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Comparing Distributions

FoundationHigherAQAEdexcel

Practise Comparing Distributions for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Distributions are compared using a measure of average and a measure of spread, always in context.

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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

Comparing distributions requires two things: a comparison of an average and a comparison of the spread. Giving only one earns half the marks at most.

The average shows which data set has generally higher values. Use the median for skewed data or where outliers are present, and the mean for symmetrical data. The spread shows how consistent the values are — a smaller spread means greater consistency.

Both comparisons must be written in context rather than as bare numbers. Saying "the median is higher" is weaker than "the median mark for Class A is higher, so Class A generally scored better". Interpreting what the comparison means for the situation is where the marks lie.

Revision notes

The two required comparisons

One average — the median or the mean — showing which data set has generally higher values.

One measure of spread — the interquartile range, range or standard deviation — showing which is more consistent. Both are needed for full marks.

Choosing which statistics

Use the median with the interquartile range when the data is skewed or contains outliers.

Use the mean with the standard deviation for symmetrical data with no outliers. Keep the pair consistent — median with IQR, mean with standard deviation.

Writing in context

Refer to the actual variables rather than to abstract statistics.

Not "the median is higher" but "the median mark for Class A is higher, so Class A generally scored better". A smaller spread means the values are more consistent, which should also be stated in context.

Key points

  • Compare an average and a measure of spread.
  • Both comparisons are needed for full marks.
  • Use median with interquartile range for skewed data.
  • Use mean with standard deviation for symmetrical data.
  • A smaller spread means greater consistency.
  • Write both comparisons in context.

Worked examples

Example 1

Class A has median 62 and IQR 14. Class B has median 55 and IQR 9. Compare the two classes. [2 marks]

Working

Class A has the higher median, so Class A generally scored bettercompare the averages in context
Class B has the smaller interquartile range, so Class B's marks were more consistentcompare the spreads in context

Example 2

Explain why comparing only the medians is insufficient. [2 marks]

Working

The median shows which data set has generally higher values but says nothing about consistencystate the limitation
so a measure of spread is also needed to describe how the values are distributedexplain what is missing

Example 3

Two data sets have equal medians but different interquartile ranges. State what this tells you. [2 marks]

Working

Both data sets have the same typical valueinterpret the equal medians
but the one with the smaller interquartile range has more consistent valuesinterpret the different spreads

Common mistakes

  • Comparing only the average.

    A measure of spread is also required.

  • Giving bare numbers without context.

    Say what the comparison means for the situation described.

  • Mixing the median with the standard deviation.

    Pair median with IQR, and mean with standard deviation.

  • Saying a larger spread is better.

    A smaller spread means greater consistency, which is usually preferable.

Exam tips

  • Always make two comparisons — average and spread.
  • Write both in the context of the question.
  • Keep the statistic pairs consistent.
  • Interpret a smaller spread as greater consistency.

Key terms

Distribution
The pattern of how data values are spread.
Consistency
How close together the values are.
In context
Referring to the actual variables in the question.
Measure of spread
A statistic such as range, IQR or standard deviation.

Related topics

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