Mean, median and mode, the five-number summary, quartiles and the interquartile range, drawing and reading a box-and-whisker plot, and describing the spread of data.
Also known as: five-number summary · box plot · box and whisker · interquartile range
Grade 10 statistics is less about calculating an average and more about describing a whole data set — where its centre is, how spread out it is, and whether it leans to one side.
1Measures of central tendency
- Mean = sum ÷ how many. Uses every value, but one outlier drags it badly.
- Median = the middle value once the data is ordered. Barely affected by outliers.
- Mode = the most frequent value or category. Of the three, it is the only one that still works for categories with no natural order, such as favourite colour.
You must order the data before finding the median or the quartiles. Forgetting to sort is the most common lost mark in this topic.
2The five-number summary
Q1 is the median of the lower half and Q3 the median of the upper half. If there is an odd number of values, leave the overall median out of both halves.
- 1Already ordered; there are 9 values, so the median is the 5th: Q₂ = 22.
- 2Lower half (12, 15, 18, 20): Q₁ = (15 + 18) ÷ 2 = 16,5.Exclude the median itself.
- 3Upper half (25, 28, 30, 35): Q₃ = (28 + 30) ÷ 2 = 29.
- 4Summary: 12 · 16,5 · 22 · 29 · 35.
3Interquartile range and box plots
- The box holds the middle 50% of the data — its width is the IQR.
- The whiskers reach out to the minimum and maximum.
- The line inside the box is the median, not the mean.
The IQR ignores the extremes entirely, which makes it a much fairer measure of spread than the range when the data contains an outlier.
4Looking ahead: Grade 11 — the shape of the data
Grade 11 adds a name for the shape a box plot shows. It is not examined in Grade 10, but it costs nothing to recognise it now:
- Symmetrical — the median sits in the middle of the box and the whiskers are about equal.
- Skewed to the right (positively) — the longer whisker is on the right; mean > median.
- Skewed to the left (negatively) — the longer whisker is on the left; mean < median.
Practice exercises
Work each one out, then click to reveal the answer.
- 1The mean of 23, 31, 27, 35, 29 and x is 29. Solve for x.
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(23 + 31 + 27 + 35 + 29 + x)/6 = 29 (1)
145 + x = 29 × 6 = 174 (1)
x = 174 − 145 = 29 (1) - 2Find the mean of 4, 8, 9, 11, 13.
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45 ÷ 5 = 9 - 3Find the median of 3, 7, 9, 12, 15, 18.
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(9 + 12) ÷ 2 = 10,5 - 4Find the mode of 2, 5, 5, 7, 9, 5.
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5 - 5Find the range of 12, 15, 18, 20, 35.
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35 − 12 = 23 - 6For 12, 15, 18, 20, 22, 25, 28, 30, 35 find Q₁, Q₂ and Q₃.
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Q₁ = 16,5 · Q₂ = 22 · Q₃ = 29 - 7Using the box plot below of that summary, find the IQR.
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29 − 16,5 = 12,5 - 8What does the line inside the box represent?
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The median (Q₂) - 9What fraction of the data lies inside the box?
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The middle 50% - 10A box plot has minimum 4, Q₁ = 9, median 14, Q₃ = 21 and maximum 30. Calculate the interquartile range, and state what fraction of the data lies inside the box.
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IQR = 21 − 9 = 12; the box holds the middle 50%
Quick Quiz
5 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 10 past papers and worksheets on this topic.