Skewness
Skewness is a measure of how asymmetrical a data distribution is, that is, whether values are bunched towards one end with a longer tail stretching towards the other. A distribution is positively skewed when mean is greater than median and negatively skewed when mean is less than median; it is symmetrical when neither tail dominates. GCSE Statistics Higher tier judges skewness using summary statistics and diagrams.
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Method
- Compare the mean and median: if mean is greater than median the data is positively skewed; if mean is less than median it is negatively skewed; if they are close, the data is roughly symmetrical.
- Alternatively, compare the two halves of the interquartile range: find Q2 - Q1 (spread of the lower half) and Q3 - Q2 (spread of the upper half).
- If Q3 - Q2 is greater than Q2 - Q1, the data is positively skewed; if Q3 - Q2 is less than Q2 - Q1, it is negatively skewed; if they are equal, the distribution is symmetrical.
- On a box plot, check whether the box and whisker sections are longer on the right of the median (positive skew) or the left (negative skew).
- On a histogram, check which side has the longer tail: a tail stretching to the right means positive skew, a tail stretching to the left means negative skew.
- State the type of skew clearly (positive, negative or symmetrical), referencing the evidence used.
Worked example
A set of exam marks has minimum 20, lower quartile 45, median 58, upper quartile 64, maximum 70, and mean 54. Describe the skewness of the distribution.
- Compare the mean and median: mean = 54, median = 58, so mean is less than median.
- Compare the two halves of the interquartile range: Q2 - Q1 = 58 - 45 = 13 and Q3 - Q2 = 64 - 58 = 6.
- Since Q3 - Q2 (6) is less than Q2 - Q1 (13), the upper half of the data is less spread out than the lower half, which indicates negative skew.
- Both methods agree: the mean-median comparison and the quartile comparison both point to negative skew.
- Final answer: the distribution of exam marks is negatively skewed.
Practice questions
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Q1A distribution has mean 40 and median 40. What type of skew does this suggest?Show answer
Answer: Symmetrical (no skew), since the mean equals the median
Q2A distribution has mean 62 and median 55. What type of skew does this suggest?Show answer
Answer: Positive skew, since mean (62) is greater than median (55)
Q3A data set has Q1 = 10, median = 18, Q3 = 22. Compare the two halves of the interquartile range and state the type of skew.Show answer
Answer: Negative skew (Q2 - Q1 = 8 is greater than Q3 - Q2 = 4, so the lower half is more spread out)
Q4A box plot has a short whisker on the left of the box and a long whisker on the right, with the median line close to the left edge of the box. What type of skew is this?Show answer
Answer: Positive skew (the longer tail/whisker is on the right)
Q5A data set has minimum 5, Q1 = 12, median 15, Q3 = 30, maximum 34, and mean 18.5. Using both the mean-median comparison and the quartile comparison, describe the skewness.Show answer
Answer: Positively skewed (mean 18.5 is greater than median 15; Q3 - Q2 = 15 is greater than Q2 - Q1 = 3, so both methods agree)
Q6Class X has mean 68 and median 70. Class Y has mean 70 and median 68. Compare the skewness of the two classes' test score distributions.Show answer
Answer: Class X is negatively skewed (mean less than median); Class Y is positively skewed (mean greater than median), so they are skewed in opposite directions
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A distribution has mean 33 and median 41. State, with a reason, the type of skew shown.
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A data set of delivery times has lower quartile 8 minutes, median 13 minutes, upper quartile 15 minutes. (a) Calculate Q2 - Q1 and Q3 - Q2. (b) State, with a reason, the type of skew shown by these quartiles.
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The times, in minutes, taken by a group of runners to finish a race have mean 52, median 47, lower quartile 40 and upper quartile 50. (a) State whether the mean-median comparison suggests positive or negative skew, giving a reason. (b) Calculate Q2 - Q1 and Q3 - Q2 and state whether this comparison agrees with part (a). (c) Give one reason the two methods might disagree for a real data set.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the skewness worksheet pack - 18 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 5 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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