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Skewness
Statistics - Skewness
If dispersion measures amount of variation, then the direction of variation is measured by skewness. The most commonly used measure of skewness is Karl Pearson s measure given by the symbol Skp. It is a relative measure of skewness.
Formula
${S_{KP} = frac{Mean-Mode}{Standard Deviation}}$
When the distribution is symmetrical then the value of coefficient of skewness is zero because the mean, median and mode coincide. If the co-efficient of skewness is a positive value then the distribution is positively skewed and when it is a negative value, then the distribution is negatively skewed. In terms of moments skewness is represented as follows:
${eta_1 = frac{mu^2_3}{mu^2_2} \[7pt] Where mu_3 = frac{sum(X- ar X)^3}{N} \[7pt] , mu_2 = frac{sum(X- ar X)^2}{N}}$
If the value of ${mu_3}$ is zero it imppes symmetrical distribution. The higher the value of ${mu_3}$, the greater is the symmetry. However ${mu_3}$ do not tell us about the direction of skewness.
Example
Problem Statement:
Information collected on the average strength of students of an IT course in two colleges is as follows:
| Measure | College A | College B |
|---|---|---|
| Mean | 150 | 145 |
| Median | 141 | 152 |
| S.D | 30 | 30 |
Can we conclude that the two distributions are similar in their variation?
Solution:
A look at the information available reveals that both the colleges have equal dispersion of 30 students. However to estabpsh if the two distributions are similar or not a more comprehensive analysis is required i.e. we need to work out a measure of skewness.
${S_{KP} = frac{Mean-Mode}{Standard Deviation}}$Value of mode is not given but it can be calculated by using the following formula:
${ Mode = 3 Median - 2 Mean \[7pt] College A: Mode = 3 (141) - 2 (150)\[7pt] , = 423-300 = 123 \[7pt] S_{KP} = frac{150-123}{30} \[7pt] , = frac{27}{30} = 0.9 \[7pt] \[7pt] College B: Mode = 3(152)-2 (145)\[7pt] , = 456-290 \[7pt] , S_kp = frac{(142-166)}{30} \[7pt] , = frac{(-24)}{30} = -0.8 }$ Advertisements