Index Numbers (AQA GCSE Statistics: Higher): 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

  • $\mathrm{index}\mathrm{number}=\frac{\mathrm{current}\mathrm{price}\mathrm{of}\mathrm{item}}{\mathrm{price}\mathrm{in}\mathrm{base}\mathrm{year}}\times 100$

  • 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
    $\frac{66}{36}\times 100=183.333...=183.3\left(1\mathrm{d}.\mathrm{p}.\right)$

• 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  $14.95\times 1.123=16.78885=\pounds 16.79$  (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  $14.71÷0.984=14.9491...=\pounds 14.95$

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  $37\times 3.78=139.86=140\mathrm{p}=\pounds 1.40$

  • 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  $80÷1.315=60.8365...=\pounds 60.84$

  • 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

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

  • $\mathrm{weighted}\mathrm{index}\mathrm{number}=\frac{\mathrm{current}\mathrm{weighted}\mathrm{mean}\mathrm{price}}{\mathrm{weighted}\mathrm{mean}\mathrm{price}\mathrm{in}\mathrm{base}\mathrm{year}}\times 100$

    • 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

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

  • $\mathrm{chain}\mathrm{base}\mathrm{index}\mathrm{number}=\frac{\mathrm{current}\mathrm{price}}{\mathrm{previous}\mathrm{year}\text{'}\mathrm{s}\mathrm{price}}\times 100$

  • 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 $\mathrm{chain}\mathrm{base}\mathrm{index}\mathrm{number}=\frac{\mathrm{current}\mathrm{price}}{\mathrm{previous}\mathrm{month}\text{'}\mathrm{s}\mathrm{price}}\times 100$

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

  • quarter to quarter, week to week, etc.

What else should I know about 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  $100.6-100=0.6\%$

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

    • then the price in June 2023 would be  $35\times \frac{100.6}{100}=\pounds 35.21$

• 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