Prices, location and spread
Prices, location and spread

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Prices, location and spread

2.3 More than two numbers

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 x sub 1 and x sub 2 with corresponding weights w sub 1 and w sub 2.

  1. Multiply each number by its weight to get the products x sub 1 w sub 1 and x sub 2 w sub 2.

  2. Sum these products to get x sub 1 w sub 1 + x sub 2 w sub 2.

  3. Sum the weights to get w sub 1 + w sub 2.

  4. 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

fraction sum of { number times weight } over sum of weights end = fraction sum of products over sum of weights end .

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).

Batch 1 with mean 525.5 and batch size 6. Batch 2 with mean 468.0 and batch size 2. Batch 3 with mean 504.2 and batch size 12.

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

BatchNumber (batch mean)Weight (batch size)Number times 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

fraction sum of products over sum of weights end = fraction 10139 .4 over 20 end = 506.97.

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:

1.3 with weight 2 1.9 with weight 1 1.7 with weight 3.

This is given by

fraction open bracket 1 .3 times 2 close bracket + open bracket 1 .9 times 1 close bracket + open bracket 1 .7 times 3 close bracket over 2 + 1 + 3 end = fraction 2 .6 + 1 .9 + 5 .1 over 6 end = fraction 9 .6 over 6 end = 1.6.

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

Described image
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 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

CityPrice (p/kWh): xWeight: w Price times weight: x w

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, w, 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, x, by the weight, w, to get the entry in the last column, x w.

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

fraction sum of products open bracket price times weight close bracket over sum of weights end

or, in symbols,

fraction sum x w over sum w end .

As sum x w = 6973.436 and sum w = 1834, the weighted mean is

fraction 6973 .436 over 1834 end =3.802310 simeq 3.802.

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

CityPrice (p/kWh): xWeight: wPrice times weight: x w

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 x w column.)

Discussion

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

CityPrice (p/kWh): xWeight: wPrice times weight: x w

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 sum x w = 24854.49, sum w = 1892 and

fraction sum x w over sum w end = fraction 24854 .49 over 1892 end = 13.136623 simeq 13.14.

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

M140_1

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