Question
Download Solution PDFIf the first, second and third moment about origin are 2, 8 and 14, respectively, then Karl Pearson gamma coefficient of skeawness γ1 is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is -2.25
Key Points
μ1' = 2; μ2' = 8; μ3' = 14 (Given)
If the first, second, and third moment about the origin is 2, 8, and 14, respectively, then Karl Pearson gamma coefficient of skewness Y1 is:
Y1 = ±√B1 ____ (1)
where B1 = \(\frac{(μ_3)^2}{(μ_2)^3}\)
Now, μ2 = μ2' - (μ1')2
⇒ 8 - (2)2
⇒ 4
μ3 = μ3' - 3 (μ2') (μ1') + 2 (μ1')3
⇒ 14 - 3 (8) (2) + 2 (2)3
⇒ 14 - 48 + 16
⇒ -18
By putting the above two values in (1)
⇒ Y1 = -\(\sqrt{\frac{(-18)^2} {(4)^3}}\)
⇒ Y1 = -√5.0625
⇒ Y1 = -2.25
Additional Information
Important points about Karl Pearson coefficient of skewness:
- It measures the asymmetry of a distribution about its mean.
- A positive value of skewness indicates that the distribution is positively skewed, i.e., it has a long tail on the right side of the mean. A negative value of skewness indicates that the distribution is negatively skewed, i.e., it has a long tail on the left side of the mean.
- A value of 0 for skewness indicates that the distribution is symmetric about its mean.
- The skewness of a normal distribution is always 0.
- The skewness can be sensitive to outliers, so it should be used with caution in case of a distribution with extreme values.
- The skewness can be used in combination with other measures of central tendency and variability, such as mean, median, mode, variance, and standard deviation, to describe the shape of a distribution.
Last updated on Jul 19, 2025
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