Question

Generally, what is the range of coefficient of skewness for data where the mode is ill-defined?(A) 0 to 1 (B) -1 to +1

(C) -3 to +3 (D) 1 to 0
PLZ GIVE RIGHT ANSWER​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

The range of coefficient of skewness for data where the mode is well defined

(A) 0 to 1

(B) -1 to +1

(C) -3 to +3

(D) 1 to 0

EVALUATION

If $\sf{Q_1, Q_2,Q_3}$ are first, second, third quartiles of any statistical distribution, the coefficient of skewness due to Bowley is given by

$\displaystyle \sf{S_k(B) = \frac{Q_3 - 2Q_2 +Q_1 }{Q_3 - Q_1}}$

Now

$\displaystyle \sf{S_k(B) = \frac{(Q_3 - Q_2)- (Q_2 - Q_1) }{(Q_3 - Q_2) + (Q_2 - Q_1) }}$

Again

$\displaystyle \sf{Q_1 \leqslant Q_2 \leqslant Q_3}$

We now use the inequality

$\sf{ |a - b| \leqslant |a| + |b| }$

We now get

$\displaystyle \sf{| S_k(B)| = \bigg| \frac{(Q_3 - Q_2)- (Q_2 - Q_1) }{(Q_3 - Q_2) + (Q_2 - Q_1) }} \bigg|$

$\displaystyle \sf{ \implies \: | S_k(B)| = \frac{| (Q_3 - Q_2)- (Q_2 - Q_1) | }{ | (Q_3 - Q_2) + (Q_2 - Q_1)| }}$

$\displaystyle \sf{ \implies \: | S_k(B)| \leqslant \frac{| (Q_3 - Q_2) | + | (Q_2 - Q_1) | }{ | (Q_3 - Q_2) | + | (Q_2 - Q_1)| }}$

$\displaystyle \sf{ \implies \: | S_k(B)| \leqslant 1}$

$\displaystyle \sf{ \implies \: - 1 \leqslant S_k(B) \leqslant 1}$

So coefficient of skewness lies between - 1 & 1

FINAL ANSWER

Hence the correct option is (B) -1 to + 1

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