Relationship Among Mean, Median and Mode

When a frequency distribution is perfectly symmetrical (its two halves are mirror images of each other around the centre), all three measures of central tendency coincide exactly:

Most real economic data, however, is not perfectly symmetrical but only moderately asymmetrical (skewed) — a few unusually high or unusually low values pull the mean away from the middle of the bulk of the data, while the median and mode are left comparatively undisturbed. For such moderately skewed distributions, Karl Pearson's widely used empirical relationship connects all three:

which can equally be rearranged as $Mean - Mode = 3(Mean-Median)$, or solved for whichever of the three is unknown given the other two — a genuinely useful shortcut when one of the three is hard to compute directly from the raw data but the other two are already known.

It is important to be honest about what this relationship actually is: an empirical approximation observed to hold reasonably well for moderately skewed distributions, not an exact algebraic identity that must hold for every dataset — the worked examples in Section 8 show the empirical estimate landing close to, but not identical with, the value obtained by direct formula-based computation. …