If the curve is symmetrical, a deviation below the mean exactly equals the corresponding deviation above the mean. This is called symmetry. Here, we will discuss Skewness and Measures of Skewness.

## Table of Contents

**Skewness **is the degree of asymmetry or departure from the symmetry of a distribution. **Positive Skewness** means when the tail on the right side of the distribution is longer or fatter. The mean and median will be greater than the mode. **Negative Skewness** is when the tail of the left side of the distribution is longer or fatter than the tail on the right side.

## Measures of Skewness

### Karl Pearson Measures of Relative Skewness

In a **symmetrical distribution**, the mean, median, and mode coincide. In skewed distributions, these values are pulled apart; the mean tends to be on the same side of the mode as the longer tail. Thus, a *measure of the asymmetry* is supplied by the difference ($mean – mode$). This can be made dimensionless by dividing by a measure of dispersion (such as SD). The **Karl Pearson measure of relative skewness** is

$$\text{SK} = \frac{\text{Mean}-\text{mode}}{SD} =\frac{\overline{x}-\text{mode}}{s}$$

The value of skewness may be either positive or negative.

The **empirical formula for skewness** (called the second coefficient of skewness) is

$$\text{SK} = \frac{3(\text{mean}-\text{median})}{SD}=\frac{3(\tilde{X}-\text{median})}{s}$$

### Bowley Measures of Skewness

In a symmetrical distribution, the quartiles are equidistant from the median ($Q_2-Q_1 = Q_3-Q_2$). If the distribution is not symmetrical, the quartiles will not be equidistant from the median (unless the entire asymmetry is located in the extreme quarters of the data). The **Bowley suggested measure of skewness** is

$$\text{Quartile Coefficient of SK} = \frac{Q_(2-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}=\frac{Q_2-2Q_2+Q_1}{Q_3-Q_1}$$

This measure is always zero when the quartiles are equidistant from the median and is positive when the upper quartile is farther from the median than the lower quartile. This measure of skewness varies between $+1$ and $-1$.

### Moment Coefficient of Skewness

In any symmetrical curve, the sum of odd powers of deviations from the mean will be equal to zero. That is, $m_3=m_5=m_7=\cdots=0$. However, it is not true for asymmetrical distributions. For this reason, a measure of skewness is devised based on $m_3$. That is

\begin{align}

\text{Moment of Coefficient of SK}&= a_3=\frac{m_3}{s^3}=\frac{m_3}{\sqrt{m_2^3}}\\

&=b_1=\frac{m_3^2}{m_2^3}

\end{align}

For perfectly symmetrical curves (normal curves), $a_3$ and $b_1$ are zero.

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