Skewness

Types of skew

• Symmetrical: Q3 - Q2 = Q2 - Q1; mean = median = mode.
• Positive skew (long tail to the right): Q3 - Q2 > Q2 - Q1; usually mean > median > mode.
• Negative skew (long tail to the left): Q3 - Q2 < Q2 - Q1; usually mean < median < mode.

Justifying

Use one of these, with the actual values:

• quartiles: compare Q3 - Q2 with Q2 - Q1;
• mean and median (or mode): which is larger.

On a box plot, a longer right-hand part of the box (and whisker) suggests positive skew.

A skewness formula

Questions may give a measure such as 3(mean - median)/standard deviation, or (mean - mode)/standard deviation. Substitute and evaluate: a positive value means positive skew, negative means negative skew, and a value near 0 means roughly symmetrical. Use exactly the formula in the question; examiners note that using the median instead of the mode (or the other way round) loses marks.

Worked example

Q1 = 12, Q2 = 17, Q3 = 21. Describe the skewness.

1. Q3 - Q2 = 4.
2. Q2 - Q1 = 5.
3. 4 < 5

Answer: Negative skew, because Q3 - Q2 < Q2 - Q1.

Check yourself

1. Q1 = 12, Q2 = 17, Q3 = 21. What type of skew is shown?
   • Negative skew
   • Positive skew
   • No skew
   • It cannot be decided from quartiles
   Show answer

   Negative skew. Q3 - Q2 = 4 is less than Q2 - Q1 = 5, so negative skew.

2. A data set has mean 48 and median 41. What does this suggest?
   • Negative skew
   • A symmetrical distribution
   • Positive skew
   • The data contain no outliers
   Show answer

   Positive skew. Mean > median suggests a long tail of high values: positive skew.

3. A distribution has mean 22.4, median 20.5 and standard deviation 6.1. Evaluate 3(mean - median)/standard deviation. Give your answer to 3 significant figures.
   Show answer

   0.934. 3 × 1.9/6.1 = 0.934.

4. Mean 31, mode 35, standard deviation 4. Evaluate (mean - mode)/standard deviation.
   Show answer

   -1. (31 - 35)/4 = -1, which indicates negative skew.

5. Data are symmetrical with Q1 = 14 and Q3 = 26. Estimate the median.
   Show answer

   20. For symmetrical data the median is halfway between the quartiles: 20.

6. On a box plot, the median line is close to the left end of the box and the right whisker is long. What does this suggest?
   • Negative skew
   • Symmetry
   • A bimodal distribution
   • Positive skew
   Show answer

   Positive skew. The upper half of the data is more spread out, a long right tail.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1
• Pearson Edexcel IAL Statistics S1 (WST01) examiners' report, October 2024

Last updated 7 November 2026. Spotted a mistake? Tell us.