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

$s_x =\sqrt{s_x^2}=\sqrt{\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\dfrac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)}$

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