Five-number summary

The five-number summary describes a distribution using its minimum, lower quartile (Q1), median (Q2), upper quartile (Q3) and maximum, giving a compact picture of centre, spread and tail behaviour that underpins the box plot.

ByHoang TruongUpdated July 3, 2026

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Twenty orders' delivery times summarise to five numbers: a minimum of 1 day, a lower quartile of 3 days, a median of 5 days, an upper quartile of 8 days and a maximum of 21 days. The interquartile range is 5 days, capturing typical variation, while the wide gap to the 21-day maximum signals a right-skewed tail of a few very late shipments.

Where it fits

TopicDescriptive StatisticsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

IQR = Q_3 - Q_1

Variables

$I QR$: interquartile range (units of the data)
$Q_{1}$: first quartile (25th percentile) (units of the data)
$Q_{3}$: third quartile (75th percentile) (units of the data)

The five-number summary is {min, Q1, median, Q3, max}. IQR measures the spread of the central 50% of observations and is the primary input to box-plot fence calculation.

LaTeX

\text{Lower fence} = Q_1 - 1.5 \times IQR \qquad \text{Upper fence} = Q_3 + 1.5 \times IQR

Variables

$Q_{1}$: first quartile (units of the data)
$Q_{3}$: third quartile (units of the data)
$I QR$: interquartile range (units of the data)

Observations beyond the inner fences are plotted individually as suspected outliers in a box-and-whisker plot.

Check yourself

PracticeCORE

A dataset has the five-number summary: minimum = 2, Q1 = 8, median = 14, Q3 = 20, maximum = 45. Using the standard 1.5 × IQR rule, which of the following correctly states both the interquartile range and the upper fence for outlier detection?

Select an answer to check your understanding.