方差有两种biased和unbiased，前者除以n；后者除以n-1，叫做Bessel'scorrection，可以修正样本的variance,更精确描述样本空间。matlab采用的是后者。抄一段对两者区别的解释Instatistics,Bessel'scorrection,namedafterFriedrichBessel,istheuseofn?1insteadofnintheformulaforthesamplevarianceandsamplestandarddeviation,wherenisthenumberofobservationsinasample:itcorrectsthebiasintheestimationofthepopulationvariance,andsome(butnotall)ofthebiasintheestimationofthepopulationstandarddeviation。

Thatis,whenestimatingthepopulationvarianceandstandarddeviationfromasamplewhenthepopulationmeanisunknown,thesamplevarianceisabiasedestimatorofthepopulationvariance,andsystematicallyunderestimatesit。

Multiplyingthestandardsamplevariancebyn/(n?1)(equivalently,using1/(n?1)insteadof1/n)correctsforthis,andgivesanunbiasedestimatorofthepopulationvariance。

Thecostofthiscorrectionisthattheunbiasedestimatorhasuniformlyhighermeansquarederrorthanthebiasedestimator。Asubtlepointisthat,whilethesamplevariance(usingBessel'scorrection)isanunbiasedestimateofthepopulationvariance,itssquareroot,thesamplestandarddeviation,isabiasedestimateofthepopulationstandarddeviation;becausethesquarerootisaconcavefunction,thebiasisdownward,byJensen'sinequality。

Thereisnogeneralformulaforanunbiasedestimatorofthepopulationstandarddeviation,thoughtherearecorrectionfactorsforparticulardistributions,suchasthenormal;seeunbiasedestimationofstandarddeviationfordetails。

OnecanunderstandBessel'scorrectionintuitivelyasthedegreesoffreedomintheresidualsvector:(x_1-overline{x},,dots,,x_n-overline{x}),whereoverline{x}isthemean。

Whiletherearenindependentsamples,thereareonlyn?1independentresiduals,astheysumto0。

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