Q.For a distribution of monthly savings of employees, the mean is Rs. 38 and the median is Rs. 36. The standard deviation is Rs. 8, and the mode cannot be located directly because the distribution is bimodal. Estimate the mode using the empirical relationship, and hence compute Karl Pearson's coefficient of skewness.

Because the mode cannot be read directly, first estimate it from the mean and median using the empirical relationship, then apply Karl Pearson's formula as usual. …

Given: Mean $\bar{x}=38$, Median $M_d=36$, Standard deviation $\sigma=8$; mode not directly available (bimodal distribution).

Step 1 — Estimate the mode using the empirical relationship.

Step 2 — Karl Pearson's coefficient using the estimated mode.

Independent check — median-based formula. Since the mode used above was itself derived from the empirical relationship, algebra guarantees that $(\bar{x}-\text{Mode})=3(\bar{x}-M_d)$ exactly. Confirming directly: $3(\bar{x}-M_d)=3(38-36)=6$, and $\dfrac{6}{8}=0.75$ — identical to Step 2, as it must be whenever the mode used is the empirically estimated one (this is precisely why the median-based formula exists — it is just this substi …

The most common error is swapping the coefficients in the empirical formula — it is 3 times Median minus 2 times Mean, never 3 times Mean minus 2 times Median. Also, remember this estimated-mode route and the direct median-based formula wi …