Q.For a certain distribution of monthly family incomes, the mean is ₹48,000 and the mode is ₹35,000. Using Karl Pearson's empirical relationship, estimate the median.

Karl Pearson's empirical relationship lets the median be estimated when the mean and mode are already known. …

Step 1 — Start from the empirical relationship.

Step 2 — Substitute the given values and solve for Median.

$35000 = 3\,Median - 96000$ …

Cross-check using the alternative (rearranged) form of the same relationship, $Mean - Mode = 3(Mean-Median)$:

$48000-Median = \frac{13000}{3} \approx 4333.33$ …

Mis-substituting mean and mode into the formula the wrong way round (writing $Mean=3\,Median-2\,Mode$ instead of $Mode=3\,Median-2\,Mean$) is the most frequent error — the coefficient 3 always …