What is a box plot?
A box plot (or box-and-whisker diagram) shows the spread of a set of data using five key values, known as the five-number summary: the minimum, lower quartile (Q1), median, upper quartile (Q3), and maximum.
- The box stretches from Q1 to Q3, and represents the middle 50% of the data
- A line inside the box marks the median
- The whiskers extend from the box out to the minimum and maximum values
- The width of the box is the interquartile range (IQR) = Q3 − Q1
- The full width of the diagram, min to max, is the range
Box plot for the same runners' race times used in Cumulative Frequency: min 22, Q1 37.5, median 47, Q3 56, max 68 minutes.
Reading the five-number summary
Worked Example 1
Using the box plot above, state the five-number summary for the runners' race times.
1
The left end of the whisker gives the minimum: 22 minutes
2
The left edge of the box gives Q1 (37.5), the line inside gives the median (47), and the right edge gives Q3 (56)
3
The right end of the whisker gives the maximum: 68 minutes
AnswerMin 22, Q1 37.5, Median 47, Q3 56, Max 68
Range and interquartile range
Worked Example 2
Using the five-number summary from Worked Example 1, find the range and the interquartile range.
1
Range = maximum − minimum = 68 − 22 = 46 minutes
2
IQR = Q3 − Q1 = 56 − 37.5 = 18.5 minutes
3
The IQR is much smaller than the range, because it ignores the fastest and slowest runners and focuses on the middle 50%
AnswerRange = 46, IQR = 18.5
Constructing a box plot from raw data
Worked Example 3
Find the five-number summary for these 11 exam scores, already written in order: 34, 38, 45, 49, 52, 58, 63, 67, 70, 75, 82
1
Minimum = 34, Maximum = 82
2
Median = the middle (6th) value = 58
3
Q1 = median of the lower half (34, 38, 45, 49, 52) = 45. Q3 = median of the upper half (63, 67, 70, 75, 82) = 70
AnswerMin 34, Q1 45, Median 58, Q3 70, Max 82
Comparing distributions
Worked Example 4 — Comparing two box plots
The box plots show test scores for Class A and Class B. Compare the two classes' performance and consistency.
Class A: min 40, Q1 55, median 65, Q3 75, max 95. Class B: min 30, Q1 45, median 60, Q3 80, max 100.
1
Compare medians: Class A's median (65) is higher than Class B's (60), so Class A performed better on average
2
Compare IQRs: Class A's box is narrower (75 − 55 = 20) than Class B's (80 − 45 = 35), so Class A's scores were more consistent
AnswerClass A scored higher on average and was more consistent than Class B
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A box plot has minimum 12, Q1 20, median 28, Q3 35, maximum 48. Find the range.Foundation
Q2Using the same box plot, find the interquartile range.Foundation
Q3A box plot shows a median of 55. What percentage of the data lies below this value?Foundation
Q4True or false: the box in a box plot always represents 25% of the data. Explain your answer.Foundation
Q5A box plot has Q1 = 18 and Q3 = 30. State the width of the box, and explain what this value represents.Foundation
Higher (Grade 5–7)
Q6Nine ordered values: 5, 9, 12, 15, 18, 21, 25, 29, 33. Find the median.Higher
Q7Using the data from Q6, find Q1 (the median of the lower four values).Higher
Q8Using the data from Q6, find Q3 (the median of the upper four values).Higher
Q9Using your answers to Q7 and Q8, find the interquartile range for the data in Q6.Higher
Q10A box plot shows min 10, Q1 22, median 30, Q3 41, max 60. A student says the data is symmetric because the median is roughly in the middle of the range. Comment on whether this reasoning is correct, using the quartiles.Higher
Higher — Hard (Grade 8–9)
Q11Two box plots show delivery times (minutes). Company A: min 5, Q1 12, median 18, Q3 25, max 40. Company B: min 8, Q1 15, median 20, Q3 22, max 30. Compare the two companies' delivery times, referring to both the median and the IQR.Grade 8–9
Q12A box plot has an unusually long right-hand whisker compared to the left. What does this suggest about the skew of the data?Grade 8–9
Q13A data set has Q1 = 40, median = 48, Q3 = 52. Determine whether the data appears positively skewed, negatively skewed, or symmetric, giving a reason.Grade 8–9
Q14Explain why an outlier might not always be shown by an extended whisker in a simple box plot, and how some exam boards represent outliers differently.Grade 8–9
Q15Explain one advantage and one limitation of using a box plot to compare two distributions, rather than comparing their full cumulative frequency graphs.Grade 8–9
Answers
Foundation (Q1–Q5)
Q136(48 − 12)
Q215(35 − 20)
Q350%(the median always splits the data exactly in half)
Q4False(the box represents the middle 50% of the data, between Q1 and Q3)
Q512(30 − 18; this is the interquartile range — the spread of the middle 50% of the data)
Higher (Q6–Q10)
Q618(the 5th of 9 ordered values)
Q710.5(median of 5, 9, 12, 15 = (9+12) ÷ 2)
Q827(median of 21, 25, 29, 33 = (25+29) ÷ 2)
Q916.5(27 − 10.5)
Q10Not necessarily correct — symmetry depends on the quartiles being evenly spaced too. Here Q1 to median is 8 (22 to 30) but median to Q3 is 11 (30 to 41), and the whiskers are also uneven (12 vs 19), suggesting the data is not symmetric even though the median sits near the middle of the range
Higher — Hard (Q11–Q15)
Q11Company A had a faster median delivery time (18 vs 20 minutes), but Company B was more consistent (IQR 7 vs 13 minutes)
Q12Positive skew(a longer right whisker means a spread-out tail of higher values)
Q13Negatively skewed(Q1 to median = 8, but median to Q3 = only 4 — the lower half is more spread out)
Q14A simple box plot extends whiskers all the way to the true minimum and maximum, so one extreme value simply stretches a whisker rather than standing out. Some exam boards instead cap whiskers at 1.5 × IQR beyond the quartiles and plot any values further out as separate dots, marking them clearly as outliers
Q15Advantage: a box plot gives a quick, clear visual comparison of median and spread between two groups. Limitation: it hides the underlying shape of the distribution (e.g. it can't show if data is bimodal or has gaps), which a full cumulative frequency graph would reveal
Common mistakes
Common Mistake 1
Confusing the box with the whole data range
The box only covers the middle 50% of the data (Q1 to Q3). The full range — including the fastest/slowest, highest/lowest values — is shown by the whiskers, from minimum to maximum.
Common Mistake 2
Misreading the line inside the box as Q1 or Q3
The line inside the box is always the median, not a quartile. The edges of the box itself are Q1 (left/bottom) and Q3 (right/top).
Common Mistake 3
Finding Q1 and Q3 incorrectly for small data sets
For an odd number of values, exclude the median itself before splitting into lower and upper halves. Including it in both halves, or in only one, gives the wrong quartiles.
Common Mistake 4
Comparing only the medians when asked to compare distributions
A full comparison needs a comment on both the median (typical value) and the spread — either the range or, more usefully, the IQR (consistency). A one-sided comparison loses marks.
Exam tips
💡 Exam Tip 1
Draw box plots to scale, using a ruler
Plot each of the five values accurately against the axis before drawing the box and whiskers — a box plot that isn't to scale can lose accuracy marks even if the five-number summary is correct.
💡 Exam Tip 2
Always label your axis
Include units and a clear scale on the horizontal (or vertical) axis — a box plot without a labelled axis cannot be marked accurately, even if the shape is correct.
💡 Exam Tip 3
Use IQR rather than range when talking about consistency
The range is easily distorted by a single extreme value. When commenting on which data set is more consistent, the interquartile range is usually the stronger, more reliable statistic to quote.
💡 Exam Tip 4
Structure comparison answers in two clear parts
Write your comparison as two separate sentences: one about average (median) and one about spread (IQR or range) — examiners are looking for both points to award full marks.
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