Skewness and Kurtosis

Quartile Deviation

**MEASURE OF SKEWNESS**

Skewness refers to the lack of symmetry. Various methods are available for measuring skewness. The difference between the way items are distributed in a particular distribution and a symmetrical distribution is defined by the measure of skewness. In a nutshell, they indicate the direction and magnitude of asymmetry in a distribution.

The measure of skewness are of two types:

(1) Absolute measures of skewness

(2) Relative measures of skewness

**Absolute Measures of Skewness**

**Based upon mean, median and mode:**The absolute skewness is given by

*S*

_{k}

(1)

(2)

(1)

(1)

The skewness computed using any of these measures is stated in the unit value of the distribution, it cannot be compared to the skewness of another distribution stated in different units, absolute measurements of skewness are not very useful.

*S*_{k}= Mean $-$ Mode(2)

*S*_{k}= Mean $-$ Median**Based upon Quartiles:**The absolute skewness is given by*S*_{k}(1)

*S*_{k}= Q_{3}+Q_{1}$-$2Q_{2}(1)

*S*_{k}= Q_{3}+Q_{1}$-$2(Median)The skewness computed using any of these measures is stated in the unit value of the distribution, it cannot be compared to the skewness of another distribution stated in different units, absolute measurements of skewness are not very useful.

**#Note 1:**Because the values of mean, median, and mode are equal in asymmetrical distributions, and the mean moves away from the mode when the observations are asymmetrical, the difference between mean and mode are used to evaluate skewness. As the discrepancy between mean and mode grows, the distribution becomes increasingly asymmetric.**#Note 2:**Because the middle quartile in a symmetrical distribution is equidistant from the lower and higher quartiles and lies between them, quartiles are used to quantify the absolute skewness of a distribution.**Relative Measures of Skewness**

****The absolute measures of skewness cannot be used to compare two or more distributions. The coefficients of skewness is computed for these purposes. These exist as pure numbers independent of units of measurement. The coefficients of skewness are as follows:

(1) Karl Pearson's coefficient of skewness

(2) Bowley's coefficient of skewness

(3) Kelly's coefficient of skewness

(4) Moment-based coefficient of skewness.

**Karl Pearson's Coefficient of Skewness:**

The Karl Pearson's coefficient of skewness

*S*of a distribution is defined as

_{kp}${S}_{kp}=\frac{\mathrm{Mean}-\mathrm{Mode}}{\mathrm{Standard}\mathrm{Deviation}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},{S}_{kp}=0\Rightarrow \frac{\mathrm{Mean}-\mathrm{Mode}}{\mathrm{Standard}\mathrm{Deviation}}=0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}-\mathrm{Mode}=0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}=\mathrm{Mode}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Distribution}\mathrm{is}\mathrm{symmertical}\phantom{\rule{0ex}{0ex}}$

Thus, a distribution is symmetrical iff

*S*= 0.

_{kp}$\mathrm{Now},{S}_{kp}0\Rightarrow \frac{\mathrm{Mean}-\mathrm{Mode}}{\mathrm{Standard}\mathrm{Deviation}}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}-\mathrm{Mode}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}\mathrm{Mode}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Distribution}\mathrm{is}\mathrm{positively}\mathrm{skewed}\phantom{\rule{0ex}{0ex}}$

Thus, a distribution is positively skewed iff

$\mathrm{Now},{S}_{kp}0\Rightarrow \frac{\mathrm{Mean}-\mathrm{Mode}}{\mathrm{Standard}\mathrm{Deviation}}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}-\mathrm{Mode}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}\mathrm{Mode}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Distribution}\mathrm{is}\mathrm{negatively}\mathrm{skewed}\phantom{\rule{0ex}{0ex}}$

Thus, a distribution is negatively skewed iff

The degree of skewness is obtained from the absolute value of

When the mode is ill-defined, the Karl Pearson's coefficient of skewness cannot be employed. The following relationship connects the mean, mode, and median in moderately skewed distributions.

Mean $-$ Mode = 3(Mean $-$ Median)

Therefore, for moderately skewed distribution, we have

Theoretically, the value of this coefficient varies between $-$3 and 3. However, in practice these limits…

*S*> 0._{kp}$\mathrm{Now},{S}_{kp}0\Rightarrow \frac{\mathrm{Mean}-\mathrm{Mode}}{\mathrm{Standard}\mathrm{Deviation}}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}-\mathrm{Mode}0\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Mean}\mathrm{Mode}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{Distribution}\mathrm{is}\mathrm{negatively}\mathrm{skewed}\phantom{\rule{0ex}{0ex}}$

Thus, a distribution is negatively skewed iff

*S*< 0._{kp}The degree of skewness is obtained from the absolute value of

*S*._{kp}When the mode is ill-defined, the Karl Pearson's coefficient of skewness cannot be employed. The following relationship connects the mean, mode, and median in moderately skewed distributions.

Mean $-$ Mode = 3(Mean $-$ Median)

Therefore, for moderately skewed distribution, we have

*S*= $\frac{3(\mathrm{Mean}-\mathrm{Median})}{\mathrm{S}.\mathrm{D}.}$_{kp}Theoretically, the value of this coefficient varies between $-$3 and 3. However, in practice these limits…

To view the complete topic, please