If the first, second and third moment about origin are 2, 8 and 14, respectively, then Karl Pearson gamma coefficient of skeawness γ₁ is:

The 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.