## Five number summary for ungrouped data

A five number summary is a quick and easy way to determine the the center, the spread and outliers (if any) of a data set.

Five number summary includes five values, namely,

• minimum value ($\min$),
• first quartile ($Q_1$),
• $\text{median }$ ($Q_2$),
• third quartile ($Q_3$),
• maximum value ($\max$).

## Formula

$\min$, $Q_1$, $\text{median}$, $Q_3$ and $\max$

The five number summary of a data gives you a rough idea about the range of the data, center of the data, variation in the data and symmetry of the data. A box and whisker plot is the visual representation of five number summary.

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

## Example

The ages (in years) of 20 randomly selected students from a class are as follows:

23, 22, 21, 27, 19, 21, 18, 25, 26, 25,

29, 28, 18, 22, 20, 17, 19, 21, 24, 20.

Find the five number summary of the data.

### Solution

Arrange the data in ascending order

17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 29.

Minumum Value

The minimum age of students is $\min = 17$ years.

Maximum Value

The maximum age of students is $\max = 29$ years.

Quartiles

The formula for $i^{th}$ quartile is

$Q_i =$ Value of $\bigg(\dfrac{i(n+1)}{4}\bigg)^{th}$ observation, $i=1,2,3$

where $n$ is the total number of observations.

First Quartile $Q_1$

The first quartle $Q_1$ can be computed as follows:

\begin{aligned} Q_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(5.25\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(5\big)^{th} \text{ obs.}+0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\ &=19+0.25\big(20 -19\big)\\ &=19.25 \end{aligned}

Thus, $25$ % of the students had age less than or equal to $19.25$ years.

Median (M) (i.e., Second Quartile $Q_2$)

The median ($M$) or second quartile $Q_2$ can be computed as follows:

\begin{aligned} M &=\text{Value of }\bigg(\dfrac{2(n+1)}{4}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{2(20+1)}{4}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(10.5\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(10\big)^{th} \text{ obs.}+0.5 \big(\text{Value of } \big(11\big)^{th}\text{ obs.}-\text{Value of }\big(10\big)^{th} \text{ obs.}\big)\\ &=21+0.5\big(22 -21\big)\\ &=21.5 \end{aligned}

Thus, $50$ % of the students had age less than or equal to $21.5$ years.

Third Quartile $Q_3$

The third quartile $Q_3$ can be computed as follows:

\begin{aligned} Q_3 &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ obs.}\\ &= \text{Value of }\big(15.75\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(15\big)^{th} \text{ obs.}+0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\ &=25+0.75\big(25 -25\big)\\ &=25 \end{aligned} Thus, $75$ % of the students had age less than or equal to $25$ years.

Thus the five number summary of given data set is

$\min = 17$ years, $Q_1 = 19.25$ years, $\text{median }=21.5$ years, $Q_3=25$ years and $\max = 29$ years.