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Weighted Index Numbers

HigherHigher tier onlyAQAEdexcel

Understand Weighted Index Numbers for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. Weighted index numbers give items different importance according to how much they are used.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

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Topic overview

A weighted index number combines several index numbers into one, giving each a weight reflecting its importance. This is a Higher-only topic.

The formula is \(\frac{\sum wI}{\sum w}\), where \(I\) is each index number and \(w\) its weight. Multiply each index by its weight, add the products, then divide by the total of the weights.

This is how the RPI and CPI are constructed. Items making up a larger share of household spending receive larger weights, so their price changes influence the overall figure more. Dividing by the number of items rather than the total weight is the standard error and produces a result that is far too large.

Revision notes

The formula

Weighted index \(= \frac{\sum wI}{\sum w}\), where \(I\) is each index and \(w\) its weight.

Multiply each index by its weight to get \(wI\), add the products, then divide by the sum of the weights — not by the number of items.

Choosing the weights

Weights reflect the relative importance of each component.

For a price index, they reflect the share of spending on each item. A component with weight 5 influences the result five times as much as one with weight 1.

Interpreting the result

The weighted index is interpreted like any index number.

Above 100 means an overall increase, below 100 an overall decrease. Subtract 100 to get the overall percentage change.

Key points

  • A weighted index combines several indices.
  • \(\text{Weighted index} = \frac{\sum wI}{\sum w}\).
  • Multiply each index by its weight.
  • Divide by the total weight, not the count.
  • Weights reflect relative importance.
  • Interpret the result like any index number.

Worked examples

Example 1

Indices 120 and 105 have weights 3 and 2. Work out the weighted index. [3 marks]

Working

\(120 \times 3 = 360\) and \(105 \times 2 = 210\)multiply each index by its weight
\(\sum wI = 570\) and \(\sum w = 5\)find both totals
\(\frac{570}{5} = 114\)divide to find the weighted index

Example 2

A weighted index is 108. Interpret this. [2 marks]

Working

\(108 - 100 = 8\)subtract 100 from the index
so there has been an overall increase of 8 per cent since the base yearstate the interpretation

Example 3

Explain why weights are used when combining index numbers. [2 marks]

Working

Some components are more important than others, contributing a larger share of the totalstate the reason
so weighting ensures their price changes influence the overall index proportionallyexplain the effect

Common mistakes

  • Dividing by the number of indices.

    Divide by the sum of the weights.

  • Adding the weights to the indices.

    The weights multiply the indices.

  • Forgetting to interpret the result.

    Subtract 100 to give the percentage change.

  • Using equal weights when the question gives different ones.

    Read the weights carefully from the question.

Exam tips

  • Divide by the total weight every time.
  • Show the wI products as a separate step.
  • Check the result lies between the smallest and largest index.
  • Remember this is a Higher-only topic.

Key terms

Weighted index
An index combining several indices with weights.
Weight
A number reflecting relative importance.
\(\sum wI\)
The total of index multiplied by weight.
Component
One of the indices being combined.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.