Statistical locationMeanFromWikipedia,thefreeencyclopediaJumptonavigationJumptosearchGeneraltermfortheseveraldefinitionsofmeanvalue,thesumdividedbythecountThisarticleisaboutthemathematicalconcept.Forotheruses,seeMean(disambiguation).Forthestateofbeingmeanorcruel,seeMeanness.Forbroadercoverageofthistopic,seeAverage.Thereareseveralkindsofmeaninmathematics,especiallyinstatistics:Foradataset,thearithmeticmean,alsoknownasaverageorarithmeticaverage,isacentralvalueofafinitesetofnumbers:specifically,thesumofthevaluesdividedbythenumberofvalues.Thearithmeticmeanofasetofnumbersx1,x2,...,xnistypicallydenotedbyx¯{\displaystyle{\bar{x}}}[note1].Ifthedatasetwerebasedonaseriesofobservationsobtainedbysamplingfromastatisticalpopulation,thearithmeticmeanisthesamplemean(denotedx¯{\displaystyle{\bar{x}}})todistinguishitfromthemean,orexpectedvalue,oftheunderlyingdistribution,thepopulationmean(denotedμ{\displaystyle\mu}orμx{\displaystyle\mu_{x}}[note2]).[1][2]Inprobabilityandstatistics,thepopulationmean,orexpectedvalue,isameasureofthecentraltendencyeitherofaprobabilitydistributionorofarandomvariablecharacterizedbythatdistribution.[3]InadiscreteprobabilitydistributionofarandomvariableX,themeanisequaltothesumovereverypossiblevalueweightedbytheprobabilityofthatvalue;thatis,itiscomputedbytakingtheproductofeachpossiblevaluexofXanditsprobabilityp(x),andthenaddingalltheseproductstogether,givingμ=∑xp(x)....{\displaystyle\mu=\sumxp(x)....}.[4][5]Ananalogousformulaappliestothecaseofacontinuousprobabilitydistribution.Noteveryprobabilitydistributionhasadefinedmean(seetheCauchydistributionforanexample).Moreover,themeancanbeinfiniteforsomedistributions.Forafinitepopulation,thepopulationmeanofapropertyisequaltothearithmeticmeanofthegivenproperty,whileconsideringeverymemberofthepopulation.Forexample,thepopulationmeanheightisequaltothesumoftheheightsofeveryindividual—dividedbythetotalnumberofindividuals.Thesamplemeanmaydifferfromthepopulationmean,especiallyforsmallsamples.Thelawoflargenumbersstatesthatthe