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

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

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