2.2 Further uses of weighted means
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
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.
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
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
This is slightly less than the simple mean in (a) because the component with the lower score (TMA) has the greater weight.
(c) iCMA 65, TMA 55
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
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
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.)