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

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

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