Box Plots and Comparing Distributions
A box plot summarises the centre, spread and extremes of a numerical dataset on one scale. Use it to compare distributions, while remembering that a summary hides individual values.
Higher. Use quartiles and interquartile range and averages and spread.
Understand the five-number summary
A standard GCSE box plot uses the minimum, lower quartile Q1, median, upper quartile Q3 and maximum. The box runs from Q1 to Q3; a line marks the median. Whiskers extend to the minimum and maximum.
| Summary | Time (min) |
|---|---|
| Minimum | 4 |
| Q1 | 7 |
| Median | 11 |
| Q3 | 16 |
| Maximum | 24 |
Worked example — construct a box plot
These values come from the ordered times 4,6,8,10,12,14,18,24 using medians of halves. Draw a labelled uniform time scale. Draw the box from 7 to 16, its median line at 11, and whiskers to 4 and 24.
IQR = 16 − 7 = 9 min. Range = 24 − 4 = 20 min. Height of the box is only a drawing choice; its horizontal positions carry the statistical meaning.
Do not space the five values equally unless their numerical gaps are equal. A box plot is not divided into four equal-length sections.
Read proportions without mistaking them for widths
Quartiles mark approximately 25%, 50% and 75% of the ordered distribution. The box contains the middle half. A short section indicates that its quarter of observations occupies a narrow range of values.
Worked example — interpret the box
Between Q1 = 7 and Q3 = 16 lies the middle half of the journey times. Between the median 11 and Q3 = 16 lies roughly a quarter. The longer upper whisker does not mean more observations than the shorter lower whisker.
For a large sample of 80, the middle half represents about 40 observations. Repeated values at a quartile, small samples and the chosen quartile convention can affect exact boundary counts. Avoid treating a diagram as proof of exact counts at a repeated value.
A box plot normally does not show sample size, mode, mean, gaps or each individual value. You cannot recover these reliably from its five summary positions alone.
Compare centre and spread on a common scale
Put both box plots on the same numerical scale with the same units. Compare median with median for typical values and IQR with IQR for the middle-half spread. Explain the consequence in context.
| Measure | A | B |
|---|---|---|
| Median | 24 | 22 |
| IQR | 12 | 6 |
| Range | 34 | 30 |
Worked example — write a supported comparison
Route B is typically quicker: its median is 22 min, 2 min below A’s 24 min. B is more consistent for the middle half: its IQR is 6 min, half A’s 12 min.
The ranges also favour B: 30 min versus 34 min. However, those ranges depend on just the extremes. A’s lower quartile is 18 min, below B’s 20 min, so do not claim every A journey is slower.
A smaller spread means more consistency, not automatically better performance. For test scores, a higher median may be desirable; for journey times, a lower median may be desirable. State the context.
Use grouped estimates and recognise limits
You can construct a box plot from a cumulative-frequency graph: read Q1, median and Q3, then use the supplied minimum and maximum. If only grouped endpoints are available, the extreme values are estimates or bounds, not known individual observations.
Worked example — transfer the journey estimates
The grouped journey example gives Q1 ≈ 15, median ≈ 25 and Q3 ≈ 37.5 min. If a question asks you to use class endpoints 0 and 60 as estimated extremes, label the resulting summary as estimated. The grouped table alone does not establish that any journey took exactly 0 or 60 minutes.
Before generalising a comparison, consider how the samples were collected and whether they represent comparable populations or conditions. A box plot from a biased sample does not become reliable simply because the drawing is accurate.
This course uses the standard five-number GCSE box plot. Formal outlier fences and plots with special outlier symbols are not needed for these GCSE Mathematics specifications.
Think Like a Mathematician
Two datasets can have the same five-number summary but different individual values. A box plot is valuable precisely because it compresses data; that compression also limits what it can tell you.
Common Mistakes
Equal spacing: use a uniform number scale, not equally spaced summary marks.
Whisker length as count: length shows a range of values.
Comparing unlike summaries: compare the same measure and units.
Unsupported “better”: explain typical value and spread separately in context.
Practice
1. A dataset has minimum 3, Q1=8, median15, Q3=21 and maximum30. State where to draw the box, median and whiskers.
Box from 8 to 21, median line at 15, whiskers to 3 and 30, on a labelled uniform scale.
2. For that summary, calculate the IQR and range.
IQR = 21 − 8 = 13; range = 30 − 3 = 27.
3. About how many of 120 observations lie in the middle half?
0.5 × 120 = 60. Exact boundary counts may need care with repeated values.
4. In the route comparison, which is typically quicker and by how much?
Route B: median22 min compared with24 min, so its typical time is 2 min lower.
5. Compare the middle-half consistency of those routes.
B has IQR26 − 20 = 6 min; A has30 − 18 = 12 min. B is more consistent for the middle half.
6. A longer upper whisker is said to contain more observations than the box. Explain the error.
The upper whisker spans roughly the upper quarter, while the box spans the middle half. Horizontal length measures spread, not frequency.
7. Can the mean or sample size be read exactly from an ordinary box plot alone?
No. Its five-number summary does not normally specify either.
8. Test group A has median68 and IQR18; B has median72 and IQR24. Compare both features.
B scores higher typically by 4 marks. A is more consistent for the middle half because its IQR is 6 marks smaller. Neither statement alone proves one group is better in every respect.
Frequently Asked Questions
Do the whiskers always end at outlier cutoffs?
In the standard five-number box plots taught here, they end at the minimum and maximum. Follow any different explicit convention if a question provides one.
Is IQR always preferable to range?
They describe different things. IQR focuses on the middle half and is less affected by extreme values; range shows the full observed span. Choose a measure that answers the question.