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Range and Interquartile Range

FoundationHigherAQAEdexcel

Learn Range and Interquartile Range for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. The range and interquartile range (IQR) measure spread; the IQR is less affected by extreme values.

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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.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The range and interquartile range both measure spread, but they behave very differently when outliers are present.

The range is the largest value minus the smallest. It is quick to calculate but uses only two values, so a single outlier can make it enormously large and unrepresentative of the data as a whole.

The interquartile range is the upper quartile minus the lower quartile, so it measures the spread of the middle 50 per cent of the data. Because it ignores the top and bottom quarters entirely, it is unaffected by outliers and is usually the better measure of spread for skewed data.

Revision notes

The range

Range \(=\) largest value \(-\) smallest value.

Quick and simple, but it uses only two values from the entire data set. A single outlier can make it very large and completely unrepresentative.

The interquartile range

IQR \(= Q_3 - Q_1\), the upper quartile minus the lower quartile.

It measures the spread of the middle 50 per cent of the data, ignoring the top and bottom quarters entirely. This makes it unaffected by outliers.

Choosing between them

Use the range for a quick measure of total spread on clean data.

Use the interquartile range when outliers are present or the data is skewed, because it represents the bulk of the data rather than its extremes.

Key points

  • Range = largest − smallest.
  • The range uses only two values.
  • One outlier can make the range unrepresentative.
  • IQR = upper quartile − lower quartile.
  • The IQR covers the middle 50 per cent.
  • The IQR is unaffected by outliers.

Worked examples

Example 1

A data set has a lower quartile of 12 and an upper quartile of 28. Work out the interquartile range. [2 marks]

Working

\(\text{IQR} = Q_3 - Q_1 = 28 - 12\)subtract the lower quartile from the upper quartile
\(= 16\)work out the interquartile range

Example 2

Explain why the interquartile range is often preferred to the range. [2 marks]

Working

The range uses only the largest and smallest values, so a single outlier can make it very largestate the weakness of the range
The interquartile range covers the middle 50 per cent and is unaffected by outliers, so it better represents the spread of most of the dataexplain the advantage

Example 3

Values are 4, 7, 9, 11 and 32. Work out the range. [2 marks]

Working

Largest \(= 32\), smallest \(= 4\)identify the largest and smallest values
\(32 - 4 = 28\)subtract to find the range

Common mistakes

  • Confusing the range with the interquartile range.

    The range uses the extremes; the IQR uses the quartiles.

  • Subtracting the quartiles the wrong way round.

    IQR is upper quartile minus lower quartile, so it is always positive.

  • Saying the range is unaffected by outliers.

    It is entirely determined by the two extreme values.

  • Giving the range as two numbers.

    It is a single value — the difference between the extremes.

Exam tips

  • Give the range as one number, not a pair.
  • Subtract lower from upper for the IQR.
  • Choose the IQR when outliers are present.
  • Explain the difference through which values each uses.

Key terms

Range
Largest value minus smallest value.
Interquartile range
Upper quartile minus lower quartile.
Spread
How dispersed the data values are.
Quartile
A value dividing ordered data into quarters.

Related topics

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