Mean

In statistics, mean has two related meanings:

averageordinary English, whichmore correctly calledarithmetic mean,distinguishfrom geometric mean or harmonic mean. The averagealso called sample mean.
expected value ofrandom variable, whichalso called population mean.

Sample meanoften used as an estimator ofcentral tendency such aspopulation mean. However, other estimatorsalso used. For example,median ismore robust estimator ofcentral tendency thansample mean.

Forreal-valued random variable X,mean isexpectationX. Ifexpectation does not exist, thenrandom variable has no mean.

Fordata set,meanjustsumallobservations divided bynumberobservations. Once we have chosen this methoddescribingcommunality ofdata set, we usually usestandard deviationdescribe howobservations differ. The standard deviation issquare root ofaveragesquared deviations frommean.

The mean isunique value about whichsumsquared deviations isminimum. If you calculatesumsquared deviations from any other measurecentral tendency,will be larger than formean. This explains whystandard deviation andmeanusually cited togetherstatistical reports.

An alternative measuredispersion ismean deviation equivalent toaverage absolute deviation frommean. Itless sensitiveoutliers, but less tractable whaen combining data sets.

The mean value offunction, , on an interval, , can also be calculated (usinglimiting process ondata set definition) thus:

Note that not every probability distribution hasdefined mean or variance - seeCauchy distributionan example.

The following issummarysome ofmultiple methodscalculatingmean ofsetn numbers. Seetablemathematical symbolsexplanations ofsymbols used.

Tablecontents

1 Arithmetic Mean
2 Geometric Mean
3 Harmonic Mean
4 Generalized Mean
5 Weighted Mean
6 Interquartile mean
7 See also

Arithmetic Mean

The arithmetic mean is"standard" average, often simply called"mean". Itusedmany purposes but also often abused by incorrectly using itdescribe skewed distributions,highly misleading results. The classic exampleaverage income - usingarithmetic mean makesappearbe much higher thanin factcase. Considerscores {1, 2, 2, 2, 3, 9}. The arithmetic mean3.16, but five outsix scoresbelow this!)

Geometric Mean

The geometric meanan average whichusefulsetsnumbers whichinterpreted accordingtheir productnot their sum (as iscase witharithmetic mean). For example ratesgrowth.

Harmonic Mean

The harmonic meanan average whichusefulsetsnumbers whichdefinedrelationsome unit,example speed (distance per unittime).

Generalized Mean

The generalized meanan abstraction ofArithmetic, GeometricHarmonic Means.

By choosingappropriate value forparameter m we can getarithmetic mean (m = 1),geometric mean (m -> 0) orharmonic mean (m = -1)

This could be generalised further as

and againsuitable choicean invertible f(x) will givearithmetic meanf(x)=x,geometric mean f(x)=log(x), andharmonic meanf(x)=1/x.

Weighted Mean

The weighted meanused, if one wantscombine average values from samples ofsame populationdifferent sample sizes:

The weights representbounds ofpartial sample. In other applicationsrepresentmeasure forreliability ofinfluence uponmean by respective values.

Interquartile mean

The interquartile meanused whensetnumbers (the data) might be contaminated by inaccurate (ie. much too low or much too high) values. Thissimplyarithmetic mean after removingcertain number oflowest andhighest values. The numbervalues removedindicated aspercentagetotal numbervalues.