Worksheet on Index Numbers (Student Version)
Step 1 - What are Index Numbers?
Index numbers are used when it is necessary to show the AVERAGE change in a large number of variables. There are a number of particularly well known ones, some of which are announced on the news each evening. Fill in any that you know of in the table below:-
| Index | Shows changes in? |
An index number is usually started in a base year at a value of 100. The number itself has no meaning and has no units, it is the changes in the number that are important. These changes are expressed in percentage terms around the base. For example:-
If in year 1 the index starts at 100 and increases to 120 in year 2, then the variable has increased by 20%. If in year 3 it increases further to 150, it has increased by a further 25%. (Take the change of 30 and divide it by the figure at the start of the year (120) and multiply by 100 to get it as a percentage). Try the following examples for yourself (remember Windows has a calculator in the 'accessories'):
| Year | Retail Price Index | Percentage change(inflation) |
| 1 | 100 | |
| 2 | 115 | |
| 3 | 140 | |
| 4 | 175 | |
| 5 | 190 |
STEP 2 - Calculating Index Numbers
Let us now try an example of calculating an index number. Say we want to calculate inflation (a Retail Price Index) for four particular goods. We set the index for each good for the first year to 100 and then work out the percentage price change in each good. The first one is filled in in the table below - try the others:-
| Product | Price - year 1 | Index - year 1 | Price - year 2 | Index - year 2 |
| Bread | 40p | 100 | 60p | 150 @ |
| Beer | 200p | 100 | 220p | |
| Toothbrushes | 100p | 100 | 75p | |
| Newspapers | 50p | 100 | 60p | |
| TOTAL | 400 / 4 * | |||
| Overall index | 100 |
* We divide by the number of products to get the average change
@ The price has gone up by 50% and so the index goes from 100 to 150
The change in the overall index is the AVERAGE rate of inflation. What was the rate of inflation for these four products?
Inflation between year 1 and 2 _____________________________ ?
STEP 3 - Weighted Index Numbers
The problem with the index we have calculated above, is that it treats each price change as being EQUALLY important. However, the products in the price index are clearly not eqaully important.
Fill in below roughly how much you think the average family would normally spend on each of the products over a year (at the prices given). In the last column work out what percentage of their total spending (on these four products) that represents:-
| Product | Price | Total spending per year | Proportion of total spending (%) |
| Bread | 40p | ||
| Beer | 200p | ||
| Toothbrushes | 100p | ||
| Newspapers | 50p | ||
| TOTAL | - | 100% |
What should probably be clear from this is that the average family spend far less on tootbrushes than the other products. If this is the case then we should make the price change for toothbrushes have a much smaller overall effect on our price index. To do this we WEIGHT each price change to give it more or less importance in the overall index.
In the table below we have done this - see if you can finish off filling in the last column:-
| Product | Weights | Price - year 1 | Index - year 1 | Weighted index - year 1 | Price - year 2 | Index - year 2 | Weighted index - year 2 |
| Bread | 4 | 40p | 100 | 400 @ | 60p | 150 | 600 |
| Beer | 3 | 200p | 100 | 300 | 220p | 110 | |
| Toothbrushes | 1 | 100p | 100 | 100 | 75p | 75 | |
| Newspapers | 2 | 50p | 100 | 200 | 60p | 120 | |
| TOTAL | 10 | 1000/10 * | |||||
| Overall weighted index | 100 |
* We divide the total of the weighted index by the total of the weights to get the average figure.
@ Each weighted index figure is worked out by multiplying the basic index by the weight.
Inflation between year 1 and 2 _____________________________ ?
This figure should be very different from the original inflation figure. It should be considerably higher. This is because the most important products (the ones with the highest weights) went up in price the most. The effect of the price of toothbrushes falling on the overall index was reduced because they had a very small weight.
A weighted index gives a much better figure for inflation. After all we wouldn't want changes in the prices of shoelaces, matches and soap to have the same effect on the Retail Price Index as changes in the price of food, fuel and housing (though don't skimp on the soap buying for the sake of all around you!).
Step 4 - Further Work on Index Numbers
Go to Penn World Data tables (http://www.bized.co.uk/dataserv/penndata/pennhome.htm) and scan through the variables available. How many of these are in index form. List them below:-
Variables
Now go to the ONS data (http://www.bized.co.uk/dataserv/ons/data/data.htm) and see how many of these are indices. Again, list them below:-
Variables