Question

Chose the correct answer:
1. Measures of skewness are independent of origin and scale
2. Measures of skewness are independent of origin and not of scale
3. Measures of skewness are independent of scale and not of origin
4. Measures of skewness are dependent of origin and scale

• A. Measures of skewness are independent of origin and scale
• B. Measures of skewness are independent of origin and not of scale
• C. Measures of skewness are independent of scale and not of origin
• D. Measures of skewness are dependent of origin and scale

Hint:

Any relative/dimensionless statistical coefficient (such as the coefficient of skewness, coefficient of kurtosis, and correlation coefficient) is always independent of both origin and scale.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question addresses the fundamental properties of measures of skewness in descriptive statistics.
Specifically, it tests whether relative measures of skewness are affected by changes in the origin and scale of the data.

Step 2: Detailed Explanation:
In statistics, absolute measures of skewness (like the difference between the mean and the mode) depend on the units of measurement.
However, relative measures of skewness are designed as dimensionless coefficients so that datasets of different units or magnitudes can be compared.
The most widely used relative measure of skewness is Karl Pearson's coefficient of skewness based on moments:
\[ \beta_1 = \frac{\mu_3^2}{\mu_2^3} \quad \text{or} \quad \gamma_1 = \frac{\mu_3}{\mu_2^{3/2}} \] where \( \mu_2 \) is the variance and \( \mu_3 \) is the third central moment.
Let us consider a change of origin and scale transformation:
\[ U = \frac{X - a}{h} \implies X = hU + a \] where \( a \) is the change of origin and \( h > 0 \) is the change of scale.
The \( r \)-th central moment of \( X \) is related to that of \( U \) by:
\[ \mu_r(X) = h^r \mu_r(U) \] Thus, for the second and third central moments, we have:
\[ \mu_2(X) = h^2 \mu_2(U) \quad \text{and} \quad \mu_3(X) = h^3 \mu_3(U) \] Substituting these into the formula for \( \gamma_1 \):
\[ \gamma_1(X) = \frac{\mu_3(X)}{[\mu_2(X)]^{3/2}} = \frac{h^3 \mu_3(U)}{[h^2 \mu_2(U)]^{3/2}} = \frac{h^3 \mu_3(U)}{h^3 [\mu_2(U)]^{3/2}} = \gamma_1(U) \] This mathematical proof demonstrates that the relative coefficient of skewness is completely unaltered by changes in both origin (\( a \)) and scale (\( h \)).
The same property holds true for other relative measures, such as Bowley's coefficient of skewness based on quartiles.

Step 3: Final Answer:
Relative measures of skewness are independent of both origin and scale.
Therefore, the correct choice is Option (A).