Index Numbers (Edexcel GCSE Statistics)

Revision Note

Author

Roger B

Expertise

Maths

Index Numbers

How do I calculate index numbers?

Index numbers are a way of comparing how the prices of items change over time
- Prices are compared to what they were in a base year
  - This is a year chosen to be the 'starting point' for the index numbers
  - The index number for the base year is 100
The index number is calculated by the formula
- $index number = \frac{current price of item}{price in base year} \times 100$
- Note that this is very similar to calculating a percentage
  - But percentage signs are not used with index numbers
- This formula will not be given on the exam, so you need to remember it
For example, the price of a pint of milk in 2024 is 66p
- If we use 2007 as the base year, when the price of a pint of milk was 36p
- then the 2024 index number for milk would be
  $\frac{66}{36} \times 100 = 183.333 . . . = 183.3 (1 d . p .)$
Note that:
- An index number greater than 100 means an increase in price
- An index number less than 100 means a decrease in price
A base year doesn't have to be at the 'beginning' of a set of index numbers
- For example, you could choose 2015 as the base year
- And use that to calculate the index number for 2012

What else can I do with index numbers?

Index numbers can be used to find the percentage change in prices between the base year and another year
- Subtract 100 from the index number for the other year
  - The result is the percentage change compared to the base year
  - A positive result means a percentage increase, a negative result means a percentage decrease
- For example the base year for widgets is 2020, and the index numbers for 2022 and 2023 are 112.3 and 98.4 respectively
  - The percentage change from 2020 to 2022 is $112.3 - 100 = 12.3 %$ (a 12.3% increase)
  - The percentage change from 2020 to 2023 is $98.4 - 100 = - 1.6 %$ (a 1.6% decrease)
Index numbers can be used to calculate a changed price between the base year and another year
- Divide the index number for the other year by 100 to find the 'percentage change multiplier' for the change
  - Multiply the price in the base year by that multiplier to find the price in the other year
  - Or divide the price in the other year by that multiplier to find the price in the base year
- For the widgets example above, the multiplier for 2020 to 2022 is $\frac{112.3}{100} = 1.123$
  - If a widget cost £14.95 in 2020
  - then a widget in 2022 cost $14.95 \times 1.123 = 16.78885 = £ 16.79$ (round money to the nearest penny!)
- The multiplier for 2020 to 2023 is $\frac{98.4}{100} = 0.984$
  - If a widget cost £14.71 in 2023
  - then a widget in 2020 cost $14.71 \div 0.984 = 14.9491 . . . = £ 14.95$

What is the retail price index (RPI)?

The retail price index (RPI) is an index number calculated to show the rate of change of prices for things people use in everyday life
- It includes such things as food, heating, petrol prices and mortgage payments
- The RPI is used by the UK government to set the interest rates paid on student loans
You can use RPI index numbers just like you use other index numbers
- For example, if the RPI base year is 1987 and the RPI for 2024 is 378.0
  - then the 'percentage change multiplier' from 1987 to 2024 is $\frac{378.0}{100} = 3.78$
- If a litre of petrol cost 37p per litre in 1987
  - then an estimate for its price in 2024 would be $37 \times 3.78 = 139.86 = 140 p = £ 1.40$
- This is an estimate because the RPI is calculated based on the prices of lots of different things that people buy
  - It is not just an index number for petrol

What is the consumer price index (CPI)?

The consumer price index (CPI) is also an index number calculated to show the rate of change of prices for things people use in everyday life
- It is similar to the RPI but does not include mortgage payments
- The CPI is used by the UK government to determine benefits and state pension payments
You can use CPI index numbers just like you use other index numbers
- For example, if the CPI base year is 2015 and the CPI for 2024 is 131.5
  - then the 'percentage change multiplier' from 2015 to 2024 is $\frac{131.5}{100} = 1.315$
- If a basket of shopping costs £80 in 2024
  - then an estimate for its price in 2015 would be $80 \div 1.315 = 60.8365 . . . = £ 60.84$
- This is an estimate because the CPI is calculated based on the prices of lots of different things that people buy
  - It is not just an index number for the items in that basket of shopping

What is gross domestic product (GDP)?

Gross domestic product (GDP) is the total value of all the goods and services produced by a country during a particular time period
- Goods are things that are made or produced (food, manufactured items, etc.)
- Services are things that people do for other people (e.g. haircuts, bus rides, repair work, medical care)
GDP figures in the UK are usually reported quarterly
- If a country's GDP falls (i.e. decreases) for two or more quarters in a row, then the country's economy is said to be in recession
Note that GDP is not an index number

Exam Tip

The index number formula is not given on the exam, so make sure you remember it
Remember that RPI and CPI are both index numbers and can be used like other index numbers
- But they are based on the prices of lots of different items
- So you can only estimate the prices of specific items using them

Worked Example

(a) The following table shows the prices of widgets from 2012 to 2015

Year	2012	2013	2014	2015
Price of widget (£)	12.95	13.50	13.85	14.10
Index number

Using 2013 as the base year, calculate the index numbers for the other years and complete the table. Give your answers correct to 1 decimal place.

The index number for the base year is always 100, so that is the index number for 2013
For the other years use $index number = \frac{current price of item}{price in base year} \times 100$
Note that it doesn't matter that 2012 is before the base year 2013!

$2012 : \frac{12.95}{13.50} \times 100 = 95.9259 . . . = 95.9 (1 d . p .) 2014 : \frac{13.85}{13.50} \times 100 = 102.5925 . . . = 102.6 (1 d . p .) 2015 : \frac{14.10}{13.50} \times 100 = 104.4444 . . . = 104.4 (1 d . p .)$

Year	2012	2013	2014	2015
Price of widget (£)	12.95	13.50	13.85	14.10
Index number	95.9	100	102.6	104.4

The base year for calculating index numbers for thingummies is 2012. The index number for 2015 is 97.4 and the index number for 2020 is 107.2.

(b) Find the percentage change in the price of thingummies from 2012 to 2015. Be sure to specify whether it is an increase or a decrease.

Subtract 100 from the index number for 2015 to find the percentage change

$97.4 - 100 = - 2.6$

That is negative, so it's a decrease

2.6% decrease

In 2012 a thingummy cost £56.50.

Divide the 2020 index number by 100 to find the 'percentage change multiplier'

$\frac{107.2}{100} = 1.072$

Multiply the 2012 price of a thingummy by this to find the 2020 price

$56.50 \times 1.072 = 60.568$

It's money, so round to the nearest penny

£60.57

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