The Empirical Relation Between Mean, Median and Mode

For a moderately (asymmetrically) skewed distribution — one that is not perfectly symmetric but is not wildly lopsided either — Karl Pearson observed that the three measures of central tendency maintain an approximate relationship:

equivalently, rearranged: $Mean - Mode = 3(Mean - Median)$.

This relation is used in two directions in the Odisha CHSE Business Mathematics and Statistics syllabus: (i) to estimate the mode when the mean and median are known but the raw modal-class data is not (Worked Example 11), and (ii) as an approximate cross-check on a mode already computed by the direct interpolation formula of §8. Because it is an empirical (observation-based) relation and not an exact algebraic identity, a mode computed directly and a mode estimated through this relation will usually be close but need not match exactly — a small gap between the two is expected and is not, by itself, evidence of an arithmetic mistake (Worked Example 10 illustrates exactly this small, expected gap). The relation holds only f …