## Bowley’s Coefficient of Skewness for Ungrouped data

Skewness is a measure of symmetry. The meaning of skewness is “lack of symmetry”. Skewness gives us an idea about the concentration of higher or lower data values around the central value of the data.

For a symmetric distribution, the two quartiles namely $Q_1$ and $Q_3$ are equidistance from the median (i.e. $Q_2$). That is for symmetric distribution $Q_3 - Q_2 = Q_2 -Q_1$.

If the distriution is not symmetric (i.e., skewed) then the distance $Q_3-Q_2$ is not equal to the distance $Q_2-Q_1$. That is for asymetric distribution $Q_3-Q_2\neq Q_2-Q1$.

The absolute measure of skewness is $(Q_3-Q2)-(Q_2-Q1)= Q_3+Q_1-2*Q2$.

## Formula

Bowley’s coefficient of skewness is the relative measure of skewness. It is denoted by $S_b$ and is defined as

`$S_b = \dfrac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}$`

where,

- $Q_1$ is the first quartile,
- $Q_2$ is the second quartile,
- $Q_3$ is the third quartile,

## Types of Skewness

- If $S_b<0$, i.e., $Q_3-Q_2<Q_2-Q1$ then the distriution is
**negatively skewed**. - If $S_b=0$, i.e., $Q_3-Q_2=Q_2-Q1$ then the distriution is
**Symmetric**or**not skewed**. - If $S_b>0$, i.e., $Q_3-Q_2>Q_2-Q1$ then the distriution is
**positively skewed**.

### Bowley’s coefficient of skewness ranges from -1 to +1.

### Proof

We know that, if $a>0$ and $b>0$, then $|a-b|\leq |a+b|$,
```
$$
\begin{aligned}
& \text{i.e., } \bigg|\dfrac{a-b}{a+b} \bigg| \leq 1
\end{aligned}
$$
```

Now, taking $a= Q_3 - Q_2$ and $b= Q_2-Q_1$ in above inequality, we get
```
$$
\begin{aligned}
& \bigg|\dfrac{(Q_3 - Q_2)-(Q_2-Q_1)}{(Q_3 - Q_2)+(Q_2-Q_1)}\bigg| \leq 1\\
&\Rightarrow \bigg|\dfrac{Q_3 + Q_1-2Q_2}{Q_3 -Q_1}\bigg| \leq 1\\
& \Rightarrow |S_b|\leq 1\\
& \Rightarrow -1\leq S_b\leq 1.
\end{aligned}
$$
```

Thus, Bowley’s coefficient of skewness ranges from -1 and +1.

## Example 1

A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.

`5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12`

.

Find Bowley’s coefficient of skewness.

### Solution

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.

**Arrange the data in ascending order**

`5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15`

**First Quartile $Q_1$**

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

```
$$
\begin{aligned}
Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{1(15+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{Value of }\big(4\big)^{th} \text{ obs.}\\
&=9 \text{ days}.
\end{aligned}
$$
```

Thus, lower $25$ % of the patients had length of stay in the hospital less than or equal to $9$ days.

**Second Quartile $Q_2$**

The second quartile $Q_2$ can be computed as follows:

```
$$
\begin{aligned}
Q_{2} &=\text{Value of }\bigg(\dfrac{2(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{2(15+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{Value of }\big(8\big)^{th} \text{ obs.}\\
&=10 \text{ days}.
\end{aligned}
$$
```

Thus, lower $50$ % of the patients had length of stay in the hospital less than or equal to $10$ days.

**Third Quartile $Q_3$**

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

```
$$
\begin{aligned}
Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{3(15+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{Value of }\big(12\big)^{th} \text{ obs.}\\
&=13 \text{ days}.
\end{aligned}
$$
```

Thus, lower $75$ % of the patients had length of stay in the hospital less than or equal to $13$ days.

**Bowley’s coefficient of skewness**

Bowley’s coefficient of skewness is

```
$$
\begin{aligned}
S_b &= \frac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}\\
&=\frac{13+9 - 2* 10}{13 - 9}\\
&=0.5
\end{aligned}
$$
```

As the coefficient of skewness $S_b > 0$, the data is $\text{positively skewed}$.

## Example 2

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

`75, 89, 72, 78, 87, 85, 73, 75, 97, 87`

`84, 76, 73, 79, 99, 86, 83, 76, 78, 73`

.

Find the Bowley’s coefficient of skewness.

### Solution

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.

**Arrange the data in ascending order**

72, 73, 73, 73, 75, 75, 76, 76, 78, 78, 79, 80, 82, 83, 84, 85, 86, 87, 97, 99

**First Quartile $Q_1$**

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

```
$$
\begin{aligned}
Q_{1} &=\text{Value of }\bigg(\dfrac{1(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{1(20+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{ Value of }\big(5.25\big)^{th} \text{ obs.}\\
&= \text{Value of }\big(5\big)^{th} \text{ obs.}+0.25 \big(\text{Value of } \big(6\big)^{th}\text{ obs.}-\text{Value of }\big(5\big)^{th} \text{ obs.}\big)\\
&=75+0.25\big(75 -75\big)\\
&=75 \text{ mg/dl}.
\end{aligned}
$$
```

Thus, $25$ % of the patients had blood sugar level less than or equal to $75$ mg/dl.

**Second Quartile $Q_2$**

The second quartile $Q_2$ can be computed as follows:

```
$$
\begin{aligned}
Q_{2} &=\text{Value of }\bigg(\dfrac{2(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{2(20+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{ Value of }\big(10.5\big)^{th} \text{ obs.}\\
&= \text{Value of }\big(10\big)^{th} \text{ obs.}+0.5 \big(\text{Value of } \big(11\big)^{th}\text{ obs.}-\text{Value of }\big(10\big)^{th} \text{ obs.}\big)\\
&=78+0.5\big(79 -78\big)\\
&=78.5 \text{ mg/dl}.
\end{aligned}
$$
```

Thus, $50$ % of the patients had blood sugar level less than or equal to $78.5$ mg/dl.

**Third Quartile $Q_3$**

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

```
$$
\begin{aligned}
Q_{3} &=\text{Value of }\bigg(\dfrac{3(n+1)}{4}\bigg)^{th} \text{ obs.}\\
&=\text{Value of }\bigg(\dfrac{3(20+1)}{4}\bigg)^{th} \text{ obs.}\\
&= \text{ Value of }\big(15.75\big)^{th} \text{ obs.}\\
&= \text{Value of }\big(15\big)^{th} \text{ obs.}+0.75 \big(\text{Value of } \big(16\big)^{th}\text{ obs.}-\text{Value of }\big(15\big)^{th} \text{ obs.}\big)\\
&=84+0.75\big(85 -84\big)\\
&=84.75 \text{ mg/dl}.
\end{aligned}
$$
```

Thus, $75$ % of the patients had blood sugar level less than or equal to $84.75$ mg/dl.

**Bowley’s coefficient of skewness**

Bowley’s coefficient of skewness is

```
$$
\begin{aligned}
S_b &= \frac{Q_3+Q_1 - 2Q_2}{Q_3 -Q_1}\\
&=\frac{84.75+75 - 2* 78.5}{84.75 - 75}\\
&=0.2820513
\end{aligned}
$$
```

As the coefficient of skewness $S_b > 0$, the data is $\text{positively skewed}$.