Editing Unimodality (section)

====Mode, median and mean====

Gauss also showed in 1823 that for a unimodal distribution<ref name=Gauss1823>Gauss C.F. Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia</ref>

: <math>\sigma \le \omega \le 2 \sigma</math>

and

: <math>|\nu - \mu| \le \sqrt{\frac{3}{4}} \omega ,</math>

where the [[median]] is ''ν'', the mean is ''μ'' and  ''ω'' is the [[root mean square deviation]] from the mode.

It can be shown for a unimodal distribution that the median ''ν'' and the  mean ''μ'' lie within (3/5)<sup>1/2</sup> ≈ 0.7746 [[standard deviation]]s of each other.<ref name="unimodal">{{cite journal | url=http://epubs.siam.org/doi/pdf/10.1137/S0040585X97975447 | doi=10.1137/S0040585X97975447 | title=The Mean, Median, and Mode of Unimodal Distributions: A Characterization | year=1997 | last1=Basu | first1=S. | last2=Dasgupta | first2=A. | journal=Theory of Probability & Its Applications | volume=41 | issue=2 | pages=210–223 }}</ref> In symbols,

: <math>\frac{|\nu - \mu|}{\sigma} \le \sqrt{\frac{3}{5}}</math>

where | . | is the [[absolute value]].

In 2020, Bernard, Kazzi, and Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average <math>\frac{ q_\alpha  + q_{(1-\alpha)} }{ 2 } </math> and the mean,<ref name="unimodalbounds">{{cite journal | doi=10.1016/j.insmatheco.2020.05.013 | title=Range Value-at-Risk bounds for unimodal distributions under partial information | year=2020 | last1=Bernard | first1=Carole | last2=Kazzi | first2=Rodrigue | last3=Vanduffel | first3=Steven | journal=Insurance: Mathematics and Economics | volume=94 | pages=9–24 | doi-access=free }}</ref>

: <math>\frac{ \left| \frac{ q_\alpha  + q_{(1-\alpha)} }{2}  - \mu \right| }{ \sigma } \le \left\{
	\begin{array}{cl}
	\frac{\sqrt[]{\frac{4}{9(1-\alpha)}-1} \text{ } + \text{ } \sqrt[]{\frac{1-\alpha}{1/3+\alpha}}}{2}
	& \text{for }\alpha \in  \left[\frac{5}{6},1\right)\!, \\ 
	\frac{\sqrt[]{\frac{3 \alpha}{4-3\alpha}} \text{ } + \text{ } \sqrt[]{\frac{1-\alpha}{1/3+\alpha}}}{2}
	& \text{for }\alpha \in \left(\frac{1}{6},\frac{5}{6}\right)\!,\\
	\frac{\sqrt[]{\frac{3 \alpha}{4-3\alpha}} \text{ } + \text{ } \sqrt[]{\frac{4}{9 \alpha} -1}}{2}
	& \text{for }\alpha \in  \left(0,\frac{1}{6}\right]\!.
	\end{array}
	\right.</math>

The maximum distance is minimized at <math>\alpha=0.5</math> (i.e., when the symmetric quantile average is equal to  <math>q_{0.5} = \nu</math>), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when <math>\alpha = 0.5</math>, the bound is equal to <math>\sqrt{3/5}</math>, which is the maximum distance between the median and the mean of a unimodal distribution.

A similar relation holds between the median and the mode ''θ'': they lie within 3<sup>1/2</sup> ≈ 1.732 standard deviations of each other:

: <math>\frac{|\nu - \theta|}{\sigma} \le \sqrt{3}.</math>

It can also be shown that the mean and the mode lie within 3<sup>1/2</sup> of each other:

: <math>\frac{|\mu - \theta|}{\sigma} \le \sqrt{3}.</math>