## 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.