MISSION · Data and probability
Quartiles and Box Plots
What a box plot shows that an average can't
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The idea in one sentence
Sort the data, cut it into four equal-sized groups, and the three cut points plus the two ends tell you nearly everything about the shape.
The five-number summary
- Minimum — the smallest value.
- Lower quartile (Q1) — a quarter of the way through the sorted data.
- Median (Q2) — the middle.
- Upper quartile (Q3) — three quarters of the way through.
- Maximum — the largest value.
Each gap between consecutive markers holds a quarter of the data. That’s the key to reading one: a wide section means those values are spread out, a narrow section means they’re bunched — and both hold the same number of data points.
A box plot draws this directly. The box runs from Q1 to Q3 with the median marked inside, and whiskers reach out to the extremes.
The interquartile range
\[IQR = Q3 - Q1\]The IQR is the width of the middle half of the data, and it’s the spread measure to use when the data has outliers. The full range uses the two most extreme values, so one freak result ruins it. The IQR throws away the top and bottom quarters entirely and can’t be moved by them.
That’s the same trade-off as the median against the mean: resistance to extremes, bought by ignoring some of the data.
Spotting outliers
The standard rule draws the line at one and a half IQRs beyond the box:
\[\text{below } Q1 - 1.5 \times IQR \quad \text{or} \quad \text{above } Q3 + 1.5 \times IQR\]Points past those fences are plotted individually, and the whiskers stop at the last value inside them. The 1.5 is a convention chosen to flag roughly the most extreme 1% of normally distributed data — not a law, but a widely agreed default.
Reading shape from the box
The position of the median inside the box tells you about skew:
- Median in the middle, whiskers similar — roughly symmetric.
- Median near the left, long right whisker — right-skewed, a tail of high values.
- Median near the right, long left whisker — left-skewed.
Box plots are at their best side by side. Four distributions on one axis, compared at a glance for both centre and spread — which is exactly what a bar chart of averages cannot show you, because it has thrown the spread away.
What it hides
A box plot can’t show you whether the data has two separate clusters. Two peaks with a gap between them produce a perfectly ordinary-looking box. If bimodality matters, plot the points or a histogram as well.
Common mix-ups
- Forgetting to sort first. Quartiles of unsorted data are meaningless.
- Reading a long section as more data. Every section holds a quarter, however wide.
- Using the range when outliers are present. That’s what the IQR is for.
- Extending whiskers to the outliers. They stop at the last value inside the fences.
Check yourself
For the data 3, 5, 6, 8, 9, 12, 15, find the median, the quartiles, and the IQR.
The median is 8. The lower half is 3, 5, 6, so Q1 = 5; the upper half is 9, 12, 15, so Q3 = 12. The IQR is \(12 - 5 = 7\).
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