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How did we get the Empirical Formula?
This formula has been etched into us in school but I want to know how this formula came about and to know if it's applicable for all statistical distributions or can only be used with large enough data.
$$(\text{mean})-(\text{mode})=3(\text{mean}-\text{median})$$
The relevant papers are:
[1] Hall, P. (1980). On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables. The Annals of Probability, 8(3), 419-430.
[2] Haldane, J. B. S. "The Mode and Median of a Nearly Normal Distribution with Given Cumulants." Biometrika 32, no. 3/4 (1942): 294-99.
[3] Pearson, Karl. "Mathematical Contributions to the Theory of Evolution. II. Skew Variation in Homogeneous Material. [Abstract]." Proceedings of the Royal Society of London 57 (1894): 257-60.
The result is attributed to Pearson (1894). Hall (1980) credits Haldane (1942) for providing a "satisfactory explanation" of the result and Hall provides a very concise explanation of this result:
Let $X_1, X_2, \dots$ be iid random variables with $E(X) = 0$, $E(X^2) = 1$, and $E(X^3) = \tau$ (assumed to exist), and set $S_n = \sum_1^nX_j$, $M_n = \text{mode}(S_n)$, and $m_n = \text{median}(S_n)$, assuming that these quantities are uniquely defined. Haldane showed that $M_n \to -\dfrac{1}{2}\tau$ and $m_n \to -\dfrac{1}{6}\tau$ as $n \to \infty$.
Hall states that Haldane shows that the formula $$\text{mean} - \text{mode} \sim 3(\text{mean} - \text{median})$$ holds true (note: approximately) when $S_n$ has a density which admits a convergent Edgeworth expansion, and Hall's paper weakens these assumptions. I recommend consulting Hall's paper for the lengthy details.
Long story short, this obviously doesn't hold true in all cases.
