## Karl Pearson coefficient of skewness for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be the observed frequency distribution.

## 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

`$\overline{x} =\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$`

## Sample Mode

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

`$\text{Mode } = l + \bigg(\dfrac{f_m - f_1}{2f_m-f_1-f_2}\bigg)\times h$`

where

- $l$, the lower limit of the modal class
- $f_m$, frequency of the modal class
- $f_1$, frequency of the class pre-modal class
- $f_2$, frequency of the class post-modal class
- $h$, the class width

## Sample Median

`$\text{Median } = l + \bigg(\dfrac{\frac{N}{2} - F_<}{f}\bigg)\times h$`

where

- $N$, total number of observations
- $l$, the lower limit of the median class
- $f$, frequency of the median class
- $F_<$, cumulative frequency of the pre median class
- $h$, the class width

## Sample Standard deviation

Sample standard deviation is given by