Question

Karl Pearson’s coefficient of skewness of a distribution is 0.32, its S.D is 6.5 and mean is 29.6. The mode and median of the distribution are
$\left( A \right)$ 27.52, 28.91
$\left( B \right)$ 26.92, 27.23
$\left( C \right)$ 25.67, 26.34
$\left( D \right)$ None of these

Answer 1

Hint – In this particular question use the concept that the Karl Pearson’s coefficient of skewness is the ratio of the difference of the mean and the mode to the standard deviation (S.D) and the Karl Pearson’s coefficient of skewness is the ratio of the thrice of the difference of the mean and the median to the standard deviation (S.D) so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
Karl Pearson’s coefficient of skewness of a distribution is 0.32
Let it be denoted by D.
Therefore, D = 0.32
S.D = 6.5
And mean = 29.6
Now as we know that Karl Pearson’s coefficient of skewness is the ratio of the difference of the mean and the mode to the standard deviation (S.D).
Therefore, Karl Pearson’s coefficient of skewness of a distribution, D = $\dfrac{{{\text{mean}} - {\text{mode}}}}{{S.D}}$
Now substitute all the values in the above equation we have,
 $ \Rightarrow 0.32 = \dfrac{{29.6 - {\text{mode}}}}{{6.5}}$
Now simplify this equation we have,
$ \Rightarrow 0.32\left( {6.5} \right) = 29.6 - {\text{mode}}$
$ \Rightarrow {\text{mode}} = 29.6 - 0.32\left( {6.5} \right)$
$ \Rightarrow {\text{mode}} = 29.6 - 2.08 = 27.52$
So the mode is 27.52.
Now we also know that Karl Pearson’s coefficient of skewness is the ratio of the thrice of the difference of the mean and the median to the standard deviation (S.D).
Therefore, Karl Pearson’s coefficient of skewness of a distribution, D = $\dfrac{{3\left( {{\text{mean}} - {\text{median}}} \right)}}{{S.D}}$
Now substitute all the values in the above equation we have,
 $ \Rightarrow 0.32 = \dfrac{{3\left( {29.6 - {\text{median}}} \right)}}{{6.5}}$
Now simplify this equation we have,
$ \Rightarrow 0.32\left( {6.5} \right) = 3\left( {29.6 - {\text{median}}} \right)$
$ \Rightarrow 3{\text{median}} = 3\left( {29.6} \right) - 0.32\left( {6.5} \right)$
$ \Rightarrow 3{\text{median}} = 88.8 - 2.08 = 86.72$
Now divide by 3 throughout we have,
$ \Rightarrow {\text{median}} = \dfrac{{86.72}}{3} = 28.906 \simeq 28.91$
So the median is 28.91.
So this is the required answer.
Hence the option (A) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember the formula of the Karl Pearson’s coefficient of skewness in terms of mean, mode and standard deviation also in the terms of mean, median and standard deviation which is all stated above, then simplify substitute the values in these equations and solve for mode and median as above which is the required answer.