Editing Mode (statistics) (section)

==Comparison of mean, median and mode==
{{See also|Mean|Median}}
[[File:visualisation mode median mean.svg|thumb|upright|Geometric visualisation of the mode, median and mean of an arbitrary probability density function.<ref>{{cite web|title=AP Statistics Review - Density Curves and the Normal Distributions|url=http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|access-date=16 March 2015|archive-url=https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|archive-date=2 April 2015|url-status=dead}}</ref>]]

{| class="wikitable"
|+ Comparison of common [[average]]s of values { 1, 2, 2, 3, 4, 7, 9 }
! Type
! Description
! Example
! Result
|-
| align="center" | [[Arithmetic mean]]
| ''Sum of values of a data set divided by number of values''
| align="center" | (1+2+2+3+4+7+9) / 7
| align="center" | '''4'''
|-
| align="center" | [[Median]]
| ''Middle value separating the greater and lesser halves of a data set''
| align="center" | 1, 2, 2, '''3''', 4, 7, 9
| align="center" | '''3'''
|-
| align="center" | Mode
| ''Most frequent value in a data set''
| align="center" | 1, '''2''', '''2''', 3, 4, 7, 9
| align="center" | '''2'''
|}

===Use===
Unlike mean and median, the concept of mode also makes sense for "[[nominal data]]" (i.e., not consisting of [[Number|numerical]] values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of [[Korean name|Korean family name]]s, one might find that "[[Kim (Korean name)|Kim]]" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.

Unlike median, the concept of mode makes sense for any random variable assuming values from a [[vector space]], including the [[real number]]s (a one-[[dimension]]al vector space) and the [[integer]]s (which can be considered embedded in the reals). For example, a distribution of points in the [[plane (mathematics)|plane]] will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a [[linear order]] on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the [[geometric median]] and the [[centerpoint (geometry)|centerpoint]].

===Uniqueness and definedness===

For some [[probability distribution]]s, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain [[pathological (mathematics)|pathological]] distributions (for example, the [[Cantor distribution]]) have no defined mode at all.{{Citation needed|date=November 2010}}<ref>{{Cite web |last=Morrison |first=Kent |date=1998-07-23 |title=Random Walks with Decreasing Steps |url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |url-status=dead |archive-url=https://web.archive.org/web/20151202055102/http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |archive-date=2015-12-02 |access-date=2007-02-16 |website=Department of Mathematics, California Polytechnic State University}}</ref> For a finite data sample, the mode is one (or more) of the values in the sample.

===Properties===
Assuming definedness, and for simplicity uniqueness, the following are some of the most interesting properties.
* All three measures have the following property: If the random variable (or each value from the sample) is subjected to the linear or [[affine transformation]], which replaces {{mvar|X}} by {{math|''aX'' + ''b''}}, so are the mean, median and mode.
* Except for extremely small samples, the mode is insensitive to "[[outliers]]" (such as occasional, rare, false experimental readings). The median is also very robust in the presence of outliers, while the mean is rather sensitive.
* In continuous [[unimodal distribution]]s the median often lies between the mean and the mode, about one third of the way going from mean to mode. In a formula, median ≈ (2 × mean + mode)/3. This rule, due to [[Karl Pearson]], often applies to slightly non-symmetric distributions that resemble a normal distribution, but it is not always true and in general the three statistics can appear in any order.<ref>{{cite web |url=http://www.se16.info/hgb/median.htm |title=Relationship between the mean, median, mode, and standard deviation in a unimodal distribution }}</ref><ref>{{cite journal |last=Hippel |first=Paul T. von |year=2005 |url=http://www.amstat.org/publications/jse/v13n2/vonhippel.html |title=Mean, Median, and Skew: Correcting a Textbook Rule |journal=Journal of Statistics Education |volume=13 |issue=2 |doi= 10.1080/10691898.2005.11910556|doi-access=free }}</ref>
* For unimodal distributions, the mode is within {{radic|3}} standard deviations of the mean, and the root mean square deviation about the mode is between the standard deviation and twice the standard deviation.<ref>{{cite journal |last=Bottomley |first=H. |year=2004 |url=http://www.se16.info/hgb/mode.pdf |title=Maximum distance between the mode and the mean of a unimodal distribution |journal=Unpublished Preprint }}</ref>

===Example for a skewed distribution===

An example of a [[Skewness|skewed]] distribution is [[Distribution of wealth|personal wealth]]: Few people are very rich, but among those some are extremely rich. However, many are rather poor.

[[Image:Comparison mean median mode.svg|thumb|300px|Comparison of [[mean]], [[median]] and mode of two [[log-normal distribution]]s with different [[skewness]].]]
A well-known class of distributions that can be arbitrarily skewed is given by the [[log-normal distribution]]. It is obtained by transforming a random variable {{mvar|X}} having a normal distribution into random variable {{math|''Y'' {{=}} ''e''<sup>''X''</sup>}}. Then the logarithm of random variable {{mvar|Y}} is normally distributed, hence the name.

Taking the mean μ of {{mvar|X}} to be 0, the median of {{mvar|Y}} will be 1, independent of the [[standard deviation]] σ of {{mvar|X}}. This is so because {{mvar|X}} has a symmetric distribution, so its median is also 0. The transformation from {{mvar|X}} to {{mvar|Y}} is monotonic, and so we find the median {{math|''e''<sup>0</sup> {{=}} 1}} for {{mvar|Y}}.

When {{mvar|X}} has standard deviation σ = 0.25, the distribution of {{mvar|Y}} is weakly skewed. Using formulas for the [[log-normal distribution]], we find:
:<math>\begin{array}{rlll}
\text{mean}   & = e^{\mu + \sigma^2 / 2} & = e^{0 + 0.25^2 / 2} & \approx 1.032 \\
\text{mode}   & = e^{\mu - \sigma^2}     & = e^{0 - 0.25^2}     & \approx 0.939 \\
\text{median} & = e^\mu                  & = e^0                & = 1
\end{array}</math>
Indeed, the median is about one third on the way from mean to mode.

When {{mvar|X}} has a larger standard deviation, {{math|σ {{=}} 1}}, the distribution of {{mvar|Y}} is strongly skewed. Now
:<math>\begin{array}{rlll}
\text{mean}   & = e^{\mu + \sigma^2 / 2} & = e^{0 + 1^2 / 2} & \approx 1.649 \\
\text{mode}   & = e^{\mu - \sigma^2}     & = e^{0 - 1^2}     & \approx 0.368 \\
\text{median} & = e^\mu                  & = e^0             & = 1
\end{array}</math>
Here, [[Skewness#Pearson's skewness coefficients|Pearson's rule of thumb]] fails.

===Van Zwet condition===

Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold.<ref name=vanZwet1979>{{cite journal | last1 = van Zwet | first1 = WR | year = 1979 | title = Mean, median, mode II | journal = Statistica Neerlandica | volume = 33 | issue = 1| pages = 1–5 | doi=10.1111/j.1467-9574.1979.tb00657.x}}</ref> The inequality

:Mode ≤ Median ≤ Mean

holds if

:F( Median - {{mvar|x}} ) + F( Median + {{mvar|x}} ) ≥ 1

for all {{mvar|x}} where F() is the [[cumulative distribution function]] of the distribution.