Skewness by Inspection
Skewness by inspection means judging the shape of a data set, such as whether it is symmetrical or lopsided, just by looking at how the data or a graph of it is spread out, without doing a formal calculation. A distribution is positively skewed if it has a long tail of higher values, negatively skewed if it has a long tail of lower values, and symmetrical if both sides are balanced. In GCSE Statistics, comparing the mean, median and mode can help decide which type of skew a data set shows.
Before you start
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Method
- Look at a graph of the data, such as a histogram, frequency polygon or box plot, to see which side has the longer tail.
- If the longer tail points towards the higher values, the data is positively skewed.
- If the longer tail points towards the lower values, the data is negatively skewed.
- If both sides look roughly balanced, mirror images of each other, the data is symmetrical (no skew).
- Check this using the averages: if mean is greater than median which is greater than mode, the data is usually positively skewed; if mean is less than median which is less than mode, it is usually negatively skewed.
- State your conclusion clearly, referring to both the shape of the graph and the position of the averages.
Worked example
A class's test scores have mean 58, median 65 and mode 70. Describe the skewness of the distribution, giving a reason.
- Compare the three averages: mean = 58, median = 65, mode = 70.
- Order them from smallest to largest: mean is less than median, which is less than mode.
- This pattern indicates a longer tail towards the lower (left-hand) values.
- Final answer: the distribution is negatively skewed, because the mean is lower than the median, which is lower than the mode, showing a tail of lower scores pulling the mean down.
Practice questions
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Q1A distribution has a long tail of high values on the right. What type of skew is this?Show answer
Answer: Positive skew.
Q2A distribution has mean = mode = median. What does this suggest about its shape?Show answer
Answer: The distribution is symmetrical (no skew).
Q3A data set has mean 12, median 12, mode 12. Describe its skew.Show answer
Answer: Symmetrical / no skew, since all three averages are equal.
Q4A box plot shows a long whisker on the left and a short whisker on the right, with the median close to the right of the box. What type of skew does this suggest?Show answer
Answer: Negative skew.
Q5A shop's till receipts have mode 5 pounds, median 8 pounds, mean 15 pounds. State the type of skew and give a reason.Show answer
Answer: Positive skew, because mean is greater than median, which is greater than mode, showing a tail of high-value receipts pulling the mean up.
Q6Describe, using the mean and median, how you could tell that a distribution is positively skewed without drawing a graph.Show answer
Answer: If the mean is noticeably greater than the median, the distribution is likely positively skewed, since a few high values pull the mean above the median.
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A data set has mean 40, median 40 and mode 40. State the type of skewness shown, and explain your reasoning.
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The frequency polygon for the ages of visitors to a theme park has a long tail towards the older ages, with most visitors being young. (a) State the type of skew shown. (b) State whether you would expect the mean to be higher or lower than the median, and why.
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A survey records the amount of pocket money, in pounds, received weekly by 9 students: 5, 5, 5, 6, 6, 7, 8, 9, 40. (a) Calculate the mean, median and mode of this data set. (b) State, with a reason, the type of skew shown.
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Free printable worksheet
Want more practice on paper? Download the skewness by inspection worksheet pack - 15 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 1 of GCSE Statistics Foundation Workbook 2, the whole course as one free printable PDF.
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