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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 Sk

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

Based upon Quartiles: The absolute skewness is given by Sk
(1) Sk = Q3 +Q1$-$2Q2
(1) Sk = Q3 +Q1$-$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.
• 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 Skp of a distribution is defined as

Thus, a distribution is symmetrical iff Skp = 0.

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

Thus, a distribution is negatively skewed iff Skp < 0.

The degree of skewness is obtained from the absolute value of Skp .
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
Skp = $\frac{3\left(\mathrm{Mean}-\mathrm{Median}\right)}{\mathrm{S}.\mathrm{D}.}$

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

• Bowley's Coefficient of Skewness

Bowley's coefficient of skewness is based on quartiles and is defined as

The first and third quartiles are equidistant from the median in a symmetrical distribution, as seen below:

Thus, the distribution is symmetrical

In a positively skewed distribution, the top 25% of the values tend to be further away from the median than the bottom 25% i.e. Q3 is farther away from the median than Q1 is from the median.

In a negatively skewed distribution, the top 25% of the values are nearer to the median than the bottom 25% i.e. Q3 is nearer to the median than Q1 is from the median.

SkB'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

Given,  SkB = 0.3, Q3 + Q1 = 30 and Median = 15. Putting these in above equation we get,
${S}_{kB}=\frac{30-2×14}{{Q}_{3}-{Q}_{1}}=0.3\phantom{\rule{0ex}{0ex}}⇒{Q}_{3}-{Q}_{1}=\frac{2}{0.3}\phantom{\rule{0ex}{0ex}}⇒{Q}_{3}-{Q}_{1}=\frac{20}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$
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:
Skp = $\frac{3\left(\mathrm{Mean}-\mathrm{Median}\right)}{\mathrm{S}.\mathrm{D}.}=\frac{3\left(200-160\right)}{80}=1.5$

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

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 -…

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