Index Numbers (Edexcel GCSE Statistics)

Revision Note

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 space number equals fraction numerator current space price space of space item over denominator price space in space base space year end fraction cross times 100

    • Note that this is very similar to calculating a percentage

      • But percentage signs are not used with index numbers

    • This formula is not on the exam formula sheet, 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
      66 over 36 cross times 100 equals 183.333... equals 183.3 space open parentheses 1 space straight d. straight p. close parentheses

  • 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 minus 100 equals 12.3 percent sign  (a 12.3% increase)

      • The percentage change from 2020 to 2023 is  98.4 minus 100 equals negative 1.6 percent sign  (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 fraction numerator 112.3 over denominator 100 end fraction equals 1.123

      • If a widget cost £14.95 in 2020

      • then a widget in 2022 cost  14.95 cross times 1.123 equals 16.78885 equals £ 16.79  (round money to the nearest penny!)

    • The multiplier for 2020 to 2023 is fraction numerator 98.4 over denominator 100 end fraction equals 0.984

      • If a widget cost £14.71 in 2023

      • then a widget in 2020 cost  14.71 divided by 0.984 equals 14.9491... equals £ 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   fraction numerator 378.0 over denominator 100 end fraction equals 3.78

    • If a litre of petrol cost 37p per litre in 1987

      • then an estimate for its price in 2024 would be  37 cross times 3.78 equals 139.86 equals 140 straight p equals £ 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   fraction numerator 131.5 over denominator 100 end fraction equals 1.315

    • If a basket of shopping costs £80 in 2024

      • then an estimate for its price in 2015 would be  80 divided by 1.315 equals 60.8365... equals £ 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

Examiner Tips and Tricks

  • The index number formula is not on the exam formula sheet, 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 space number equals fraction numerator current space price space of space item over denominator price space in space base space year end fraction cross times 100
Note that it doesn't matter that 2012 is before the base year 2013!

2012 colon space space space fraction numerator 12.95 over denominator 13.50 end fraction cross times 100 equals 95.9259... equals 95.9 space open parentheses 1 space straight d. straight p. close parentheses

2014 colon space space space fraction numerator 13.85 over denominator 13.50 end fraction cross times 100 equals 102.5925... equals 102.6 space open parentheses 1 space straight d. straight p. close parentheses

2015 colon space space space fraction numerator 14.10 over denominator 13.50 end fraction cross times 100 equals 104.4444... equals 104.4 space open parentheses 1 space straight d. straight p. close parentheses

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 minus 100 equals negative 2.6

That is negative, so it's a decrease

2.6% decrease


In 2012 a thingummy cost £56.50.

(c) Find the cost of a thingummy in 2020.

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

fraction numerator 107.2 over denominator 100 end fraction equals 1.072

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

56.50 cross times 1.072 equals 60.568

It's money, so round to the nearest penny

£60.57

Weighted Index Numbers

How do I calculate a weighted index number?

  • For index numbers based on the prices of several different items a weighted index number is used

    • A different weight is given to each item depending on how important it is

    • The weight used for each item can change from year to year

  • Weighted index numbers are calculated using the formula

    • weighted space index space number equals fraction numerator current space weighted space mean space price over denominator weighted space mean space price space in space base space year end fraction cross times 100

      • The weighted index number for the base year is 100

      • See the 'Other Types of Mean' revision note for how to calculate a weighted mean

    • This formula is not on the exam formula sheet, so you need to remember it

      • The only difference from the basic index number formula is that it uses weighted mean prices

  • The retail price index (RPI) and consumer price index (CPI) are both weighted index numbers

    • The weights used change from year to year according to changes in consumer spending habits

Worked Example

Roger orders pizza from Lenny and John's Pizzeria as often as he can. He recorded all the pizzas he ordered in 2019 and 2023, along with the price for each variety.

Variety

Price (2019)

Number ordered (2019)

Price (2023)

Number ordered (2023)

Cheese

£13.60

12

£16.20

9

White

£19.10

5

£22.00

7

Sicilian

£14.60

3

£17.20

8

Grandma

£20.00

6

£22.80

4

(a) Calculate the weighted means for 2019 and 2023.

Use  weighted space mean equals fraction numerator sum for blank of open parentheses value cross times weight close parentheses over denominator sum for blank of weights end fraction
The weights will be the 'number ordered' for each type of pizza

2019 colon space space fraction numerator 13.60 cross times 12 plus 19.10 cross times 5 plus 14.60 cross times 3 plus 20.00 cross times 6 over denominator 12 plus 5 plus 3 plus 6 end fraction equals 16.25

2023 colon space space fraction numerator 16.20 cross times 9 plus 22.00 cross times 7 plus 17.20 cross times 8 plus 22.80 cross times 4 over denominator 9 plus 7 plus 8 plus 4 end fraction equals 18.8785...

It's money, so round the 2023 value to the nearest penny

2019 weighted mean = £16.25
2023 weighted mean = £18.88

(b) Work out the weighted index number for Roger's pizzas in 2023, taking 2019 as the base year. Give your answer correct to 1 decimal place.

Use  weighted space index space number equals fraction numerator current space weighted space mean space price over denominator mean space price space in space base space year end fraction cross times 100

fraction numerator 18.88 over denominator 16.25 end fraction cross times 100 equals 116.1846...

Round to 1 decimal place

116.2

Chain Base Index Numbers

How do I calculate a chain base index number?

  • A chain base index number (or chain based index number) is an index number comparing prices in one year with prices in the previous year

    • There is no fixed 'base year'

    • Instead each year is compared with the year before

  • Chain base index numbers are calculated using the formula

    • chain space base space index space number equals fraction numerator current space price over denominator previous space year apostrophe straight s space price end fraction cross times 100

    • This formula is not on the exam formula sheet, so you need to remember it

  • You can also calculate chain base index numbers to compare prices from month to month

    • To do this change the formula to chain space base space index space number equals fraction numerator current space price over denominator previous space month apostrophe straight s space price end fraction cross times 100

  • Similarly you can calculate chain base index numbers for other time periods

    • quarter to quarter, week to week, etc.

What else should I know abut chain base index numbers?

  • You can find percentage changes and calculate (or estimate) changed prices using chain base index numbers

    • This works exactly the same as for basic index numbers

      • Except you only look at changes for one year (or month, etc.) compared to the previous year (or month, etc.)

      • The previous year (or month, etc.) acts as the 'base year (or month, etc.)'

    • For example, if the chain base index number for June 2023 is 100.6 compared to May 2023

      • then the percentage increase from May to June is  100.6 minus 100 equals 0.6 percent sign

    • And if the price of the item being tracked was £35 in May 2023

      • then the price in June 2023 would be  35 cross times fraction numerator 100.6 over denominator 100 end fraction equals £ 35.21

  • The UK government also publishes chain base versions of the retail price index (RPI) and consumer price index (CPI)

    • Both year to year and month to month versions are published

    • The year to year versions are used to define the rate of inflation

      • This is the change in 'average' prices from one year to the next

      • Usually the CPI is used

        • but sometimes the RPI is used instead

  • Figures for the UK gross domestic product (GDP) are published every quarter

    • Quarter to quarter chain base index numbers are calculated for these

      • If there is a percentage increase (chain base index number greater than 100), then the economy is growing

      • If there is a percentage decrease (chain base index number less than 100), then the economy is shrinking

Examiner Tips and Tricks

  • Read questions carefully to see if a year to year or month to month chain base index number is needed

Worked Example

The table shows the prices for gold sold by a particular dealer for four months in 2020.

Month

April 2020

May 2020

June 2020

July 2020

Price of gold (£ per gram)

£42.61

£44.17

£44.58

£45.71

(a) Calculate the chain base index numbers for April 2020 to May 2020, and for June 2020 to July 2020. Give your answers correct to 1 decimal place.

These are month to month, so use  chain space base space index space number equals fraction numerator current space price over denominator previous space month apostrophe straight s space price end fraction cross times 100

April space to space May colon space space space fraction numerator 44.17 over denominator 42.61 end fraction cross times 100 equals 103.6611...

June space to space July colon space space space fraction numerator 45.71 over denominator 44.58 end fraction cross times 100 equals 102.5347...

Round to 1 decimal place

April 2020 to May 2020:   103.7
June 2020 to July 2020:   102.5

The chain base index number for March 2020 to April 2020 is 106.2.

(b) Find the price per gram that the dealer was selling gold for in March 2020.

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

fraction numerator 106.2 over denominator 100 end fraction equals 1.062

March 2020 is the 'base month' for the chain base index number
So divide the April 2020 price by that to find the March 2020 price

fraction numerator 42.61 over denominator 1.062 end fraction equals 40.1224...

This is money so round to the nearest penny

£40.12

Last updated:

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.