Editing Median (section)

==Uses==
The median can be used as a measure of [[location parameter|location]] when one attaches reduced importance to extreme values, typically because a distribution is [[skewness|skewed]], extreme values are not known, or [[outlier]]s are untrustworthy, i.e., may be measurement or transcription errors.

For example, consider the [[multiset]]
{{block indent | em = 1.5 | text = 1, 2, 2, 2, 3, 14. }}

The median is 2 in this case, as is the [[mode (statistics)|mode]], and it might be seen as a better indication of the [[central tendency|center]] than the [[arithmetic mean]] of 4, which is larger than all but one of the values.  However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true.  At most, one can say that the two statistics cannot be "too far" apart; see {{slink||Inequality relating means and medians}} below.<ref>{{cite journal |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 |author=Paul T. von Hippel |year=2005 |access-date=2015-06-18 |archive-date=2008-10-14 |archive-url=https://web.archive.org/web/20081014045349/http://www.amstat.org/publications/jse/v13n2/vonhippel.html |url-status=dead }}</ref>

As a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.<ref name="Robson">{{cite book | last1=Robson|first1=Colin | title=Experiment, Design and Statistics in Psychology |date=1994|publisher=Penguin |isbn=0-14-017648-9|pages=42–45}}</ref>

Because the median is simple to understand and easy to calculate, while also a robust approximation to the [[mean]], the median is a popular [[summary statistic]] in [[descriptive statistics]].  In this context, there are several choices for a measure of [[variability (statistics)|variability]]: the [[Range (statistics)|range]], the [[interquartile range]], the [[mean absolute deviation]], and the [[median absolute deviation]].

For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the [[Efficiency (statistics)|efficiency]] of candidate estimators shows that the sample mean is more statistically efficient [[if and only if|when—and only when—]] data is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions.{{citation needed|date=February 2020}}  Even then, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean.<ref name="Williams 2001 165">{{cite book |last=Williams |first=D. |year=2001 |title=Weighing the Odds |url=https://archive.org/details/weighingoddscour00will_530 |url-access=limited |publisher=Cambridge University Press |isbn=052100618X |page=[https://archive.org/details/weighingoddscour00will_530/page/n184 165]}}</ref><ref>{{Cite book | last1=Maindonald|first1=John| url=https://books.google.com/books?id=8bMj8m-4RDQC&pg=PA104| title=Data Analysis and Graphics Using R: An Example-Based Approach| last2=Braun|first2=W. John |date=2010-05-06|publisher=Cambridge University Press|isbn=978-1-139-48667-5| pages=104|language=en}}</ref>