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

 

(1) S_k = Mean $-$ Mode
(2) S_k = Mean $-$ Median

Based upon Quartiles: The absolute skewness is given by S_k 
(1) S_k = Q₃ +Q₁$-$2Q₂   
(1) S_k = Q₃ +Q₁$-$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.

Thus, a distribution is positively skewed iff S_kp > 0.

$$\begin{gathered} \mathrm{Now}, S_{kp}<0 \Rightarrow \frac{\mathrm{Mean} - \mathrm{Mode}}{\mathrm{Standard} \mathrm{Deviation}}<0 \\ \Rightarrow \mathrm{Mean} - \mathrm{Mode}<0 \\ \Rightarrow \mathrm{Mean}<\mathrm{Mode} \\ \Rightarrow \mathrm{Distribution} \mathrm{is} \mathrm{negatively} \mathrm{skewed} \end{gathered}$$
Thus, a distribution is negatively skewed iff S_kp < 0.

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_kp = $\frac{3(\mathrm{Mean}-\mathrm{Median})}{\mathrm{S}.\mathrm{D}.}$

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

 S_kB's denominator is twice the quartile deviation, so the degree of skewness is quantified in relation to the distribution's dispersion. 

Bowley's skewness coefficient, often known as the quartile measure of skewness, ranges from $-$1 to 1. This method is useful for determining skewness in open-ended distributions with extreme values.


Example 1:  The frequency distribution has the coefficient of skewness based on quartiles as 0.3. If the sum of the upper and lower quartiles is 30 and the median is 15, calculate for the values of lower and upper quartiles.

Solution: 

The coefficient of skewness based on quartiles is given by 
$S_{kB}=\frac{Q_3+Q_1-2 \mathrm{Median}}{Q_3-Q_1}$
Given,  S_kB = 0.3, Q₃ + Q₁ = 30 and Median = 15. Putting these in above equation we get,
$$\begin{gathered} S_{kB}=\frac{30-2\times 14}{Q_3-Q_1}=0.3 \\ \Rightarrow Q_3-Q_1=\frac{2}{0.3} \\ \Rightarrow Q_3-Q_1=\frac{20}{3} \end{gathered}$$
Thus, we have Q₃ + Q₁ = 30 and Q₃ $-$ Q₁ = $\frac{20}{3}$ 
Solving these two equations we get Q₁ = $\frac{35}{3}$  and Q₃ = $\frac{55}{3}$.

Example 2: From the information given below compute Karl Pearson's coefficient of skewness and also Bowley's quartiles coefficient of skewness.
 

Measure	Point 1	Point 2
Mean	200	180
Median	160	146
S.D.	80	64
Third Quartile	280	250
First Quartile	100	80

Solution: 

The coefficient for Point 1 is:
S_kp = $\frac{3(\mathrm{Mean}-\mathrm{Median})}{\mathrm{S}.\mathrm{D}.}=\frac{3\left(200-160\right)}{80}=1.5$

$S_{kB}=\frac{Q_3+Q_1-2 \mathrm{Median}}{Q_3-Q_1}=\frac{280+100-2\times 160}{280-100}=0.34$

The coefficient for Point 2 is:
S_kp = $\frac{3(\mathrm{Mean}-\mathrm{Median})}{\mathrm{S}.\mathrm{D}.}=\frac{3\left(180-146\right)}{64}=1.594$

$S_{kB}=\frac{Q_3+Q_1-2 \mathrm{Median}}{Q_3-Q_1}=\frac{250+80-2\times 146}{250-80}=0.2236$

Example 3:  The weekly revenue earned by a company manufacturing 100 different products is as follows. Find Bowley's Coefficient of skewness:

Daily revenue (in Rs)	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 70	70 - 80	80 - 90	90 - 100
No of products	12	20	8	15	15	10	16	14	10

Solution: 
Calculation of coefficient of skewness 

Daily revenue (in Rs)	No of products	c.f.
10 - 20	12	12
20 - 30	20	32
30 -…