## Karl Pearson coefficient of skewness for ungrouped data

Let $x_i, i=1,2, \cdots , n$ be $n$ observations.

## Formula

The Karl Pearson’s coefficient Skewness is given by

`$S_k =\dfrac{Mean-Mode}{sd}=\dfrac{\overline{x}-Mode}{s_x}$`

OR

`$S_k =\dfrac{3(Mean-Median)}{sd}=\dfrac{3(\overline{x}-M)}{s_x}$`

where,

- $\overline{x}$ is the sample mean,
- $Mode$ is the sample mode,
- $M$ is the sample median,
- $s_x$ is the sample standard deviation.

## Sample mean

The sample mean $\overline{x}$ is given by

```
$$
\begin{eqnarray*}
\overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i
\end{eqnarray*}
$$
```

## Sample Mode

The mode is the value of $x$ that occurs maximum number of times.

## Sample Median

Arrange the data in ascending order of magnitude.

Median of $X$ is given by

```
$$
\begin{equation*}
Md= \left\{
\begin{array}{ll}
\text{value of }\big(\frac{n+1}{2}\big)^{th}\text{ obs.}, & \hbox{if $n$ is odd;} \\
\text{average of }\big(\frac{n}{2}\big)^{th}\text{ and }\big(\frac{n}{2}+1\big)^{th} \text{ obs.}, & \hbox{if $n$ is even.}
\end{array}
\right.
\end{equation*}
$$
```

## Sample Standard deviation

Sample standard deviation is given by

```
$$
\begin{aligned}
s_x &=\sqrt{s_x^2}\\
&=\sqrt{\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)}
\end{aligned}
$$
```