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.
This first subsection looks at how a mean can be calculated when two unequally weighted batches are combined.
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
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.
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
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.
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.
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.]
Video content is not available in this format.
Screencast 2 Calculating a weighted mean
We shall now look at another similar problem about mean prices – one which is perhaps closer to your everyday experience.
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.)
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:
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.
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.
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
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.
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.
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
(b) iCMA 40, TMA 60
(c) iCMA 65, TMA 55
(d) iCMA 25, TMA 75
(e) iCMA 30, TMA 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.)
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.
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.
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 ![]() |
---|---|---|---|
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
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 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 ![]() ![]() |
---|---|---|---|
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.
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.)
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: ![]() |
Price ![]() ![]() |
---|---|---|---|
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.)
The following exercises provide extra practice on the topics covered in Section 2.
Find the mean price of the batch formed by combining the following two batches, and
, of camera prices.
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.