⚖️ Empirical Relation & Effects

For a moderately skewed distribution, Karl Pearson's relation ties the three together:

$\text{Mode}=3\,\text{Median}-2\,\text{Mean}$

Rearrange to find whichever is missing: $\text{Median}=\dfrac{\text{Mode}+2\,\text{Mean}}{3}$ and $\text{Mean}=\dfrac{3\,\text{Median}-\text{Mode}}{2}$. A neat shortcut: $(\text{Mean}-\text{Mode})=3(\text{Mean}-\text{Median})$, so the Mean–Mode gap is always three times the Mean–Median gap. Memory hook: the bigger coefficient $3$ rides with Median (the longer word).

• Mean — add / remove / replace / misread: everything runs on $\text{total}=\text{mean}\times n$. Fix the total, then divide by the right count. For a replacement the count stays the same and the mean shifts by $\dfrac{\text{new}-\text{old}}{n}$; for a misread, corrected total $=$ wrong total $-$ wrong value $+$ correct value.

• Median — positional: it depends only on the middle position, not on extreme sizes. A change moves the median only if it touches the middle value, or if adding/removing an item shifts the middle position. Otherwise the median is unmoved — even when the mean changes.

• Linear transformation of the mean: if every value becomes $y=ax+b$, the mean follows the identical rule, $\bar{y}=a\bar{x}+b$, and the count is irrelevant. Apply the same operations in the same order to the old mean.