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Skewness

HigherHigher tier onlyAQAEdexcel

Learn Skewness for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. Skewness describes whether a distribution's tail is longer on the higher or lower side of the average.

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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.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Skewness describes the shape of a distribution and whether it is symmetrical or has a longer tail in one direction. This is a Higher-only topic.

A symmetrical distribution has the mean, median and mode all equal. Positive skew has a longer tail to the right, and the mean is greater than the median, which is greater than the mode. Negative skew has a longer tail to the left, and the order reverses: mean less than median less than mode.

Skewness can be measured using \(\frac{3(\text{mean} - \text{median})}{\text{standard deviation}}\). A positive result indicates positive skew, a negative result negative skew, and a result near zero indicates a roughly symmetrical distribution.

Revision notes

Recognising skew

Symmetrical: mean \(=\) median \(=\) mode.

Positive skew: longer tail to the right, and mean \(>\) median \(>\) mode. Negative skew: longer tail to the left, and mean \(<\) median \(<\) mode.

Why the mean moves

The mean is pulled towards the tail because it uses every value.

The median is less affected and the mode not at all, which is why their order reveals the direction of skew. Remembering that the mean follows the tail makes the order easy to reconstruct.

Measuring skewness

Skewness \(= \frac{3(\text{mean} - \text{median})}{\text{standard deviation}}\).

A positive value indicates positive skew, negative indicates negative skew, and a value near zero indicates approximate symmetry. From a box plot, a median nearer the left edge suggests positive skew.

Key points

  • Symmetrical distributions have equal averages.
  • Positive skew has a longer right tail.
  • Positive skew gives mean > median > mode.
  • Negative skew has a longer left tail.
  • The mean is pulled towards the tail.
  • Skewness = 3(mean − median) ÷ standard deviation.

Worked examples

Example 1

A distribution has mean 52, median 48 and standard deviation 12. Work out the skewness. [3 marks]

Working

\(3(52 - 48) = 3 \times 4 = 12\)work out three times the difference
\(\frac{12}{12}\)divide by the standard deviation
\(= 1\), which is positive, so the distribution has positive skewstate the value and interpret it

Example 2

A distribution has mean 30 and median 35. State the type of skew. [2 marks]

Working

The mean is less than the mediancompare the two averages
so the distribution has negative skewstate the type of skew

Example 3

Explain why the mean is pulled towards the tail of a skewed distribution. [2 marks]

Working

The mean uses every value in the data setstate the property
so the extreme values in the tail pull it in that direction, whereas the median depends only on positionexplain the effect

Common mistakes

  • Reversing the skew directions.

    Positive skew has a longer RIGHT tail, with mean greater than median.

  • Forgetting the factor of 3 in the formula.

    It is 3(mean − median) divided by the standard deviation.

  • Not interpreting the sign.

    State whether the result indicates positive or negative skew.

  • Saying the mode is pulled towards the tail.

    The mode is unaffected; it is the mean that moves.

Exam tips

  • Remember the mean follows the tail.
  • Include the factor of 3 in the formula.
  • Always interpret the sign of your answer.
  • Remember this is a Higher-only topic.

Key terms

Skewness
A measure of asymmetry in a distribution.
Positive skew
A distribution with a longer right tail.
Negative skew
A distribution with a longer left tail.
Symmetrical
Having equal mean, median and mode.

Related topics

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