2.2 Further uses of weighted means

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.)

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.

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