Editing Sufficient statistic (section)

==Mathematical definition==

A statistic ''t''&nbsp;=&nbsp;''T''(''X'') is '''sufficient for underlying parameter ''θ''''' precisely if the [[conditional probability distribution]] of the data ''X'', given the statistic ''t''&nbsp;=&nbsp;''T''(''X''), does not depend on the  parameter ''θ''.<ref name="CasellaBerger">{{cite book | last = Casella | first = George |author2=Berger, Roger L.  | title = Statistical Inference, 2nd ed | publisher=Duxbury Press | year = 2002}}</ref>

Alternatively, one can say the statistic&nbsp;''T''(''X'') is sufficient for ''θ'' if, for all prior distributions on ''θ'', the [[mutual information]] between ''θ'' and ''T(X)'' equals the mutual information between ''θ'' and ''X''.<ref>{{Cite book|last=Cover|first=Thomas M.|title=Elements of Information Theory|date=2006|publisher=Wiley-Interscience|others=Joy A. Thomas|isbn=0-471-24195-4|edition=2nd|location=Hoboken, N.J.|pages=36|oclc=59879802}}</ref> In other words, the [[data processing inequality]] becomes an equality:

:<math>I\bigl(\theta ; T(X)\bigr) = I(\theta ; X)</math>

===Example===
As an example, the sample mean is sufficient for the (unknown) mean ''μ'' of a [[normal distribution]] with known variance. Once the sample mean is known, no further information about ''μ'' can be obtained from the sample itself.  On the other hand, for an arbitrary distribution the [[median]] is not sufficient for the mean: even if the median of the sample is known, knowing the sample itself would provide further information about the population mean.  For example, if the observations that are less than the median are only slightly less, but observations exceeding the median exceed it by a large amount, then this would have a bearing on one's inference about the population mean.