Five-number summary, box plots and outlier fences

Minimum, Q1, median, Q3 and maximum summarise a distribution; Tukey's fences at 1.5·IQR beyond the quartiles flag potential outliers.

\text{inner fences: } Q_1 - 1.5\,\text{IQR},\ \ Q_3 + 1.5\,\text{IQR} \qquad \text{outer fences: } Q_1 - 3\,\text{IQR},\ \ Q_3 + 3\,\text{IQR}

Tukey's fences (quartiles by type 7 unless stated).

$Q_1,\ Q_3$: First and third quartiles.
$\text{IQR}$: $Q_3 - Q_1$ .

When to use

A compact numerical summary of a quantitative variable, robust to extreme values.
Box plots for comparing several groups side by side.
Screening for unusual values before analysis.

How to interpret

The box spans $Q_1$ to $Q_3$ with a line at the median; whiskers reach the most extreme values inside the inner fences; points beyond are drawn individually. Values beyond the outer fences are "extreme". A long upper whisker and a median near the bottom of the box indicate right skew. Some software (R's fivenum) uses Tukey's hinges instead of type 7 quartiles, so box edges can differ slightly.

Common mistakes

Deleting points just because they are beyond a fence. An outlier is a prompt to investigate (entry error? different population? real extreme?), not a licence to remove. Report any removal and its effect.
Applying the 1.5·IQR rule to heavily skewed data and calling the long tail "outliers" — in such data many genuine values lie beyond the fence.
Forgetting that a box plot hides bimodality and sample size.

Example

Data 1, 3, 4, 5, 6, 7, 8, 30. Five-number summary (type 7): 1, 3.75, 5.5, 7.25, 30. IQR = 3.5, inner fences −1.5 and 12.5, outer fences −6.75 and 17.75. The value 30 lies beyond the outer fence: an extreme outlier worth checking.