4/7/2015 StandardDeviationandVariance

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StandardDeviationandVariance

Deviationjustmeanshowfarfromthenormal

Standard DeviationTheStandardDeviationisameasureofhowspreadoutnumbersare.

Itssymbolis(thegreeklettersigma)

Theformulaiseasy:itisthesquarerootoftheVariance.Sonowyouask,"WhatistheVariance?"

VarianceTheVarianceisdefinedas:

TheaverageofthesquareddifferencesfromtheMean.

Tocalculatethevariancefollowthesesteps:

WorkouttheMean(thesimpleaverageofthenumbers)

Thenforeachnumber:subtracttheMeanandsquaretheresult(thesquareddifference).

Thenworkouttheaverageofthosesquareddifferences.(WhySquare?)

ExampleYouandyourfriendshavejustmeasuredtheheightsofyourdogs(inmillimeters):

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Theheights(attheshoulders)are:600mm,470mm,170mm,430mmand300mm.

FindouttheMean,theVariance,andtheStandardDeviation.

YourfirststepistofindtheMean:

Answer:

Mean=600+470+170+430+300

=1970

=3945 5

sothemean(average)heightis394mm.Let'splotthisonthechart:

Nowwecalculateeachdog'sdifferencefromtheMean:

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TocalculatetheVariance,takeeachdifference,squareit,andthenaveragetheresult:

So,theVarianceis21,704.

AndtheStandardDeviationisjustthesquarerootofVariance,so:

StandardDeviation:=21,704=147.32...=147(tothenearestmm)

AndthegoodthingabouttheStandardDeviationisthatitisuseful.NowwecanshowwhichheightsarewithinoneStandardDeviation(147mm)oftheMean:

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So,usingtheStandardDeviationwehavea"standard"wayofknowingwhatisnormal,andwhatisextralargeorextrasmall.

Rottweilersaretalldogs.AndDachshundsareabitshort...butdon'ttellthem!

Nowtrythe StandardDeviationCalculator .

But ... there is a small change with Sample DataOurexamplewasforaPopulation(the5dogsweretheonlydogswewereinterestedin).

ButifthedataisaSample(aselectiontakenfromabiggerPopulation),thenthecalculationchanges!

Whenyouhave"N"datavaluesthatare:

ThePopulation:dividebyNwhencalculatingVariance(likewedid)

ASample:dividebyN1whencalculatingVariance

Allothercalculationsstaythesame,includinghowwecalculatedthemean.

Example:ifour5dogswerejustasampleofabiggerpopulationofdogs,wewoulddivideby4insteadof5likethis:

SampleVariance=108,520/4 =27,130

SampleStandardDeviation=27,130=164(tothenearestmm)

Thinkofitasa"correction"whenyourdataisonlyasample.

Formulas

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Herearethetwoformulas,explainedat StandardDeviationFormulas ifyouwanttoknowmore:

The"PopulationStandardDeviation":

The"SampleStandardDeviation":

Lookscomplicated,buttheimportantchangeistodividebyN1(insteadofN)whencalculatingaSampleVariance.

*Footnote:Whysquarethedifferences?

Ifwejustaddedupthedifferencesfromthemean...thenegativeswouldcancelthepositives:

4+444

=04

Sothatwon'twork.Howaboutweuse absolutevalues ?

|4|+|4|+|4|+|4|

=4+4+4+4

=44 4

Thatlooksgood(andisthe MeanDeviation ),butwhataboutthiscase:

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|7|+|1|+|6|+|2|

=7+1+6+2

=44 4

OhNo!Italsogivesavalueof4,Eventhoughthedifferencesaremorespreadout!

Soletustrysquaringeachdifference(andtakingthesquarerootattheend):

42+42+42+42

=64

=44 4

72+12+62+22

=90

=4.74...4 4

Thatisnice!TheStandardDeviationisbiggerwhenthedifferencesaremorespreadout...justwhatwewant!

Infactthismethodisasimilarideato distancebetweenpoints ,justappliedinadifferentway.

Anditiseasiertousealgebraonsquaresandsquarerootsthanabsolutevalues,whichmakesthestandarddeviationeasytouseinotherareasofmathematics.

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