Home / Analysis of Data / Box Plots
3.1c · Analysis of Data · Sub-skill
The five-number summary and how to build, read and compare box-and-whisker plots.
Build it up, step by step
Click each step below to reveal it — work through them in order the first time round.
A box plot is built from five values: the minimum, lower quartile ($Q_1$), median, upper quartile ($Q_3$), and maximum. Together these five numbers summarise the whole distribution of a data set in one compact picture.
For a list of $n$ ordered values, the median splits the data in half; $Q_1$ is the median of the lower half, and $Q_3$ is the median of the upper half. For grouped data (from a cumulative frequency graph), read off the value at $\frac{n}{4}$ for $Q_1$, $\frac{n}{2}$ for the median, and $\frac{3n}{4}$ for $Q_3$ on the horizontal axis.
The ‘box’ runs from $Q_1$ to $Q_3$ (this is the interquartile range, IQR), with a line inside at the median. ‘Whiskers’ extend out to the minimum and maximum — unless a value is flagged as an outlier (more than $1.5\times IQR$ beyond $Q_1$ or $Q_3$), in which case the whisker stops at the last non-outlier value and the outlier is marked separately with a cross or dot.
When comparing box plots, always compare both position (medians — which data set is typically higher/lower) and spread (IQR/box width, and overall range — which is more consistent). Also comment on skew: if the median is off-centre in the box, or one whisker is much longer than the other, the data is skewed in that direction. Always write your comparison in context.
Two box plots show the times (minutes) for Route A and Route B to complete a fun run. Route A: min 18, Q1 22, median 25, Q3 29, max 40. Route B: min 20, Q1 24, median 26, Q3 28, max 33. Compare the two routes.
Route A has a slightly lower median (25 vs 26), so runners were typically a little faster on Route A. However, Route A has a much larger IQR (29−22=7 vs 28−24=4) and a wider overall range (40−18=22 vs 33−20=13), so times on Route A were far more variable/inconsistent than on Route B, which had a more consistent spread of times.
Test yourself
A gym records the resting heart rate (bpm) of members before and after a 6-week fitness programme, shown as box plots.
Before: min 58, Q1 68, median 74, Q3 82, max 96.
After: min 54, Q1 62, median 66, Q3 70, max 80.
Using the box plots, compare the heart rates before and after the programme. [4 marks]
The median heart rate is lower after the programme (66 bpm) than before (74 bpm), suggesting the programme has, on average, reduced members' resting heart rate.
The IQR is also smaller after the programme (70−62 = 8) than before (82−68 = 14), so heart rates became more consistent across members after the programme, as well as generally lower.
Practice
Five short questions on box plots. Work through them, then reveal the mark scheme to check.