Common mistake 1 with Box plot

Treating the whiskers as a confidence interval. They are a spread of the data, not of the estimate.

Common mistake 2 with Box plot

Boxes for very small groups, or for a variable with only a few distinct values.

Common mistake 3 with Box plot

Reading a box plot of a bimodal variable as though it were unimodal - the median lands in the valley between the humps, where no observations are.

Common mistake 4 with Box plot

Using boxes where the individual points would fit. With fewer than about twenty observations per group, showing the points is strictly more informative and no harder to read.

EconStatsCraft › Box plot

Box plot

Several distributions side by side, each reduced to its median, quartiles and outliers. The chart to reach for when the question is whether groups differ in spread as well as in level.

What it needs

Variable

The intuition

A box plot compresses a distribution into five numbers - minimum, lower quartile, median, upper quartile, maximum - with outliers marked separately. It is unbeatable for comparing many groups at once, because five numbers per group stack neatly where five histograms would not. What it cannot show is the one thing a histogram can: whether the distribution has one hump or two.

When to use it

Whenever the question involves groups and the answer might be about spread rather than level. Two groups can share a mean and be entirely different distributions.

Where the estimator comes from

The box spans the interquartile range [Q_1,Q_3] with the median marked inside. Whiskers extend to the most extreme observations within Q_1-1.5\,\mathrm{IQR} and Q_3+1.5\,\mathrm{IQR}, and anything beyond is drawn individually. The 1.5 is Tukey's convention, chosen so that for Normal data about 0.7% of observations fall outside - roughly 1 in 150. It follows that in a sample of 1,000 Normal observations, about seven points will be flagged as outliers by construction, and none of them is anomalous.

What it assumes

Each group has enough observations for quartiles to mean anything. A box drawn on six points is decoration.
The five-number summary is enough. It hides bimodality completely, which is why the violin variant exists.
The groups are comparable in what they measure, not merely in units. Boxes side by side invite a comparison that assumes the same variable was collected the same way in each.
The reader knows which whisker convention is in use. The 1.5 IQR rule, the 5th-95th percentile and min-max all produce boxes that look identical and mean different things.

Statistical properties

A flagged outlier is not necessarily an error. At the 1.5 IQR rule roughly 0.7% of Normal data is flagged, so in a large clean sample there will always be some.
The box plot cannot show bimodality. Two distributions - one with a single central hump, one with two humps at the quartiles - can produce identical boxes. A violin plot or a strip plot alongside solves it.
It hides the sample size completely. A box drawn from 6 observations looks as authoritative as one from 6,000, so n should be printed or encoded in the box width.
The median, not the mean, is the line inside the box. On skewed data these differ substantially and readers frequently assume the mean.
Whisker conventions vary between packages - 1.5 IQR, min/max, or the 5th and 95th percentiles - so the caption must say which is drawn or the plot is ambiguous.
For comparing many groups it is the most efficient chart available: twenty boxes are readable where twenty histograms are not.

How to read the output

Compare medians for level and box heights for spread. A group with a similar median and a much taller box is a different phenomenon, not a noisier measurement of the same one.
Overlapping boxes do not mean "no difference" and non-overlapping boxes are not a test. Run the test.
The outliers are data, not errors. In economics they are often the interesting observations - the megacity, the outlier firm.
If a box looks symmetric but the violin shows two humps, believe the violin.
The box shows the middle half and nothing about how the rest is distributed. Two groups with identical boxes can have very different tails, which is often the interesting part.
Width carries no information unless it has been set to encode sample size, and the module says whether it has.
A median line near one end of the box is skewness, and it is the most reliable thing a box plot tells you.

A worked reading

Take 1,000 draws from a clean Normal distribution and the box plot will mark about seven points as outliers. Nothing is wrong with them: the 1.5 IQR rule flags roughly 0.7% of Normal data by design, so the "outliers" are the expected tail of a well-behaved sample. Treating flagged points as errors to be removed - which happens routinely - deletes the tail of the distribution and biases every subsequent variance estimate downward.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Many points flagged as outliers	Expected at 1.5 IQR, or a heavy-tailed variable	Not automatically errors
A box with no whisker on one side	The quartile equals the extreme	Common with small n or bounded data
Two groups with identical boxes	The shapes may still differ	Overlay the points or use a violin
A very small box	Concentrated data, or the wrong scale	Check against a histogram
Comparing groups of very different sizes	The plot hides n	Print n under each box
Median at the edge of the box	Strong skew	Real, and worth saying explicitly

Common mistakes

Treating the whiskers as a confidence interval. They are a spread of the data, not of the estimate.
Boxes for very small groups, or for a variable with only a few distinct values.
Reading a box plot of a bimodal variable as though it were unimodal - the median lands in the valley between the humps, where no observations are.
Using boxes where the individual points would fit. With fewer than about twenty observations per group, showing the points is strictly more informative and no harder to read.

Where to read this properly

Tukey, Exploratory Data Analysis (1977).
McGill, Tukey and Larsen (1978), "Variations of Box Plots", The American Statistician 32(1).
Hintze and Nelson (1998), "Violin Plots", The American Statistician 52(2).

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Box plot

What it needs

The intuition

When to use it

Where the estimator comes from

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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