quartiles is to put the prices in order. But if you have a look up at the top there, luckily, somebody’s already done that for us. They’re in order.

So we can bash straight on with the next step, which is actually finding the quartiles. Now, what I’m going to do to begin with is to use the notation that we use in defining what the quartiles are, just so you can relate what I’m doing to what’s in the unit on that. Remember how that works. We take the first number, the smallest number in the batch, which in this case is 53, right over the left there.

And we write that as x, within brackets, 1. The 1 in brackets means it’s the smallest value in the batch. And the next one, which is 60, that’s x, brackets, 2. And then you just carry on like that.

And the last one is x10. That means there’s 10 values in this bunch. We need to know that.

And in fact, that’s the next thing we need to worry about. There are 10 observations, 10 camera prices. So in the notation, we call that n. n’s 10.

Now, remember what you’ve got to do to work out this. We want for the quartiles a number that’s 1/4 of the way along the batch when we’ve arranged it in order and another number that’s 3/4 of the way along the batch. But of course, it’s a bit more complicated than that. You don’t look at 1/4 of n. You look at n plus 1 divided by 4.

And that gives you the position of the lower quartiles. n plus 1 divided by 4 is 11/4, which is 2 and 3/4. And that defines where the position of the lower quartile is in a way that I’ll come to.

And to get the upper quartile – well, this time, you’ve got to take 3/4 of n plus 1. That is 3 times n plus 1, over 4. And that comes to n plus 1 is 11.

So it’s 33 – 3 times 11 – over 4. And if you work that out, that’s 8 and 1/4. And that defines the position of the upper quartile.

OK, so moving on, then, Q1 is the lower quartile. And it’s defined by this number here, this 2 and 3/4 number. And what that tells us is it’s between the second number in order and the third number. And it’s 3/4 of the way between them.

So I’ll just write down what that is. Q1 is 3/4 of the way from x2 to x3. And you can work that out just in arithmetic – but it’s probably easier if we have a look and see what it looks like graphically.

Here is a bit of a number line. x2 is going to be there. And x2 is 60.

And x3’s going to be up here somewhere. And x3, if we look up here, is 65. So that one’s 65.

And what we want is a number that’s 3/4 of the way from 60 to 65. Well, that’s halfway there. That’s going to be 62 and 1/2. And 3/4 of the way is about here somewhere. And that’s going to be 63.75.

OK, but how do we write that down in algebra? The way to do it is like this. We say you start at x2, which is 60. And then you go 3/4 of the way from x2 to x3.

Well, how far is it from x2 to x3? Well, again, if you look on this, it’s 65 to 60. That is 5.

And you can calculate that as 65 minus 60. So it’s 3/4 of 65 minus 60. So that’s 60 plus – we’ve got 3 times 5 over 4. That is 15/4. And if you work that out on your calculator or something, it comes to 63.75.

So we’ve worked out the lower quartile. It’s 63.75. And luckily, that’s the same as I showed you over here.

So now we’ve got to go and find the upper quartile. Remember, we denote that as Q3. And if you look at the thing over here, that says it’s at position 8 and 1/4, which means it’s 1/4 of the way from the eighth one to the ninth one. That is, it is 1/4 of the way from x8 to x9.

And again, it’s helpful to draw what’s going on here. So here is a number line. And here we’ve got x8, which is 81. And here we’ve got x9, which is 85.

And again, we want the number that’s 1/4 of the way along, bit different. So here’s halfway. That’s going to be 83. And here is 1/4 of the way, halfway between 81 and 83. That’s 82.

So it’s going to be 82. Let’s just see how we can write that down as a bit of algebra. So again, you start from x8, which is 81. And then you go 1/4 of the way up to x9.

Now this time, the distance between x8 and x9 is 85 – that’s x9 – minus 81, which is x8. So the arithmetic’s actually a lot easier in this case. That’s 81 plus 1/4 of 4. And that just comes to 81 plus 1, which is 82. Again, it’s the same.

So just one final step to do here. We’ve got to write down the answers in an appropriate way. Now, what we have to do is round them to an appropriate accuracy. That’s all we haven’t done.

And what we’ve got is that Q1 – what accuracy do we use? Well, the usual thing we do is to round quartiles to the accuracy of the original data that we had. And the original data we had up at the top there is in whole pounds. It’s rounded to whole pounds. So we need the quartiles in whole pounds, as well.

And if you look at this one, it’s not in whole pounds. It’s 63.75. So we’ve got to round it to the nearest pound. The bit in pence, essentially, here that we’re going to get rid of in the rounding is bigger than 1/2.

So we’ve got to round up. That is, this one is– let’s write in the pound sign. It’s £64 to the nearest pound.

And what about Q3? Well, you’ve got to think about rounding that, as well. But actually, you needn’t bother. It’s already a whole number of pounds, so no rounding involved. We can just write down it’s 82. And that’s that.

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Activity 10 Finding more quartiles

(a) Find the lower and upper quartiles of the batch of 15 coffee prices in Figure 17. (This batch of coffee prices was first introduced in Table 1 of Subsection 1.1.)

Figure 17 Stemplot of 15 coffee prices

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A stemplot with 11 levels, which start at 26 and end at 36. At level 26 there are five leaves 8, 8, 8, 8, 9. At level 27 there two leaves 5, 9. At level 28 there are no leaves. Level 29 has five leaves, 5, 5, 5, 5, 9. Level 30 has one leaf, 5 and level 31 also has one leaf, 5. Levels 32, 33, 34, and 35 have no leaves. Level 36 has one leaf, 9. Beneath the stemplot is written n = 15, followed by 26 vertical line 8 represents 268 pence. The vertical line sits horizontally between the 26 and the 8.

Stemplot of 15 coffee prices

Discussion

Here, because , an appropriate picture of the data would be Figure 12 (Subsection 3.2). To find the lower and upper quartiles, and , of this batch, first find and . Therefore and .

(b) Find the lower and upper quartiles of the batch of 14 gas prices in Figure 18. (This batch of gas prices was first introduced in Table 3 of Subsection 1.2.)

Figure 18 Stemplot of 14 gas prices

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A stemplot with 8 levels, which start at 374 and end at 381. Level 374 has three leaves, 0, 0, 3. Level 375 has no leaves. Level 376 has two leaves, 0, 7. Level 377 has one leaf, 6. Level 378 also has one leaf, 4. Level 379 has two leaves, 5, 6. Level 380 has four leaves, 1, 1, 4, 5. Level 381 has one leaf, 8. Beneath the stemplot is written n = 14, followed by 374 vertical line 0 represents 3.740 pence per kilowatt hour. The vertical line sits horizontally between the 374 and the 0.

Stemplot of 14 gas prices

Discussion

For this batch, so and .

and

So the lower quartile is 3.756 p per kWh and the upper quartile is 3.802p per kWh.

A measure of spread

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Now we can define a new measure of spread based entirely on the lower and upper quartiles.

The interquartile range

The interquartile range (sometimes abbreviated to IQR) is the distance between the lower and upper quartiles:

Note that this value is independent of the sizes of and .

Example 16 The prices of small televisions, yet again!

For the batch of 20 television prices in Example 14 (Subsection 3.2),

So the interquartile range is £50.

Activity 11 Coffee prices again

Calculate both the range and the interquartile range of the batch of 15 coffee prices, last seen in Figure 17 (Subsection 3.2).

Discussion

The range is the distance between the extremes:

The interquartile range is the distance between the quartiles:

Activity 12 Interquartile range of gas prices

In Activity 10(b) (Subsection 3.2) you found the quartiles of the 14 gas prices from Activity 2 (Subsection 1.2). Find the interquartile range.

Discussion

The quartiles, before rounding, are and . So

and the interquartile range is 0.046p per kWh.

You may be wondering why you are being asked to learn a new measure of spread when you already know the range. As a measure of spread, the range is not very satisfactory because it is not resistant to the effects of unrepresentative extreme values. (Resistant measures were explained in Subsection 1.4.) The interquartile range, by contrast, is a highly resistant measure of spread (because it is not sensitive to the effects of values lying outside the middle 50% of the batch) and it is generally the preferred choice.

Example 17 Comparing the resistance of the range and the IQR

Suppose the price of the most expensive jar of coffee is reduced from 369p to 325p. How does this affect the range and the interquartile range of the batch of coffee prices in Figure 17 (Subsection 3.2)?

The new range is

a lot less than the original value of 101p (found in Activity 11). The interquartile range is unchanged.

3.3 The five-figure summary and boxplots

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

As well as giving us a new measure of spread – the interquartile range – the quartiles are important figures in themselves. Our -shaped diagram, Figure 19, gives five important points which help to summarise the shape of a distribution: the median, the two quartiles and the two extremes.

Figure 19 Values in a five-figure summary

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Values in a five-figure summary. There are four lines, forming the shape of a letter M with sloping sides. At the base of the line on the left is capital E subscript capital L, indicating the lower extreme, the lowest value in the data. At the top of that line is capital Q subscript 1, indicating the lower quartile. The line then slopes down and to the right. At the end of the second line is capital M, indicating the median. This is on the same horizontal level as capital E subscript capital L. The line then rises and slopes to the right. It ends at capital Q subscript 3, indicating the upper quartile, which is at the same horizontal level as capital Q subscript 1. The line then falls, sloping to the right. At the end of the line is capital E subscript capital U, indicating the upper extreme, the highest value in the data. This is at the same horizontal level as capital M.

Values in a five-figure summary

These are conveniently displayed in the following form, called the five-figure summary of the batch.

Five-figure summary

Figure 20

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A five-figure summary which is a diagrammatic representation showing the batch size, n, the median capital M, the lower quartile capital Q subscript 1, the upper quartile capital Q subscript 3, the lower extreme capital E subscript capital L, and the upper extreme capital E subscript capital U. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n. Towards the bottom of the line and to its right is written capital E subscript capital L and capital Q subscript 1, with capital Q subscript 1 being above capital E subscript capital L. Beneath the middle of the horizontal line is written capital M. To the left of the second vertical line but level with capital Q subscript 1 is written capital Q subscript 3. Below that, and level with capital E subscript capital L, is written capital E subscript capital U.

Example 18 Five-figure summary for television price data

For the television price data, we have , , , , and . (You last saw these data in Figure 16, Subsection 3.2.)

Therefore, the five-figure summary of this batch is

Figure 21

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A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 20. Towards the bottom of the left vertical line and to its right is written 90 and above that 130. Beneath the middle of the horizontal line is written 150. To the left of the second vertical line and level with 130 is written 180. Below that and level with 90 is written 270.

This diagram contains the following information about the batch of prices.

The general level of prices, as measured by the median, is £150.
The individual prices vary from £90 to £270.
About 25% of the prices were less than £130.
About 25% of the prices were more than £180.
About 50% of the prices were between £130 and £180.

We hope you agree that the five-figure summary is quite an efficient way of presenting a summary of a batch of data.

The five values in a five-figure summary can be very effectively presented in a special diagram called a boxplot. For the 14 gas prices (Figure 15, Subsection 3.2) the diagram looks like Figure 22.

Figure 22 Boxplot of batch of 14 gas prices

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A boxplot with a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pence per kilowatt hour. The scale starts just before the first marked point of 3.74 and is then marked in four intervals of 0.02, ending with 3.82. The boxplot is drawn above and parallel to the line. The first whisker starts at the lower extreme, level with 3.74.The box starts at the lower quartile, 3.756 and ends at the upper quartile, 3.802. Within the box but nearer the right-hand end is a vertical line indicating the median at 3.790. The second whisker starts at the midpoint of the right-hand end of the box and stretches to 3.818, as far as the upper extreme.

Boxplot of batch of 14 gas prices

The central feature of this diagram is a box – hence the name boxplot. The box extends from the lower quartile (at the left-hand edge of the box) to the upper quartile (the right-hand edge). This part of the diagram contains 50% of the values in the batch. The length of this box is thus the interquartile range.

Outside the box are two whiskers. (Boxplots are sometimes called box-and-whisker diagrams.) In many cases, such as in Figure 22, the whiskers extend all the way out to the extremes. Each whisker then covers the end 25% of the batch and the distance between the two whisker-ends is then the range. (You will see examples later where the whiskers do not go right out to the extremes.)

So far we have dealt with four figures from the five-figure summary: the two quartiles and the two extremes. The remaining figure is perhaps the most important: it is the median, whose position is shown by putting a vertical line through the box.

Thus a boxplot shows clearly the division of the data into four parts: the two whiskers and the two sections of the box; these are the four parts of the -shaped diagram and each contains (approximately) 25% of values in the batch (see Figure 21).

John W. Tukey (1915–2000), inventor of the five-figure summary and boxplot

John Tukey was a prominent and prolific US statistician, based at Princeton University and Bell Laboratories. As well as working in some very technical areas, he was a great promoter of simple ways of picturing and summarising data, and invented both the five-figure summary and the boxplot (except that he called them the ‘five-number summary’ and the ‘box-and-whisker plot’).

He had what has been described as an ‘unusual’ lecturing style. The statistician Peter McCullagh describes a lecture he gave at Imperial College, London in 1977:

Tukey ambled to the podium, a great bear of a man dressed in baggy pants and a black knitted shirt. These might once have been a matching pair, but the vintage was such that it was hard to tell. …The words came …, not many, like overweight parcels, delivered at a slow unfaltering pace. …Tukey turned to face the audience …. ‘Comments, queries, suggestions?’ he asked …. As he waited for a response, he clambered onto the podium and manoeuvred until he was sitting cross-legged facing the audience. …We in the audience sat like spectators at the zoo waiting for the great bear to move or say something. But the great bear appeared to be doing the same thing, and the feeling was not comfortable. …After a long while, …he extracted from his pocket a bag of dried prunes and proceeded to eat them in silence, one by one. The war of nerves continued …four prunes, five prunes. …How many prunes would it take to end the silence?

(Source: McCullagh, P. (2003) ‘John Wilder Tukey’, Biographical Memoirs of Fellows of the Royal Society, vol. 49, pp. 537–55.)

Figure 23 A standard boxplot with annotation

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A boxplot which consists of a line, a rectangle, known as a box, and a second line. The first line, known as a whisker, starts at capital E subscript capital L, the lower extreme and leads to the midpoint of the left side of the box. Immediately above and at the corner of the box is written capital Q subscript 1. This point is the lower quartile. The box stretches for some way to the right, ending at the upper quartile, with capital Q subscript 3 written above the end of the box and on the same horizontal level as capital Q subscript 1. A second horizontal line, also known as a whisker, starts at the midpoint of the right-hand end of the box and stretches as far as capital E subscript capital U, the upper extreme. The median is marked by capital M above the box at the appropriate position and by a vertical line within the box. Beneath the boxplot are 4 horizontal curly brackets, each of which has 25% written under it. The first stretches from immediately beneath capital E subscript capital L to the start of the box. The second bracket stretches from the start of the box to level with the vertical line level indicating the position of the median. The third stretches from the line level with the median to level with capital Q subscript 3 and the last stretches from level with capital Q subscript 3 to level with capital E subscript capital U.

A standard boxplot with annotation

A typical boxplot looks something like Figure 23 because in most batches of data the values are more densely packed in the middle of the batch and are less densely packed in the extremes. This means that each whisker is usually longer than half the length of the box. This is illustrated again in the next example.

Example 19 Boxplot for the prices of small televisions

The boxplot for the batch of 20 television prices (last worked with in Example 18) is shown in Figure 24.

Figure 24 Boxplot of batch of 20 television prices

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The horizontal scale is marked from just before 100 to 275 in intervals of 25 units. The scale is labelled pounds sterling. The boxplot shows that the lower extreme is less than 100. The whisker leads to the lower quartile, at the start of the box. This occurs just past 125. The box contains a vertical line, indicating the position of the median. This occurs at 150. The box ends at the upper quartile, which occurs just after 175. The right-hand whisker ends at 250. There is then a gap. Farther on, but at the same level as the whisker, is an asterisk.

Boxplot of batch of 20 television prices

You can see that each whisker is longer than half the length of the box.

However, this boxplot has a new feature. The whisker on the left goes right down to the lower extreme. But the whisker on the right does not go right to the upper extreme. The highest extreme data value, 270, which might potentially be regarded as an outlier, is marked separately with a star. Then the whisker extends only to cover the data values that are not extreme enough to be regarded as potential outliers. The highest of these values is 250.

(This course does not describe the rule to decide which data values (if any) can be regarded as potential outliers that are plotted separately on the diagram. This is another issue that may be dealt with differently by different authors and different software.)

Example 19 is the subject of the following screencast. [Note that the reference to ‘Unit 2’ should be ‘this course’ and ‘Figure 18’ should be ‘Figure 23’. Unit 2 and Figure 18 are references to the Open University course from which this material is adapted.]

Download this video clip.Video player: Interpreting a boxplot

Screencast 4 Interpreting a boxplot

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Transcript: Screencast 4 Interpreting a boxplot

INSTRUCTOR

In this screencast, I’m going to talk about interpreting a boxplot. And what we have here is an example of a boxplot. And it happens to be Figure 18 from Subsection 3.3 of Unit 2. And it’s a boxplot of the small television prices. And you can see here that television prices are given in pounds, and they go from just under £100 up to £275.

The first thing on the boxplot to look at is the box itself – in particular, to look at the ends of the box. The end on the left hand side shows us where the lower quartile is. And here is about £130. The end on the right hand side shows us where the upper quartile is – Q3. And this translates to about £180. So the lower quartile is about £130, and the upper quartile is about £180.

The line in the middle of the box shows us where the median is. And this is about £150. So the price of the median small television is £150. Notice in this example, it is quite clear where the line is in the middle of the box. There are some examples where the median is the same as the lower quartile or where the median is the same as the upper quartile. And then you won’t actually see a line in the box. The line will be at one end or the other.

The other thing to notice on the boxplot are the two whiskers. So there’s a whisker on the right hand side and a whisker on the left hand side. The whisker on the right hand side shows us where the values that are high but not too high are. Similarly, on the left hand side, the whisker on the left hand side shows us where the values are low but not too low are.

And finally, notice there’s one point here marked all by itself. And this is a value that we wonder whether it’s too high. In other words, we’re marking this one out as a potential outlier. This shows all the elements that are on a boxplot. And one thing we can use these elements for is to say something about the symmetry of the data.

And one thing we can look at is where the median is relative to the two ends of the box. So here we notice that the left hand side is short relative to the right hand side. We can look at the whiskers in the same way, and notice that the whisker on the left hand side is relatively short. And the whisker on the right hand side is relatively long.

And both these observations together suggest that the data are right-skew. The data tends to be more spread out on the right hand side of the median relative to the data on the left hand side the median. And notice, in doing this, we haven’t actually taken account of the outlier. If we took the outlier into account as well, this would only emphasise more that the data are right-skew. Because this adds to the impression that the data are more spread out to the right of the median relative to the left of the median.

End transcript: Screencast 4 Interpreting a boxplot

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Screencast 4 Interpreting a boxplot

Interactive feature not available in single page view (see it in standard view).

One important use of boxplots is to picture and describe the overall shape of a batch of data.

Example 20 Skew televisions

The stemplot of small television prices, last seen in Figure 16 (Subsection 3.2), shows a lack of symmetry. Since the higher values are more spread out than the lower values, the data are right-skew.

The boxplot of these data, given in Figure 22, also shows this right-skew fairly clearly. In the box, the right-hand part (corresponding to higher prices) is rather longer than the left-hand part, and the right-hand whisker is longer than the left-hand whisker.

Activity 13 Skew gas prices?

A stemplot of the gas price data from Activity 2 (Subsection 1.2) is shown, yet again, in Figure 25.

Figure 25 Stemplot of 14 gas prices

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Stemplot of 14 gas prices

(a) Prepare a five-figure summary of the batch.

Discussion

All the necessary figures have already been calculated. You found the median (3.790) in Activity 2 and the quartiles (, ) in Activity 10. The extremes (, ) and the batch size () are clearly shown in the stemplot.

So the five-figure summary is as follows:

Figure 26

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A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 14. Towards the bottom of the line and to the right is written 3.740 and above that 3.756. Beneath the middle of the horizontal line is written 3.790. To the left of the second vertical line and level with 3.756 is written 3.802. Below that, level with 3.740, is written 3.818.

(b) Figure 27 shows the boxplot of these data that you have already seen in Figure 22. What do the stemplot and boxplot tell us about the symmetry and/or skewness of the batch?

Figure 27 Boxplot of batch of 14 gas prices

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Boxplot of batch of 14 gas prices, previously shown as Figure 17. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pence per kilowatt hour. The scale starts just before the first marked point of 3.74 and is then marked in four intervals of 0.02, ending with 3.82. The boxplot is drawn above and parallel to the line. The first whisker starts at the lower extreme, level with 3.74.The box starts at the lower quartile, 3.756 and ends at the upper quartile, 3.802. Within the box but nearer the right-hand end is a vertical line indicating the median at 3.790. The second whisker starts at the midpoint of the right-hand end of the box and stretches to 3.818, as far as the upper extreme.

Boxplot of batch of 14 gas prices

Discussion

Looking at the stemplot, on the whole the lower values are more spread out, indicating that the data are not symmetric and are left-skew.

The central box of the boxplot again shows left skewness, with the left-hand part of the box being clearly longer than the right-hand part. However, this skewness does not show up in the lengths of the whiskers in this batch – they are both the same length.

Example 21 Camera prices: skew or not?

In Example 20 and Activity 13 you saw how boxplots look for batches of data that are right-skew or left-skew. What happens in a batch that is more symmetrical?

For the small batch of camera prices from Table 2 (Subsection 1.2), a (stretched) stemplot is shown in Figure 28.

Figure 28 Stemplot of ten camera prices

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A stemplot with 9 levels, which are 5, 5, 6, 6, 7, 7, 8, 8, 9. The first level 5 has one leaf, 3. The second level 5 has no leaves. The first level 6 has one leaf, 0. The second level 6 has one leaf, 5. The first level 7 has three leaves, 0, 0, 4. The second level 7 has one leaf, 9. The first level 8 has one leaf, 1 and the second level 8 also has one leaf, 5. Level 9 has one leaf, 0. Beneath the stemplot is written n = 10, followed by 5 vertical line 3 represents 53 pounds sterling. The vertical line sits horizontally between the 5 and the 3.

Stemplot of ten camera prices

The stemplot looks reasonably symmetric.

A boxplot of the data, Figure 29, confirms the impression of symmetry. The two parts of the box are roughly equal in length, and the two whiskers are also roughly equal in length.

Figure 29 Boxplot of batch of ten camera prices

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A boxplot with a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pounds sterling. The scale starts just before the first marked point of 50 and then marked in five intervals of 10, ending with 90. The boxplot is drawn above and parallel to the line. The first whisker starts with capital E subscript capital L at 53 and ends at capital Q subscript 1, the lower quartile, 65. The box ends at capital Q subscript 3, the upper quartile, 81. Near the centre of the box is a vertical line indicating the median at 72. The second whisker starts at the midpoint of the right-hand end of the box and stretches to capital E subscript capital U at 90.

Boxplot of batch of ten camera prices

You have now spent quite a lot of time looking at various ways of investigating prices and, in particular, at methods of measuring the location and spread of the prices of particular commodities.

In order to begin to answer our question, Are people getting better or worse off?, we need to know not just location (and spread) of prices but also how these prices are changing from year to year. That is the subject of the rest of this course.

Exercises on Section 3

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

The following exercises provide extra practice on the topics covered in Section 3.

Exercise 6 Finding quartiles and the interquartile range

(a) For the arithmetic scores in Exercise 1 (Section 1), find the quartiles and calculate the interquartile range. The stemplot of the scores is given below.

Figure 30 Stemplot of arithmetic stores

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A stemplot with 11 levels. The numbers in the stem start at 0, increase in steps of one and end at 10. At level 0 there is one leaf, 7. Level 1 has one leaf, 5. Level 2 has no leaves, while level 3 has two leaves, 3, 5. Level 4 has three leaves, 2, 2, 3. Level 5 has two leaves, 5, 8. Level 6 has three leaves, 4, 6, 8. Level 7 has five leaves, 1, 1, 6, 8, 9. Level 8 has nine leaves, 0, 1, 1, 3, 4, 5, 5, 6, 9. Level 9 has five leaves, 1, 1, 3, 5, 9. Level 10 has two leaves which are both 0. Beneath the stemplot is written n = 33, followed by 0 vertical line 7 represents a score of 7 per cent.

Figure 30 Stemplot of arithmetic stores

Discussion

For the arithmetic scores, so and .

The lower quartile is therefore

The upper quartile is

The interquartile range is

(b) For the television prices in Exercise 1, find the quartiles and calculate the interquartile range. The table of prices is given below.

170	180	190	200	220	229	230	230	230
230	250	269	269	270	279	299	300	300
315	320	349	350	400	429	649	699

Discussion

For the television prices, so and .

The lower quartile is therefore

The upper quartile is

The interquartile range is

Exercise 7 Some five-figure summaries

Prepare a five-figure summary for each of the two batches from Exercise 1.

(a) For the arithmetic scores, the median is 79% (found in Exercise 1), and you found the quartiles and interquartile range in Exercise 6.

Discussion

Arithmetic scores:

From the stemplot, , and .

Figure 31 Five-figure summary of arithmetic scores

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A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 33. Towards the bottom of the line and to its right is written 7 and above that 57. Beneath the middle of the horizontal line is written 79. To the left of the second vertical line and level with 57 is written 88. Below that and level with 7 is written 100.

Five-figure summary of arithmetic scores

(b) For the television prices, the median is £270 (found in Exercise 1), and you found the quartiles and interquartile range in Exercise 6.

Discussion

Television prices:

From the data table, , and .

Figure 32 Five-figure summary of television prices

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A five-figure summary. The diagram forms three sides of a rectangle, with the bottom line missing. It therefore has a vertical line to the left, a horizontal line across the top and a vertical line to the right. To the left of the left vertical line is written n = 26. Towards the bottom of the line and to its right is written 170 and above that 230. Beneath the middle of the horizontal line is written 270. To the left of the second vertical line and level with 230 is written 327. Below that and level with 170 is written 699.

Five-figure summary of television prices

Exercise 8 Boxplots and the shape of distributions

Boxplots of the two batches used in Exercises 1, 6 and 7 are shown in Figures 33 and 34. On the basis of these diagrams, comment on the symmetry and/or skewness of these data.

Figure 33 Boxplot of batch of 33 arithmetic scores

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A boxplot of a batch of 33 arithmetic scores. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled percentage. The scale starts at 0 and is then marked in five intervals of 20, ending with 100. Above the line, and level with the boxplot is an asterisk at 7. The first whisker on the boxplot starts at 15. The capital Q subscript 1 is at about 58, capital M is at about 79 and capital Q subscript 3 is at about 86. The second whisker stretches to the upper extreme at 100.

Boxplot of batch of 33 arithmetic scores

Figure 34 Boxplot of batch of 26 television prices

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A boxplot of a batch of 26 television prices. There is a horizontal scale with an arrow head, pointing right, at the right-hand end. It is labelled pounds sterling. The scale starts at 100 and is then marked in six intervals of 100, ending with 700. The first whisker on the boxplot starts at 170. The box starts at capital Q subscript 1, which is about 230 and capital M is at about 270. The box ends at capital Q subscript 3 at about 320. The second whisker ends at 429. There are two asterisks. One is at 649 and the other is at 699.

Boxplot of batch of 26 television prices

Discussion

For the boxplot of arithmetic scores, the left part of the box is longer than the right part, and the left whisker is also considerably longer than the right. This batch is left-skew.

For the boxplot of television prices, the right part of the box is rather longer than the left part. The right whisker is also rather longer than the left, and if one also takes into account the fact that two potential outliers have been marked, the top 25% of the data are clearly much more spread out than the bottom 25%. This batch is right-skew.

4 A simple chained price index

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

You have already seen that it is not a simple task to measure the price of even a single commodity at a fixed time and place. Measuring the change in price of a single commodity from one year to the next will be even more complicated but, as was said in Subsection 1.1, to answer our question it is necessary to measure the changes in the prices of the whole range of goods and services which people use. Moreover, since we wish to know how all the different changes in the prices of these goods and services affect people, we need to take into account those people’s consumption patterns. For example, a large increase in the price of high-quality caviar will not affect most people’s budgets since most households’ shopping lists do not include this commodity!

This makes the task of measuring price changes and examining how they affect us seem exceedingly difficult; but such a task is carried out in the UK regularly each month, organised by the Office for National Statistics. (Most of the prices are actually collected by a market research company under contract to the Office for National Statistics.) The results of their data collection and subsequent calculations are summarised in two measures called the Consumer Prices Index (CPI) and the Retail Prices Index (RPI).

These indices do not measure prices. (‘Indices’ is the plural of ‘index’.) Each is an index of price changes over time, and one or both of these indices are commonly used when people make comparisons about the cost of living. They are highly relevant measures for those engaged in wage bargaining.

The RPI and the CPI are both ‘chained’ in the sense that the index value for each year is linked to the year before. The very first link in the chain is called the base year and it is given an index value of 100.

Figure 35 A chained index

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A chained index. There are two rows. The first is labelled Index value. There is then the number 100, followed by 7 sets of 3 dots. Above the 100 is an arrow, leading to the first set of dots. Then from that set is another arrow linking it to the next set, and so on to the end of the line. This gives 7 arrows in total. The next row is labelled Year. There is then a chain of 7 oval shapes, each overlapping slightly with the next one. The oval under the 100 in the first row contains the year 2007, the next one, under the first set of dots, contains 2008. The pattern continues in this way, the year increasing by 1 each time. The last entry, under the sixth set of dots, contains 2013, though it is clear that the numbers can continue further. Under the table is a cloud containing the words ‘2007 is the base year’. From it an arrow points to 2007 in the first link of the chain.

A chained index

4.1 A two-commodity price index

The Open University — Wed, 20 Jul 2016 10:11:48 GMT

Section 5 includes an outline of how the information used to calculate the official UK price indices is collected, and describes how the indices are calculated. To introduce ideas, in this section we describe a very much simpler example of a price index calculation. It uses exactly the same basic method of calculation as the actual Retail Prices Index. (Not every index is calculated in this way.)

The context is a mythical computing company, Gradgrind Ltd.

Gradgrind Ltd uses both gas and electricity in its operations. Table 7 shows the price they paid for each fuel in 2007 and 2008. The prices are shown in £ per megawatt hour (MWh). (It is more usual, in the UK, for prices to be quoted in pence per kilowatt hour (p/kWh). Here, £/MWh have been used simply to make some of the later calculations a little more straightforward. Because there are 100 pence in £1 and 1000 kilowatts in a megawatt, £10/MWh is exactly the same price as 1p/kWh – so Gradgrind’s gas price in 2007, for instance, was 2.4p/kWh.)

Table 7 Gradgrind’s energy prices in 2007 and 2008
Energy type	2007	2008
Gas (£/MWh)	24	29
Electricity (£/MWh)	76	87

If we were interested in looking at the change in price of just one of these fuels, say gas, things would be relatively straightforward. For instance, it might well be appropriate to look at the increase in price as a percentage of the price in 2007.

Activity 14 Gradgrind’s gas price increase

Work out the increase in Gradgrind’s gas price between 2007 and 2008 as a percentage of the 2007 price.

Discussion

The increase (in £/MWh) is . This is as a proportion of the 2007 price. That is, of the 2007 price. Or you might have worked this out by finding that the 2008 price is of the 2007 price, so that again the increase is 20.8% of the 2007 price.

So we could say that, for this company at least, gas has gone up by 20.8%. In other words, for every £1 they spent on gas in 2007, they would have spent £1.208 in 2008 if they had bought the same amount of gas in each year. Or putting it another way, for every 100 units of money (pence, pounds, whatever) they spent in 2007, they would have spent 120.8 units of money in 2008 if they had bought the same amount. So a way of representing this price change would have been to define an index for the gas price such that it takes the value 100 for 2007, and 120.8 for 2008.

Notice that the value of the gas price index for 2008 could be calculated as

That is, the value of the index in one year is the value of the index in the previous year multiplied by a price ratio, in this case the gas price ratio for 2008 relative to 2007. This ratio, as a number, is 1.208.

But Gradgrind did not only use gas, they used electricity as well, and the aim here is to find a representation of their overall fuel price change, not just the change in gas prices.

An electricity price ratio for 2008 relative to 2007 can be worked out, like the gas price ratio. It is .

Activity 15 Gradgrind’s electricity price index

Use the electricity price ratio above to find the increase in Gradgrind’s electricity price between 2007 and 2008 as a percentage of the 2007 price. What would the 2008 value be for a price index of Gradgrind’s electricity price alone, calculated in the same way as the gas price index (with 2007 as the base year)?

Discussion

The 2008 electricity price is of the 2007 price, so that the increase is 14.5% of the 2007 price.

The 2008 value of the electricity price index is

But this has got us no further in finding a price index that simultaneously covers both fuels.

One possibility might be to look at how Gradgrind’s total expenditure on these two fuels changed from 2007 to 2008. The expenditures are given in Table 8.

Table 8 Gradgrind’s energy expenditure (£) in 2007 and 2008
Energy type	2007	2008
Gas	9 298	8 145
Electricity	3 205	2 991
Total	12 503	11 136

This seems not to have helped. The total expenditure went down, but you have already seen that the prices of both gas and electricity went up.

Activity 16 How much fuel did Gradgrind use?

Use the data in Tables 7 and 8 to find the quantity of each fuel that Gradgrind used in 2007 and 2008 (in MWh). Hence explain why the energy expenditure fell.

Discussion

The expenditure on a particular fuel in a particular year can be calculated as . Therefore, if the expenditure and price are known, the quantity used can be calculated as

In 2007, Gradgrind’s gas cost £24 per MWh, and they spent £9298 on gas, so the amount of gas they used in MWh was

The other amounts, in MWh, are found in a similar way, and all are shown in the following table.

Energy type	2007	2008
Gas	387.4	280.9
Electricity	42.2	34.4

The reason that the expenditures went down is simply that Gradgrind used less of each fuel in 2008 than in 2007.

Remember the aim is to produce a measure of price changes. So looking at expenditure changes does not do the right thing, since expenditure depends on the amount of fuel consumed as well as the price.

One possibility might be as follows. We could work out how much Gradgrind would have spent on fuel in 2008 if the consumptions of both fuels had not changed from 2007. That would remove the effect of any changes in consumption. Then we could calculate an overall energy price ratio for 2008 relative to 2007 by dividing the total expenditure on energy for 2008 (using the 2007 consumption figures) by the total expenditure on energy for 2007 (again using the 2007 consumption figures).

You should have found, in Activity 16, that the quantities of gas and electricity consumed in 2007 were, respectively, 387.4 MWh and 42.2 MWh. To buy those quantities at 2008 prices would have cost (in £): for the gas and for the electricity, giving a total expenditure of

So a reasonable overall energy price ratio for 2008 relative to 2007 can be found by dividing this total by the 2007 total expenditure, again calculated using the 2007 consumptions. The appropriate figure for 2007 is just the actual total expenditure, which (in £) was (see Table 8). This gives an overall energy price ratio for 2008 relative to 2007 as

Now we have an appropriate price ratio, the Gradgrind energy price index can be set as 100 for the base year, 2007, and the value of the 2008 index is found by multiplying the 2007 index value by the price ratio:

This is indeed how a chained index of this kind is calculated – but the calculations are rather messy. You might be wondering whether it would be simpler to calculate the overall energy price ratio as a weighted mean of the two price ratios for the two fuels, in much the same way that weighted means were used to combine prices in Section 2. If you did think this, you would be right – and furthermore, the resulting overall energy price ratio is exactly the same as has just been found, if we make the right choice of weights. The overall energy price ratio for 2008 relative to 2007 is just a weighted mean of the two price ratios for gas and electricity, with the 2007 expenditures as weights.

Just to show it really does come to the same thing, let us see how it works with the numbers, using the formula for weighted means in Subsection 2.3.

Energy type	Price ratio (2008 relative to 2007):	Weight (2007 expenditure):
Gas	1.208	9298
Electricity	1.145	3205

The weighted average of these price ratios is

giving the same value for the overall energy price ratio for 2008 relative to 2007 as we found earlier. (And this is not some sort of fluke that applies only to these particular numbers; it can be shown mathematically that it always works.)

Activity 17 Gradgrind’s energy price ratio for 2009 relative to 2008

Table 9 Gradgrind’s energy prices and expenditures for 2008 and 2009
Energy type: price and expenditure	2008	2009
Gas price (£/MWh)	29	30
Gas expenditure (£)	8 145	23 733
Electricity price (£/MWh)	87	98
Electricity expenditure (£)	2 991	2 275

(a) Using the data in Table 9, calculate the price ratios for gas and for electricity, in each case for 2009 relative to 2008.

Discussion

The gas price ratio for 2009 relative to 2008 is

The electricity price ratio for 2009 relative to 2008 is

(Over this year, electricity prices rose a lot more than gas prices.)

(b) With the 2008 expenditures as weights, use your answers to part (a) to calculate the overall energy price ratio for 2009 relative to 2008.

Discussion

The overall energy price ratio for 2009 relative to 2008 is

(c) Now see what happens if you use the 2009 expenditures as weights to calculate the overall energy price ratio for 2009 relative to 2008. How do the results of the calculation differ from what you got in part (b)?

Discussion

Using the 2009 expenditures for weights instead of the 2008 expenditures, the overall energy price ratio for 2009 relative to 2008 is

This price ratio is considerably less than the one found in part (b).

(Note that if full calculator accuracy is retained throughout the calculations, the price ratio is 1.043 to three decimal places.)

The reason that the price ratios you calculated in parts (b) and (c) in Activity 17 were so different is that Gradgrind’s ‘energy mix’ changed a lot over the year. Compared with 2008, in 2009 they spent a great deal more on gas but less on electricity. The weighted mean of the gas and electricity price ratios is, in both cases, nearer the price ratio for gas than that for electricity – this is Rule 2 for weighted means – but it is even nearer the gas weighted mean when the 2009 expenditures are used. This is because the weight for gas is proportionally much greater than it is when the 2008 expenditures are used as weights.

This all shows that it does make a difference which expenditures are used as weights. In practice, it is much more common to use the expenditures from the earlier year – 2008 in this case – as weights. In some circumstances, though, there are good reasons for using the later year, or indeed some more complicated set of weights that depend on both expenditures. However, in this course we shall use the expenditures from the earlier year to provide the weights, partly because that matches more closely what is done in calculating the official UK price indices.

Another possibility for weights would have been to continue to use the 2007 expenditures. These were used to find the overall energy price ratio for 2008 relative to 2007 and could be used for later years as well. Again, in some circumstances this would make sense, but here the pattern of Gradgrind’s fuel expenditure has changed a lot over time, and weights should change in consequence. To continue to use the 2007 expenditures for all later years would mean that this change in the relative importance to Gradgrind of the two fuels would never be taken into account. Instead, to obtain the overall energy price ratio from one year to the next, we use the fuel expenditures in the earlier year as weights, so each year the weights change.

That determines the choice of weights in forming an overall price ratio. Now, how is that used to find the energy price index? Here we simply continue the ‘chaining’ that started when finding the 2008 index: the 2009 index is found by multiplying the value of the index for the previous year, 2008, by the overall energy price ratio for 2009 relative to 2008. The value of the index for 2008 was calculated earlier as 119.2, and (using the weights from the previous year) the overall energy price ratio for 2009 relative to 2008 was found in Activity 17(b) as 1.059. So the value of Gradgrind’s energy price index for 2009 is

(So, in a particular kind of average way, Gradgrind’s energy prices for 2009 have risen by 26.2% since the base year, 2007.)

In general, the value index for a particular year is found by multiplying the value of the index for the previous year by the overall energy price ratio for that year relative to the previous year. This is illustrated in Figure 36.

Figure 36 Determining a chained price index

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Determining a chained price index. There are 3 rows. The first is labelled Price ratios, the second is labelled Index and the third is labelled Base year. The row headed Index contains three numbers. These are 100, 119.2 and 126.2. Above these, the row labelled Price ratios contains two arrows. The first goes from the index number 100 to the index number 119.2. Above the arrow is written times 1.192. The second arrow goes from the index number 119.2 to the index number 126.2. Above the arrow is written times 1.059. The last row has three ovals each containing a number. The first contains 2007, below the index number 100. The second contains 2008, below the index number 119.2. The third and last contains 2009 below the index number 126.2.

Determining a chained price index

In the process of chaining, the overall price ratio is calculated anew each year, looking back only at the previous year. The ratio is used to ‘chain’ to earlier years and hence determine the value of the index. This method of calculating a chained price index is summarised below. Although there were only two commodities (gas and electricity) in Gradgrind’s index, this summary is not restricted to two commodities.

Procedure used to calculate a chained price index

For each year calculate the following.
- The price ratio for each commodity covered by the index:
- The weighted mean of all these price ratios, using as weights the expenditure on each commodity in the previous year. This weighted mean is called the all-commodities price ratio.
For each year, the value of the index is
The value of the index in the first year is set at 100; this date is the base date of the index.

Activity 18 Gradgrind’s energy price index for 2010

Use the data in Table 10, and other necessary numbers from previous calculations, to calculate the value of Gradgrind’s energy price index for 2010.

Table 10 Gradgrind’s energy prices and expenditures for 2009 and 2010
Energy type: price and expenditure	2008	2009
Gas price (£/MWh)	30	28
Gas expenditure (£)	23 733	23 969
Electricity price (£/MWh)	98	88
Electricity expenditure (£)	2 275	2 920

Discussion

The gas price ratio for 2010 relative to 2009 is

The electricity price ratio for 2010 relative to 2009 is

(Both price ratios are less than 1 because, over this year, Gradgrind’s gas and electricity prices both fell.)

The overall energy price ratio for 2010 relative to 2009 is

Then the value of the index for 2010 is found by multiplying the 2009 value of the index by this overall price ratio, giving

The Retail Prices Index (RPI), published by the UK Office for National Statistics, is calculated once a month rather than once a year, but the method used is basically that outlined above, though with far more than two commodities. The process of finding the weights in the Retail Prices Index is also more complicated, because it involves taking into account the expenditures of millions of people as measured in a major survey. However, the principles are the same as for Gradgrind. The calculation each January follows exactly this method. In the other 11 months of the year, the calculation is very similar but uses only the increases in prices since the previous January. (See Subsection 5.2 for the details of these calculations.) In the next section, you will learn more about how all this works.

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