Editing Beta distribution (section)

====Mean, mode and median relationship====
If 1 < ''α'' < ''β'' then mode ≤ median ≤ mean.<ref name=Kerman2011>{{cite arXiv | eprint=1111.0433 | last1=Kerman | first1=Jouni | title=A closed-form approximation for the median of the beta distribution | date=2011 | class=math.ST }}</ref> Expressing the mode (only for ''α'', ''β'' > 1), and the mean in terms of ''α'' and ''β'':

: <math> \frac{ \alpha - 1 }{ \alpha + \beta - 2 } \le \text{median} \le \frac{ \alpha }{ \alpha + \beta } ,</math>

If 1 < ''β'' < ''α'' then the order of the inequalities are reversed. For ''α'', ''β'' > 1 the absolute distance between the mean and the median is less than 5% of the distance between the maximum and minimum values of ''x''. On the other hand, the absolute distance between the mean and the mode can reach 50% of the distance between the maximum and minimum values of ''x'', for the ([[Pathological (mathematics)|pathological]]) case of ''α'' = 1 and ''β'' = 1, for which values the beta distribution approaches the uniform distribution and the [[information entropy|differential entropy]] approaches its [[Maxima and minima|maximum]] value, and hence maximum "disorder".

For example, for ''α'' = 1.0001 and ''β'' = 1.00000001:
* mode   = 0.9999;   PDF(mode) = 1.00010
* mean   = 0.500025; PDF(mean) = 1.00003
* median = 0.500035; PDF(median) = 1.00003
* mean − mode   = −0.499875
* mean − median = −9.65538 × 10<sup>−6</sup>

where PDF stands for the value of the [[probability density function]].

[[File:Mean Median Difference - Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px]]
[[File:Mean Mode Difference - Beta Distribution for alpha and beta from 1 to 5 - J. Rodal.jpg|325px]]