Q.For a distribution the mean is $45$, the mode is $39$ and the standard deviation is $6$. Compute Karl Pearson's coefficient of skewness and state the type of skewness.

Since the mode is given, use the mode form of Karl Pearson's coefficient, $Sk_p = (\bar{x} - Z)/\sigma$.

Here $\bar{x} - Z = 45 - 39 = 6$, so $Sk_p = 6/6 = 1$. As $\bar{x} > Z$, the coefficient is positive.

Given: Mean $\bar{x} = 45$, Mode $Z = 39$, Standard deviation $\sigma = 6$.

Karl Pearson's coefficient (mode form) is $Sk_p = \frac{\bar{x} - Z}{\sigma} = \frac{45 - 39}{6} = \frac{6}{6} = 1.$

Because the value is positive, the distribution is positively (right) skewed.

Verification (dual solve). Using the empirical relationship $Z = 3M - 2\bar{x}$ the given mode $39$ corresponds to a median $M$ satisfying $39 = 3M - 90$, i.e. $M = 43$. The median form must then give the same coefficient: $Sk_p = \frac{3(\bar{x} - M)}{\sigma} = \frac{3(45 - 43)}{6} = \frac{6}{6} = 1,$ which agrees with the mode form. Also $\bar{x} = 45 > M = 43 > Z = 39$, the order Mean > Median > Mode expected of a positively skewed distribution.

Writing (Mode - Mean) instead of (Mean - Mode) reverses the sign and reports the wrong direction of skewness. Always keep the order mean first, mode second; the sign of the answer is part of the answer, not an optional detail.