Box Plots
Two classes sit the same test. Class A's marks and Class B's marks are each a long list of
numbers — hard to compare at a glance. But squeeze each class down to five key
numbers and draw them as a little box with two whiskers, and suddenly you can see
everything at once: who scored higher on average, and whose marks were more spread out.
That compact picture is a box plot.
A single average hides a lot — two classes can have the same mean yet look completely
different. A box plot refuses to hide it. Stack two box plots above one number line and the
differences in typical value and in consistency jump straight out. That is
exactly why scientists reach for them when comparing groups.
The five-number summary as a picture
A box plot (also called a box-and-whisker plot) draws five
numbers on a single scale: the minimum, the lower quartile
(Q_1), the median (Q_2),
the upper quartile (Q_3) and the
maximum.
The box stretches from Q_1 to
Q_3, so its width is the
interquartile range.
A line drawn inside the box marks the median, and the two whiskers reach out
to the smallest and largest values.
At a glance the box shows where the central half of the data sits, while the whiskers show how
far the tails stretch.
Reading a box plot
For a data set with five-number summary min, Q_1, median,
Q_3, max:
- the box runs from Q_1 to Q_3, with the median marked by a line inside it;
- the box's width is the interquartile range, \text{IQR} = Q_3 - Q_1;
- the whiskers reach out to the two extremes — the minimum and the maximum;
- the full range is \text{max} - \text{min};
- box plots make it easy to compare two data sets' spread and average side by side.
Worked example — draw a box plot from five numbers
A test out of 50 gives the five-number summary
\text{min} = 12, Q_1 = 22,
\text{median} = 30, Q_3 = 38,
\text{max} = 48. To draw it:
- draw a number line covering at least 12 to 48;
- draw a box from 22 to 38;
- draw the median line inside the box at 30;
- draw whiskers from the box out to 12 and 48.
Read straight off: the interquartile range is
\text{IQR} = 38 - 22 = 16 and the full range is
48 - 12 = 36. Notice the median (30) sits almost exactly in the
middle of the box, so this data is fairly symmetric.
Worked example — comparing two groups
Here are two classes' test marks (out of 100), drawn on one shared scale so we can line them
up directly.
-
Average (median): Class B's median (70) sits to
the right of Class A's (65), so Class B typically scored higher.
-
Spread (IQR): Class A's box is narrow —
\text{IQR} = 72 - 55 = 17 — while Class B's is wide —
\text{IQR} = 85 - 50 = 35. Class A is far more
consistent.
So the honest comparison is: "Class B scored higher on average, but Class A's marks were much
more consistent." Comparing both the middle and the spread tells the real story — a
single average would have hidden half of it.
-
The box is the MIDDLE 50%, not the whole data. Everything between
Q_1 and Q_3 — the central half of the
values — lives inside the box; a quarter of the data hides in each whisker.
-
The median line is usually NOT in the centre of the box. If it is pushed
toward one end, that is telling you the data is skewed — real, useful
information, not a wonky drawing. A median near Q_1 means the
low values are bunched up and the high ones stretched out.
-
When you compare groups, compare BOTH the median AND the spread. "Which is
higher on average?" uses the median; "which is more consistent?" uses the IQR or range.
Reporting only one is half an answer.
The box plot was invented by the American statistician John Tukey in the
1970s — the same restless mind behind the stem-and-leaf plot
and the word "software". His goal was exploratory: a quick sketch you could draw by
hand to see the shape of data before doing any heavy maths.
It became the go-to tool for comparing distributions across real science — test scores between
schools, rainfall between cities, reaction times between treatment groups. Two box plots
stacked together instantly reveal differences in both typical value and consistency
that a lone average would smother — which is exactly what you need when
comparing distributions.
See it explained