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

Question

Question

This question was previously asked in

SSC CGL Tier-II ( JSO ) 2021 Official Paper ( Held On: 10 August 2022 )

Answer 1

The correct answer is 1/125

Key PointsGiven:

μ1' = 1; μ2' = 6; μ3' = 15

Calculation:

B1 = \(\frac{(μ_3)^2}{(μ_2)^3}\) ____ (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 = \(\frac{(-1)^2}{(5)^3}\)

⇒ 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