Index numbers, weighting and rebasing

Overview

Statistical agency rebase a price index every few years and two index numbers not simply be added together.

What you need to know

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Questions from this subtopic

Each one with the model answer underneath.

1. 1

   The average price of a loaf of bread is £2.40 in the base year and £2.64 two years later. What is the price index for the later year with the base set at 100, and what does it tell you?

   Model answer

   Dividing the later price by the base price and multiplying by 100 gives 2.64 divided by 2.40 multiplied by 100, which is 110. Bread is 10 per cent dearer than in the base year. An index number has no units: it expresses a value purely as a proportion of the base, which is what allows series measured in different units to be compared on the same chart.

2. 2

   Average house prices are £180,000, £198,000 and £216,000 in three successive years. Construct an index with the first year as base and state the percentage rise between the second and third years.

   Model answer

   The base year is 100. The second year is 198,000 divided by 180,000 multiplied by 100, which is 110, and the third is 216,000 divided by 180,000 multiplied by 100, which is 120. The third year is 20 per cent above the base but only 9.1 per cent above the second year, since 120 divided by 110 is 1.091. The index rose by ten points between the two years, which is not the same as a ten per cent rise, because the change must be measured against 110 rather than 100.

3. 3

   A basket of goods has three groups with weights of 300, 200 and 500 out of 1,000, and their price indices this year stand at 112, 108 and 104. What is the overall price index, and what part do the weights play?

   Model answer

   Multiply each index by its weight and divide by the sum of the weights: (300 multiplied by 112) plus (200 multiplied by 108) plus (500 multiplied by 104) equals 33,600 plus 21,600 plus 52,000, which is 107,200, and dividing by 1,000 gives 107.2. Prices are 7.2 per cent above the base. The weights represent the share of household spending going to each group, so a sharp price rise in something households barely buy has little effect on the headline figure, while a small rise in a heavily weighted group such as housing moves it substantially.

4. 4

   In a weighted price index the group with the largest weight sees no price change while a group carrying a weight of 0.05 sees prices rise 20%. How much does that add to the overall index, and what does it show about headline inflation?

   Model answer

   The contribution of any group is its price change multiplied by its weight, so 0.05 multiplied by 20 gives 1 percentage point. A dramatic price rise in a minor part of the basket therefore adds only one point to the headline figure. This is why the published rate can remain low while particular prices soar, and why households whose spending is concentrated in the fast-rising group experience an inflation rate well above the average.

5. 5

   An index reads 100, 118, 125 and 132 in four successive years. Rebase the series so that the third year equals 100, and state what rebasing changes and what it does not.

   Model answer

   Multiply each figure by 100 divided by 125. The rebased series is 80, 94.4, 100 and 105.6. Rebasing changes only the reference point against which each value is measured; it leaves the underlying data untouched. The percentage change between any two years is identical before and after: 118 divided by 100 and 94.4 divided by 80 both give a rise of 18 per cent, and 132 divided by 125 and 105.6 divided by 100 both give 5.6 per cent.

6. 6

   Why does a statistical agency rebase a price index every few years?

   Model answer

   As an index climbs further above 100 the base year becomes an increasingly remote and less meaningful reference, and recent movements become harder to read against a large number. Rebasing resets the reference to a recent year, and the exercise is normally used at the same time to refresh the basket of goods and the weights so that the index reflects current spending patterns. It does not alter the measured inflation between any two years, only the numbers in which it is expressed.

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