Editing Median (section)

===Inequality relating means and medians===
[[File:Comparison mean median mode.svg|thumb|upright=1.35|Comparison of [[mean]], median and [[mode (statistics)|mode]] of two [[log-normal distribution]]s with different [[skewness]]]]

If the distribution has finite variance, then the distance between the median <math>\tilde{X}</math> and the mean <math>\bar{X}</math> is bounded by one [[standard deviation]].

This bound was proved by Book and Sher in 1979 for discrete samples,<ref>{{cite journal |author1=Stephen A. Book |author2=Lawrence Sher |title=How close are the mean and the median? |journal=The Two-Year College Mathematics Journal |date=1979 |volume=10 |issue=3 |pages=202–204 |doi=10.2307/3026748 |jstor=3026748 |url=https://www.jstor.org/stable/3026748 |access-date=12 March 2022}}</ref> and more generally by Page and Murty in 1982.<ref>{{cite journal |author1=Warren Page |author2=Vedula N. Murty |title=Nearness Relations Among Measures of Central Tendency and Dispersion: Part 1 |journal=The Two-Year College Mathematics Journal |date=1982 |volume=13 |issue=5 |pages=315–327 |doi=10.1080/00494925.1982.11972639 |doi-broken-date=1 November 2024 |url=https://www.tandfonline.com/doi/abs/10.1080/00494925.1982.11972639?journalCode=ucmj19 |access-date=12 March 2022}}</ref>  In a comment on a subsequent proof by O'Cinneide,<ref>{{cite journal |last1=O'Cinneide |first1=Colm Art |title=The mean is within one standard deviation of any median |journal=The American Statistician |date=1990 |volume=44 |issue=4 |pages=292–293 |doi=10.1080/00031305.1990.10475743 |url=https://www.tandfonline.com/doi/abs/10.1080/00031305.1990.10475743?journalCode=utas20 |access-date=12 March 2022}}</ref> Mallows in 1991 presented a compact proof that uses [[Jensen's inequality]] twice,<ref>{{cite journal |last=Mallows |first=Colin |title=Another comment on O'Cinneide |journal=The American Statistician |date=August 1991 |volume=45 |issue=3 |pages=257 | doi = 10.1080/00031305.1991.10475815}}</ref> as follows. Using |·| for the [[absolute value]], we have

<math display="block">\begin{align}
  \left|\mu - m\right| = \left|\operatorname{E}(X - m)\right|
  & \leq \operatorname{E}\left(\left|X -  m \right|\right) \\[2ex]
  & \leq \operatorname{E}\left(\left|X - \mu\right|\right) \\[1ex]
  & \leq \sqrt{\operatorname{E}\left({\left(X - \mu\right)}^2\right)} = \sigma.
\end{align}</math>

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex.  The second inequality comes from the fact that a median minimizes the [[absolute deviation]] function <math>a \mapsto \operatorname{E}[|X-a|]</math>.

Mallows's proof can be generalized to obtain a multivariate version of the inequality<ref name=PicheRandomVectorsSequences>{{cite book|last=Piché|first=Robert|title=Random Vectors and Random Sequences|year=2012|publisher=Lambert Academic Publishing|isbn=978-3659211966}}</ref> simply by replacing the absolute value with a [[norm (mathematics)|norm]]:
<math display="block">\left\|\mu - m\right\| \leq \sqrt{ \operatorname{E}\left({\left\|X - \mu\right\|}^2\right) }
= \sqrt{ \operatorname{trace}\left(\operatorname{var}(X)\right) }</math>

where ''m'' is a [[spatial median]], that is, a minimizer of the function <math>a \mapsto \operatorname{E}(\|X-a\|).\,</math> The spatial median is unique when the data-set's dimension is two or more.<ref name="Kemperman">{{cite journal |first=Johannes H. B. |last=Kemperman |title=The median of a finite measure on a Banach space: Statistical data analysis based on the L1-norm and related methods |journal=Papers from the First International Conference Held at Neuchâtel, August 31–September 4, 1987 |editor-first=Yadolah |editor-last=Dodge |publisher=North-Holland Publishing Co. |location=Amsterdam |pages=217–230 |mr=949228 |year=1987 }}</ref><ref name="MilasevicDucharme">{{cite journal |first1=Philip |last1=Milasevic |first2=Gilles R. |last2=Ducharme |title=Uniqueness of the spatial median |journal=[[Annals of Statistics]] |volume=15 |year=1987 |number=3 |pages=1332–1333 |mr=902264 |doi=10.1214/aos/1176350511|doi-access=free }}</ref>

An alternative proof uses the one-sided Chebyshev inequality; it appears in [[An inequality on location and scale parameters#An application - distance between the mean and the median|an inequality on location and scale parameters]].  This formula also follows directly from [[Cantelli's inequality]].<ref>[http://www.montefiore.ulg.ac.be/~kvansteen/MATH0008-2/ac20112012/Class3/Chapter2_ac1112_vfinalPartII.pdf K.Van Steen ''Notes on probability and statistics'' ]</ref>

====Unimodal distributions====
For the case of [[Unimodality|unimodal]] distributions, one can achieve a sharper bound on the distance between the median and the mean:<ref name="unimodal">{{Cite journal|title=The Mean, Median, and Mode of Unimodal Distributions:A Characterization|journal=Theory of Probability and Its Applications|volume=41|issue=2|pages=210–223|doi=10.1137/S0040585X97975447|year = 1997|last1 = Basu|first1 = S.|last2=Dasgupta|first2=A.|s2cid=54593178}}</ref>

<math display="block">\left|\tilde{X} - \bar{X}\right| \le \left(\frac{3}{5}\right)^{1/2} \sigma \approx 0.7746\sigma.</math>

A similar relation holds between the median and the mode:

<math display="block">\left|\tilde{X} - \mathrm{mode}\right| \le 3^{1/2} \sigma \approx 1.732\sigma.</math>
[[File:Proof without words- The mean is greater than the median for monotonic distributions.svg|thumb|The mean is greater than the median for monotonic distributions.]]