Q.Karl Pearson's coefficient of skewness of a distribution is $0.5$, its mode is $20$ and its standard deviation is $8$. Find the mean.

Rearrange the mode form $Sk_p = (\bar{x} - Z)/\sigma$ to make the mean the subject.

Given: $Sk_p = 0.5$, Mode $Z = 20$, Standard deviation $\sigma = 8$; the mean is required.

From the mode form $Sk_p = \frac{\bar{x} - Z}{\sigma} \quad\Rightarrow\quad \bar{x} - Z = Sk_p \times \sigma = 0.5 \times 8 = 4.$ Therefore $\bar{x} = Z + 4 = 20 + 4 = 24.$ …

Dividing by the standard deviation instead of multiplying when solving for the mean. Since $\bar{x} - Z = Sk_p \times \sigma$, the standard deviation is multiplied, n …