Question

The coefficient of skewness is 0.75. If Mean is 9 more than mode. Find its variance. (a) 12 (b) 144 (c) 36 (d) 1296

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{Mean = 9 + Mode} \\ &\sf{Coefficient \: of \: Skewness = 0.75} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:To\:find-\begin{cases} &\sf{Variance}  \end{cases}\end{gathered}\end{gathered}$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:Mean = 9 + Mode$

$\bf\implies \:Mean - Mode = 9$

and

$\rm :\longmapsto\:Coefficient \: of \: Skewness =0.75$

We know that,

Coefficient of Skewness is given by

$\rm :\longmapsto\:Coefficient \: of \: Skewness =\dfrac{Mean - Mode}{S. D.}$

$\rm :\longmapsto\:Coefficient \: of \: Skewness =\dfrac{9}{0.75}$

$\bf\implies \:S. D. = 12$

Now,

We know that

$\rm :\longmapsto\:Variance = {(S. D.)}^{2}$

$\rm :\longmapsto\:Variance = {(12)}^{2}$

$\rm :\longmapsto\:Variance = 144$

Hence, option (b) is correct.

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Additional Information :-

• The coefficient of skewness is a measure of asymmetry in the distribution.

• A positive skew indicates a longer tail to the right, while a negative skew indicates a longer tail to the left.

• A perfectly symmetric distribution, like the normal distribution, has a skew equal to zero.

Interpretation :-

• If skewness is less than −1 or greater than +1, the distribution is highly skewed.

• If skewness is between −1 and −½ or between +½ and +1, the distribution is moderately skewed.

• If skewness is between −½ and +½, the distribution is approximately symmetric.