In probability and statistics, the standard deviation is a measure of the dispersion of a collection of values. It can apply to a probability distribution, a random variable, a population or a data set. The standard deviation is usually denoted with the letter σ (lowercase sigma). It is defined as the root-mean-square (RMS) deviation of the values from their mean, or as the square root of the variance. Formulated by Galton in the late 1860s, the standard deviation remains the most common measure of statistical dispersion, measuring how widely spread the values in a data set are. If many data points are close to the mean, then the standard deviation is small; if many data points are far from the mean, then the standard deviation is large. If all data values are equal, then the standard deviation is zero. A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.
When only a sample of data from a population is available, the population standard deviation can be estimated by a modified standard deviation of the sample, explained below.
biāo zhǔn fāng chā （ standarddeviation)
jiù shì fāng chā de píng fāng gēn： yī zǔ shù jù zhōng de měi yī gè shù yǔ zhè zǔ shù jù de píng jūn shù de chā de píng fāng de hé zài chú yǐ shù jù de gè shù， qǔ píng fāng gēn jì shì。
jí： [ ∑（ Xn-X)^2]/n,(X biǎo shì zhè zǔ shù jù de píng jūn shù。）
Standarddeviationofaprobabilitydistributionorrandomvariable
Thestandarddeviationofa(univariate)probabilitydistributionisthesameasthatofarandomvariablehavingthatdistribution.
Thestandarddeviationσofareal-valuedrandomvariableXisdefinedas:
begin{array}{lcl}sigma&=&sqrt{operatorname{E}((X-operatorname{E}(X))^2)}=sqrt{operatorname{E}(X^2)-(operatorname{E}(X))^2},,end{array}
whereE(X)istheexpectedvalueofX(anotherwordforthemean),oftenindicatedwiththeGreekletterμ.
Notallrandomvariableshaveastandarddeviation,sincetheseexpectedvaluesneednotexist.Forexample,thestandarddeviationofarandomvariablewhichfollowsaCauchydistributionisundefinedbecauseitsE(X)isundefined.
[edit]Standarddeviationofacontinuousrandomvariable
Continuousdistributionsusuallygiveaformulaforcalculatingthestandarddeviationasafunctionoftheparametersofthedistribution.Ingeneral,thestandarddeviationofacontinuousreal-valuedrandomvariableXwithprobabilitydensityfunctionp(x)is
sigma=sqrt{int(x-mu)^2,p(x),dx},,
where
mu=intx,p(x),dx,,
andwheretheintegralsaredefiniteintegralstakenforxrangingovertherangeofX.
[edit]Standarddeviationofadiscreterandomvariableordataset
Thestandarddeviationofadiscreterandomvariableistheroot-mean-square(RMS)deviationofitsvaluesfromthemean.
IftherandomvariableXtakesonNvaluestextstylex_1,dots,x_N(whicharerealnumbers)withequalprobability,thenitsstandarddeviationσcanbecalculatedasfollows:
1.Findthemean,scriptstyleoverline{x},ofthevalues.
2.Foreachvaluexicalculateitsdeviation(scriptstylex_i-overline{x})fromthemean.
3.Calculatethesquaresofthesedeviations.
4.Findthemeanofthesquareddeviations.Thisquantityisthevarianceσ2.
5.Takethesquarerootofthevariance.
Thiscalculationisdescribedbythefollowingformula:
sigma=sqrt{frac{1}{N}sum_{i=1}^N(x_i-overline{x})^2},,
wherescriptstyleoverline{x}isthearithmeticmeanofthevaluesxi,definedas:
overline{x}=frac{x_1+x_2+cdots+x_N}{N}=frac{1}{N}sum_{i=1}^Nx_i,.
Ifnotallvalueshaveequalprobability,buttheprobabilityofvaluexiequalspi,thestandarddeviationcanbecomputedby:
sigma=sqrt{frac{sum_{i=1}^Np_i(x_i-overline{x})^2}{sum_{i=1}^Np_i}},,and
s=sqrt{frac{N'sum_{i=1}^Np_i(x_i-overline{x})^2}{(N'-1)sum_{i=1}^Np_i}},,
where
overline{x}=frac{sum_{i=1}^Np_ix_i}{sum_{i=1}^Np_i},,
andN'isthenumberofnon-zeroweightelements.
Thestandarddeviationofadatasetisthesameasthatofadiscreterandomvariablethatcanassumepreciselythevaluesfromthedataset,wherethepointmassforeachvalueisproportionaltoitsmultiplicityinthedataset. |