Statistics: Five-Number Summary & Box Plots (Grade 10)

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 and skewness of data.

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. The only average that works for categories.

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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

minimum · Q₁ (lower quartile) · Q₂ (median) · Q₃ (upper quartile) · maximum

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.

Worked ExampleFind the five-number summary of 12, 15, 18, 20, 22, 25, 28, 30, 35

1
Already ordered; there are 9 values, so the median is the 5th: Q₂ = 22.
2
Lower half (12, 15, 18, 20): Q₁ = (15 + 18) ÷ 2 = 16,5.
Exclude the median itself.
3
Upper half (25, 28, 30, 35): Q₃ = (28 + 30) ÷ 2 = 29.
4
Summary: 12 · 16,5 · 22 · 29 · 35.

3Interquartile range and box plots

IQR = Q₃ − Q₁ · Range = maximum − minimum

A box-and-whisker plot of that data. The box spans Q₁ to Q₃ (the middle 50%) with the median marked inside it.

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.

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The IQR ignores the extremes entirely, which makes it a much fairer measure of spread than the range when the data contains an outlier.

4Describing the shape

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.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Find the mean of 4, 8, 9, 11, 13.
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45 ÷ 5 = 9
2
Find the median of 3, 7, 9, 12, 15, 18.
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(9 + 12) ÷ 2 = 10,5
3
Find the mode of 2, 5, 5, 7, 9, 5.
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5
4
Find the range of 12, 15, 18, 20, 35.
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35 − 12 = 23
5
For 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
6
Using that summary, find the IQR.
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29 − 16,5 = 12,5
7
What does the line inside the box represent?
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The median (Q₂)
8
What fraction of the data lies inside the box?
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The middle 50%
9
A box plot has a much longer right whisker. Describe the skewness.
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Skewed to the right (positively skewed)