Five Number Summary

Formula and examples for calculating the five number summary of a data series

The five number summary is a way to show the statistical dispersion. This summary consists of the minimum, lower quartile, median, upper quartile, and the maximum. If data are placed in order, then the lower quartile is central to the lower half of the data and the upper quartile is central to the upper half of the data.

Calculating the five-number summary

This example finds the five-number summary for the following data series

\(2, 5, 4, 8, 3, 7, 9, 3, 1, 6\)

Determine the number of numbers

Determine the number of numbers by counting all the numbers in the data set.

Number of numbers \( n = 10\)

Determine the smallest and largest numbers

Sort the data in ascending order.

\(\color{#44F}{\bf 1}\ 2\ 3\ 3\ 4\ 5\ 6\ 7\ 8\ \color{#44F}{\bf 9}\)

The smallest number is: \(\color{#44F}{\bf 1}\)
The largest number is: \(\color{#44F}{\bf 9}\)

Berechnung des unteren Quartils

Calculate the position of the lower quartile in the data set

\( \displaystyle \frac{1}{4}\cdot (n+1)=\frac{1}{4}\cdot(10+1)=2.75\)

The lower quartile is at position 2.75, i.e. between the 2nd and 3rd numbers in the data set.

\(1 \ \color{#44F}{\bf 2 \ 3} \ 3\ 4\ 5\ 6\ 7\ 8\ 9\)

The lower quartile is calculated from the values at this position

Lower quartile = \(\displaystyle \frac{2+3}{2}=\color{blue}{2.5}\)

Calculating the median

Calculate the position of the median in the data set

\( \displaystyle \frac{2}{4}\cdot (n+1)=\frac{2}{4}\cdot(10+1)=5.5\)

The median is at position 5.5, i.e. between the 5th and 6th numbers in the data set.

\(1 \ 2\ 3\ 3 \ \color{#44F}{\bf 4 \ 5} \ 6\ 7\ 8\ 9\)

The median is calculated from the values at this position

The median = \(\displaystyle \frac{4+5}{2}= \color{blue}{4.5}\)

Calculating the upper quartile

Calculate the position of the upper quartile in the data set

\( \displaystyle \frac{3}{4}\cdot (n+1)=\frac{3}{4}\cdot(10+1)=8.25\)

The upper quartile is at position 8.25, between the 8th and 9th numbers in the data set.

\(1 \ 2 \ 3 \ 3\ 4\ 5\ 6\ \color{#44F}{\bf 7 \ 8}\ 9\)

The upper quartile is calculated from the values in this position

Upper quartile= \(\displaystyle \frac{7+8}{2}= \color{blue}{7.5}\)