Q.The mean of a distribution is $50$, its median is $48$ and its standard deviation is $10$. The mode cannot be determined reliably. Find Karl Pearson's coefficient of skewness.

When the mode is ill-defined, use the median form $Sk_p = 3(\bar{x} - M)/\sigma$.

Here $3(\bar{x} - M) = 3(50 - 48) = 6$, so $Sk_p = 6/10 = 0.6$, a positive value.

Given: Mean $\bar{x} = 50$, Median $M = 48$, Standard deviation $\sigma = 10$; mode ill-defined.

As the mode is unreliable, use the median form of Karl Pearson's coefficient: $Sk_p = \frac{3(\bar{x} - M)}{\sigma} = \frac{3(50 - 48)}{10} = \frac{6}{10} = 0.6.$ The positive value shows the distribution is positively (right) skewed.

Verification (dual solve). Recover the mode from the empirical relationship: $Z = 3M - 2\bar{x} = 3(48) - 2(50) = 144 - 100 = 44.$ Substituting this mode into the mode form gives $Sk_p = \frac{\bar{x} - Z}{\sigma} = \frac{50 - 44}{10} = \frac{6}{10} = 0.6,$ identical to the median-form value — confirming the result.

Forgetting the factor of 3 in the median form and computing (Mean - Median)/SD instead of 3(Mean - Median)/SD understates the coefficient threefold. The 3 comes from the empirical relationship and must never be dropped.