Box plots (also called box-and-whisker diagrams) are a Higher-tier topic that features prominently on AQA, Edexcel and OCR GCSE Maths papers. They provide a visual summary of data using five key values and are especially useful for comparing two distributions. You need to draw box plots, read values from them, calculate the interquartile range, and compare data sets using median, spread and skewness. This guide covers all these skills. For an overview of every topic, see our complete GCSE Maths topics list.
What Is a Box Plot?
A box plot displays a data set using the five-number summary:
- Minimum — the smallest value.
- Lower quartile (Q1) — the value one quarter of the way through the ordered data.
- Median (Q2) — the middle value.
- Upper quartile (Q3) — the value three quarters of the way through the ordered data.
- Maximum — the largest value.
Key Formulas
The IQR measures the spread of the middle 50% of the data and is not affected by outliers. A smaller IQR means the data is more consistent.
Skewness
- Positive skew — the median is closer to Q1, and the right whisker is longer.
- Negative skew — the median is closer to Q3, and the left whisker is longer.
- Symmetrical — the median is roughly central in the box.
Step-by-Step Method
- Order the data from smallest to largest (if not already done).
- Find the minimum and maximum.
- Find the median (Q2) — the middle value.
- Find Q1 — the median of the lower half of the data.
- Find Q3 — the median of the upper half of the data.
- Draw a number line with an appropriate scale.
- Draw the box from Q1 to Q3, with a line at the median.
- Draw whiskers from the box to the minimum and maximum.
Worked Example 1 — Foundation Level
Question: Draw a box plot for this data set: 4, 7, 9, 12, 15, 18, 21, 24, 28.
Working:
Data is already ordered. n = 9.
Minimum = 4. Maximum = 28.
Median (Q2) = 5th value = 15.
Lower half: 4, 7, 9, 12. Q1 = (7 + 9) ÷ 2 = 8.
Upper half: 18, 21, 24, 28. Q3 = (21 + 24) ÷ 2 = 22.5.
Draw a number line from 0 to 30. Box from 8 to 22.5 with a median line at 15. Whiskers to 4 and 28.
Answer: Five-number summary: 4, 8, 15, 22.5, 28.
Worked Example 2 — Higher Level
Question: Two box plots compare the test scores of Class A and Class B.
Class A: Min = 30, Q1 = 42, Median = 55, Q3 = 68, Max = 85.
Class B: Min = 40, Q1 = 50, Median = 60, Q3 = 66, Max = 78.
Compare the two distributions.
Working:
Class A: Median = 55, IQR = 68 − 42 = 26, Range = 85 − 30 = 55.
Class B: Median = 60, IQR = 66 − 50 = 16, Range = 78 − 40 = 38.
Answer: Class B has a higher median (60 vs 55), suggesting that, on average, Class B scored higher. Class B also has a smaller IQR (16 vs 26) and a smaller range (38 vs 55), meaning the scores in Class B were more consistent and less spread out than in Class A.
Worked Example 3 — Exam Style
Question: From a box plot, read off: Min = 12, Q1 = 20, Median = 35, Q3 = 42, Max = 50. (a) Find the IQR. (b) Find the range. (c) Describe the skewness.
Working:
(a) IQR = Q3 − Q1 = 42 − 20 = 22.
(b) Range = 50 − 12 = 38.
(c) The median (35) is closer to Q3 (42) than to Q1 (20). The left whisker (12 to 20 = 8) is shorter than the right whisker (42 to 50 = 8) — they are actually equal. However, within the box, the median is to the right of centre (distance from Q1 to median is 15, from median to Q3 is 7). This suggests negative skew — the data is bunched towards the higher values.
Answer: (a) IQR = 22. (b) Range = 38. (c) Negative skew — the median is closer to Q3 within the box.
Common Mistakes
- Confusing Q1 and Q3 with the minimum and maximum. The box shows Q1 to Q3; the whiskers extend to the min and max.
- Not ordering data before finding quartiles. Quartiles only work on ordered data.
- Using range instead of IQR for comparison. The IQR is usually more useful because it ignores outliers and focuses on the central 50%.
- Only comparing one measure. When comparing distributions, always comment on both an average (median) and a measure of spread (IQR or range).
Exam Tips
- When comparing box plots, always make two comparison points: one about the median (average) and one about the IQR or range (spread). Use context — for example, "Class B scored higher on average and their marks were more consistent."
- Read box plot values carefully from the scale — misreading by one division is a common error.
- If asked about skewness, look at where the median sits within the box and compare whisker lengths.
- For related cumulative frequency, see cumulative frequency and box plots. For key formulas, visit our GCSE Maths formulas page.
Practice Questions
Q1 (Foundation): A box plot has Q1 = 15 and Q3 = 35. What is the IQR?
Q2 (Foundation): From a five-number summary of 10, 18, 25, 33, 45, state the median and the range.
Q3 (Higher): Box plot A has median 50, IQR 12. Box plot B has median 45, IQR 20. Compare the two distributions.
Practise drawing and comparing box plots free on GCSEMathsAI.
Related Topics
Summary
- A box plot displays data using five values: minimum, Q1, median, Q3 and maximum.
- The interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of data.
- When comparing two box plots, comment on both the median (average) and the IQR or range (spread), and relate your answer to the context.
- Skewness is identified by the position of the median within the box and the relative lengths of the whiskers.
- Box plots are drawn on a scaled number line with a box from Q1 to Q3 and whiskers to the minimum and maximum.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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