# Empirical Relationship between mean, median and mode.(Derivation)

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.

$$(\text{mean})-(\text{mode})=3(\text{mean}-\text{median})$$

• I think you intend: mean - mode = $3\times$(mean- median). It's just an approximation that works fairly well if the distribution is nearly symmetric.
– lulu
Feb 2, 2021 at 17:45
• @lulu thankyou . Feb 2, 2021 at 19:19

The relevant papers are:

 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.

 Haldane, J. B. S. "The Mode and Median of a Nearly Normal Distribution with Given Cumulants." Biometrika 32, no. 3/4 (1942): 294-99.

 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.

• I am just a curious undergrad and stats is not exactly my major. but thank you for all these resources. May 15, 2021 at 3:35