If first, second, and third moment about origin are 1, 6, and 15 respectively, then Karl Pearson beta coefficient of skewness β1 is

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SSC CGL Tier-II ( JSO ) 2021 Official Paper ( Held On: 10 August 2022 )
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The correct answer is 1/125

Key PointsGiven:

μ1' = 1; μ2' = 6; μ3' = 15

Calculation:

B1 =  ____ (1)

Now, μ2 = μ2' - (μ1')2 

⇒ 6 - (1)2  

⇒ 5 

μ3 = μ3' - 3 (μ2') (μ1') + 2 (μ1')3

⇒ 15 - 3 (6) (1) + 2 (1)3

⇒ 15 - 18 + 2

⇒ -1

By putting the above two values in (1)

⇒ B1 =

⇒ B1 = 1/125

Additional Information

  • The Karl Pearson's coefficient of skewness is a measure of the degree of skewness or asymmetry of a probability distribution. 
  • It is calculated as the third standardized moment about the mean, which is the ratio of the third central moment and the cube of the standard deviation.
  • The coefficient of skewness has the following properties:
    • If β1 = 0, then the distribution is symmetric and has no skewness.
    • If β1 > 0, then the distribution is positively skewed and has a long right tail.
    • If β1 < 0, then the distribution is negatively skewed and has a long left tail.
  • The coefficient of skewness is an important descriptive statistic that provides information about the shape of a distribution.
  • The coefficient of skewness should be used in conjunction with other measures such as the mean, median, and standard deviation to describe the distribution and understand its properties 

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