measures of shape_ skewness and kurtosis — math200 (tc3, brown)

TC3 → Stan Brown → Statistics → Measures of Shape

Summary:

Seealso:

Contents:

revised 26 Apr 2011 (What’s New?)

Copyright©2008–2011byStanBrown,OakRoadSystems

You’velearnednumericalmeasuresofcenter,spread,andoutliers,butwhataboutmeasuresofshape?Thehistogramcangiveyouageneralideaoftheshape,buttwonumericalmeasuresofshapegiveamorepreciseevaluation:skewnesstellsyoutheamountanddirectionofskew(departurefromhorizontalsymmetry),andkurtosistellsyouhowtallandsharpthecentralpeakis,relativetoastandardbellcurve.

Whydowecare?Oneapplicationistestingfornormality:manystatisticsinferencesrequirethatadistributionbenormalornearlynormal.Anormaldistributionhasskewnessandexcesskurtosisof0,soifyourdistributionisclosetothosevaluesthenitisprobablyclosetonormal.

MATH200BProgram—ExtraStatisticsUtilitiesforTI-83/84hasaprogramtodownloadtoyourTI-83orTI-84.Amongotherthings,theprogramcomputesalltheskewnessandkurtosismeasuresinthisdocument,exceptcon idenceintervalofskewnessandtheD’Agostino-Pearsontest.

SkewnessComputingExample1:CollegeMen’sHeightsInterpretingInferringEstimating

KurtosisVisualizingComputingInferring

AssessingNormalityExample2:SizeofRatLittersWhat’sNew

Skewness

The irstthingyouusuallynoticeaboutadistribution’sshapeiswhetherithasonemode(peak)ormorethanone.Ifit’sunimodal(hasjustonepeak),likemostdatasets,thenextthingyounoticeiswhetherit’ssymmetricorskewedtooneside.Ifthebulkofthedataisattheleftandtherighttailislonger,wesaythatthedistributionisskewedrightorpositivelyskewed;ifthepeakistowardtherightandthelefttailislonger,wesaythatthedistributionisskewedleftornegativelyskewed.

Lookatthetwographsbelow.Theybothhaveμ=0.6923andσ=0.1685,buttheirshapesaredifferent.

Measures of Shape: Skewness and Kurtosis — MATH200 (TC3, Brown) http://www.tc3.edu/instruct/sbrown/stat/shape.htm

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Beta(α=4.5,β=2)skewness=−0.5370

1.3846−Beta(α=4.5,β=2)skewness=+0.5370

The irstoneismoderatelyskewedleft:thelefttailislongerandmostofthedistributionisattheright.Bycontrast,theseconddistributionismoderatelyskewedright:itsrighttailislongerandmostofthedistributionisattheleft.

Youcangetageneralimpressionofskewnessbydrawingahistogram(MATH200Apart1),buttherearealsosomecommonnumericalmeasuresofskewness.Someauthorsfavorone,somefavoranother.ThisWebpagepresentsoneofthem.Infact,thesearethesameformulasthatExcelusesinits“DescriptiveStatistics”toolinAnalysisToolpak.

Youmayrememberthatthemeanandstandarddeviationhavethesameunitsastheoriginaldata,andthevariancehasthesquareofthoseunits.However,theskewnesshasnounits:it’sapurenumber,likeaz-score.

Computing

Themomentcoef icientofskewnessofadatasetisskewness:g1=m3/m2

3/2

wherem3=∑(x−x̄)3/nandm2=∑(x−x̄)2/n

x̄isthemeanandnisthesamplesize,asusual.m3iscalledthethirdmomentofthedataset.m2isthevariance,thesquareofthestandarddeviation.

You’llrememberthatyouhavetochooseoneoftwodifferentmeasuresofstandarddeviation,dependingonwhetheryouhavedataforthewholepopulationorjustasample.Thesameistrueofskewness.Ifyouhavethewholepopulation,theng1aboveisthemeasureofskewness.Butifyouhavejustasample,youneedthesampleskewness:

sampleskewness:

source:D.N.JoanesandC.A.Gill.“ComparingMeasuresofSampleSkewnessandKurtosis”.TheStatistician47(1):183–189.

Exceldoesn’tconcernitselfwithwhetheryouhaveasampleorapopulation:itsmeasureofskewnessisalwaysG1.


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Height(inches)

ClassMark,x

Frequ-ency,f

59.5–62.5 61 5

62.5–65.5 64 18

65.5–68.5 67 42

68.5–71.5 70 27

71.5–74.5 73 8

Example 1: College Men’s Heights

Herearegroupeddataforheightsof100randomlyselectedmalestudents,adaptedfromSpiegel&Stephens,TheoryandProblemsofStatistics3/e(McGraw-Hill,1999),page68.

Ahistogramshowsthatthedataareskewedleft,notsymmetric.

Buthowhighlyskewedarethey,comparedtootherdatasets?Toanswerthisquestion,youhavetocomputetheskewness.

Beginwiththesamplesizeandsamplemean.(Thesamplesizewasgiven,butitneverhurtstocheck.)n=5+18+42+27+8=100x̄=(61×5+64×18+67×42+70×27+73×8)÷100x̄=9305+1152+2814+1890+584)÷100x̄=6745÷100=67.45

Now,withthemeaninhand,youcancomputetheskewness.(Ofcourseinreallifeyou’dprobablyuseExcelorastatisticspackage,butit’sgoodtoknowwherethenumberscomefrom.)

ClassMark,x Frequency,f xf (x−x̄) (x−x̄)²f (x−x̄)³f

61 5 305 -6.45 208.01 -1341.68

64 18 1152 -3.45 214.25 -739.15

67 42 2814 -0.45 8.51 -3.83

70 27 1890 2.55 175.57 447.70

73 8 584 5.55 246.42 1367.63

∑ 6745 n/a 852.75 −269.33

x̄,m2,m3 67.45 n/a 8.5275 −2.6933

Finally,theskewnessisg1=m3/m2

3/2=−2.6933/8.52753/2=−0.1082

Butwait,there’smore!Thatwouldbetheskewnessiftheyouhaddataforthewholepopulation.Butobviouslytherearemorethan100malestudentsintheworld,oreveninalmostanyschool,sowhatyouhavehereisasample,notthepopulation.Youmustcomputethesampleskewness:

=[√(100×99)/98][−2.6933/8.52753/2]=−0.1098


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Interpreting

Ifskewnessispositive,thedataarepositivelyskewedorskewedright,meaningthattherighttailofthedistributionislongerthantheleft.Ifskewnessisnegative,thedataarenegativelyskewedorskewedleft,meaningthatthelefttailislonger.

Ifskewness=0,thedataareperfectlysymmetrical.Butaskewnessofexactlyzeroisquiteunlikelyforreal-worlddata,sohowcanyouinterprettheskewnessnumber?Bulmer,M.G.,PrinciplesofStatistics(Dover,1979)—aclassic—suggeststhisruleofthumb:

Ifskewnessislessthan−1orgreaterthan+1,thedistributionishighlyskewed.Ifskewnessisbetween−1and−½orbetween+½and+1,thedistributionismoderatelyskewed.Ifskewnessisbetween−½and+½,thedistributionisapproximatelysymmetric.

Withaskewnessof−0.1098,thesampledataforstudentheightsareapproximatelysymmetric.Caution:Thisisaninterpretationofthedatayouactuallyhave.Whenyouhavedataforthewhole

population,that’s ine.Butwhenyouhaveasample,thesampleskewnessdoesn’tnecessarilyapplytothewholepopulation.Inthatcasethequestionis,fromthesampleskewness,canyouconcludeanythingaboutthepopulationskewness?Toanswerthatquestion,seethenextsection.

Inferring

Yourdatasetisjustonesampledrawnfromapopulation.Maybe,fromordinarysamplevariability,yoursampleisskewedeventhoughthepopulationissymmetric.Butifthesampleisskewedtoomuchforrandomchancetobetheexplanation,thenyoucanconcludethatthereisskewnessinthepopulation.

ButwhatdoImeanby“toomuchforrandomchancetobetheexplanation”?Toanswerthat,youneedtodividethesampleskewnessG1bythestandarderrorofskewness(SES)togettheteststatistic,whichmeasureshowmanystandarderrorsseparatethesampleskewnessfromzero:

teststatistic:Zg1=G1/SESwhere

Thisformulaisadaptedfrompage85ofCramer,Duncan,BasicStatisticsforSocialResearch(Routledge,1997).(Someauthorssuggest√(6/n),butforsmallsamplesthat’sapoorapproximation.Andanyway,we’veallgotcalculators,soyoumayaswelldoitright.)

ThecriticalvalueofZg1isapproximately2.(Thisisatwo-tailedtestofskewness≠0atroughlythe0.05signi icancelevel.)

IfZg1<−2,thepopulationisverylikelyskewednegatively(thoughyoudon’tknowbyhowmuch).IfZg1isbetween−2and+2,youcan’treachanyconclusionabouttheskewnessofthepopulation:itmightbesymmetric,oritmightbeskewedineitherdirection.IfZg1>2,thepopulationisverylikelyskewedpositively(thoughyoudon’tknowbyhowmuch).

Don’tmixupthemeaningsofthisteststatisticandtheamountofskewness.Theamountofskewnesstellsyouhowhighlyskewedyoursampleis:thebiggerthenumber,thebiggertheskew.Theteststatistictellsyouwhetherthewholepopulationisprobablyskewed,butnotbyhowmuch:thebiggerthenumber,thehighertheprobability.


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Estimating

GraphPadsuggestsacon idenceintervalforskewness:95%con idenceintervalofpopulationskewness=G1±2SES

I’mnotsosureaboutthat.JoanesandGillpointoutthatsampleskewnessisanunbiasedestimatorofpopulationskewnessfornormaldistributions,butnotothers.SoIwouldsay,computethatcon idenceinterval,buttakeitwithseveralgrainsofsalt—andthefurtherthesampleskewnessisfromzero,themoreskepticalyoushouldbe.

Forthecollegemen’sheights,recallthatthesampleskewnesswasG1=−0.1098.Thesamplesizewasn=100andthereforethestandarderrorofskewnessis

SES=√[(600×99)/(98×101×103)]=0.2414Theteststatisticis

Zg1=G1/SES=−0.1098/0.2414=−0.45Thisisquitesmall,soit’simpossibletosaywhetherthepopulationissymmetricorskewed.Sincethesampleskewnessissmall,acon idenceintervalisprobablyreasonable:

G1±2SES=−.1098±2×.2414=−.1098±.4828=−0.5926to+0.3730.Youcangivea95%con idenceintervalofskewnessasabout−0.59to+0.37,moreorless.

Kurtosis

Ifadistributionissymmetric,thenextquestionisaboutthecentralpeak:isithighandsharp,orshortandbroad?Youcangetsomeideaofthisfromthehistogram,butanumericalmeasureismoreprecise.

Theheightandsharpnessofthepeakrelativetotherestofthedataaremeasuredbyanumbercalledkurtosis.Highervaluesindicateahigher,sharperpeak;lowervaluesindicatealower,lessdistinctpeak.Thisoccursbecause,asWikipedia’sarticleonkurtosisexplains,higherkurtosismeansmoreofthevariabilityisduetoafewextremedifferencesfromthemean,ratherthanalotofmodestdifferencesfromthemean.

BalandaandMacGillivraysaythesamethinginanotherway:increasingkurtosisisassociatedwiththe“movementofprobabilitymassfromtheshouldersofadistributionintoitscenterandtails.”(KevinP.BalandaandH.L.MacGillivray.“Kurtosis:ACriticalReview”.TheAmericanStatistician42:2[May1988],pp111–119,drawntomyattentionbyKarlOveHufthammer)

Youmayrememberthatthemeanandstandarddeviationhavethesameunitsastheoriginaldata,andthevariancehasthesquareofthoseunits.However,thekurtosishasnounits:it’sapurenumber,likeaz-score.

Thereferencestandardisanormaldistribution,whichhasakurtosisof3.Intokenofthis,oftentheexcesskurtosisispresented:excesskurtosisissimplykurtosis−3.Forexample,the“kurtosis”reportedbyExcelisactuallytheexcesskurtosis.

Anormaldistributionhaskurtosisexactly3(excesskurtosisexactly0).Anydistributionwithkurtosis≈3(excess≈0)iscalledmesokurtic.Adistributionwithkurtosis<3(excesskurtosis<0)iscalledplatykurtic.Comparedtoanormaldistribution,itscentralpeakislowerandbroader,anditstailsareshorterandthinner.Adistributionwithkurtosis>3(excesskurtosis>0)iscalledleptokurtic.Comparedtoanormaldistribution,itscentralpeakishigherandsharper,anditstailsarelongerandfatter.


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Visualizing

Kurtosisisunfortunatelyhardertopicturethanskewness,buttheseillustrations,suggestedbyWikipedia,shouldhelp.Allthreeofthesedistributionshavemeanof0,standarddeviationof1,andskewnessof0,andallareplottedonthesamehorizontalandverticalscale.Lookattheprogressionfromlefttoright,askurtosisincreases.

Uniform(min=−√3,max=√3)kurtosis=1.8,excess=−1.2

Normal(μ=0,σ=1)kurtosis=3,excess=0

Logistic(α=0,β=0.55153)kurtosis=4.2,excess=1.2

Movingfromtheillustrateduniformdistributiontoanormaldistribution,youseethatthe“shoulders”havetransferredsomeoftheirmasstothecenterandthetails.Inotherwords,theintermediatevalueshavebecomelesslikelyandthecentralandextremevalueshavebecomemorelikely.Thekurtosisincreaseswhilethestandarddeviationstaysthesame,becausemoreofthevariationisduetoextremevalues.

Movingfromthenormaldistributiontotheillustratedlogisticdistribution,thetrendcontinues.Thereisevenlessintheshouldersandevenmoreinthetails,andthecentralpeakishigherandnarrower.

Howfarcanthisgo?Whatarethesmallestandlargestpossiblevaluesofkurtosis?Thesmallestpossiblekurtosisis1(excesskurtosis−2),andthelargestis∞,asshownhere:

Discrete:equallylikelyvalueskurtosis=1,excess=−2

Student’st(df=4)kurtosis=∞,excess=∞

Adiscretedistributionwithtwoequallylikelyoutcomes,suchaswinningorlosingonthe lipofacoin,hasthelowestpossiblekurtosis.Ithasnocentralpeakandnorealtails,andyoucouldsaythatit’s“allshoulder”—it’sasplatykurticasadistributioncanbe.Attheotherextreme,Student’stdistribution


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withfourdegreesoffreedomhasin initekurtosis.Adistributioncan’tbeanymoreleptokurticthanthis.

Computing

Themomentcoef icientofkurtosisofadatasetiscomputedalmostthesamewayasthecoef icientofskewness:justchangetheexponent3to4intheformulas:

kurtosis:a4=m4/m22andexcesskurtosis:g2=a4−3

wherem4=∑(x−x̄)4/nandm2=∑(x−x̄)2/n

Again,theexcesskurtosisisgenerallyusedbecausetheexcesskurtosisofanormaldistributionis0.x̄isthemeanandnisthesamplesize,asusual.m4iscalledthefourthmomentofthedataset.m2isthevariance,thesquareofthestandarddeviation.

Justaswithvariance,standarddeviation,andkurtosis,theaboveisthe inalcomputationifyouhavedataforthewholepopulation.Butifyouhavedataforonlyasample,youhavetocomputethesampleexcesskurtosisusingthisformula,whichcomesfromJoanesandGill:

sampleexcesskurtosis:

Exceldoesn’tconcernitselfwithwhetheryouhaveasampleorapopulation:itsmeasureofkurtosisisalwaysG2.

Example:Let’scontinuewiththeexampleofthecollegemen’sheights,andcomputethekurtosisofthedataset.n=100,x̄=67.45inches,andthevariancem2=8.5275in²werecomputedearlier.

ClassMark,x Frequency,f x−x̄ (x−x̄)4f

61 5 -6.45 8653.84

64 18 -3.45 2550.05

67 42 -0.45 1.72

70 27 2.55 1141.63

73 8 5.55 7590.35

∑ n/a 19937.60

m4 n/a 199.3760

Finally,thekurtosisisa4=m4/m2²=199.3760/8.5275²=2.7418

andtheexcesskurtosisisg2=2.7418−3=−0.2582

Butthisisasample,notthepopulation,soyouhavetocomputethesampleexcesskurtosis:G2=[99/(98×97)][101×(−0.2582)+6)]=−0.2091

Thissampleisslightlyplatykurtic:itspeakisjustabitshallowerthanthepeakofanormal


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distribution.

Inferring

Yourdatasetisjustonesampledrawnfromapopulation.Howfarmusttheexcesskurtosisbefrom0,beforeyoucansaythatthepopulationalsohasnonzeroexcesskurtosis?

Theanswercomesinasimilarwaytothesimilarquestionaboutskewness.Youdividethesampleexcesskurtosisbythestandarderrorofkurtosis(SEK)togettheteststatistic,whichtellsyouhowmanystandarderrorsthesampleexcesskurtosisisfromzero:

teststatistic:Zg2=G2/SEKwhere

Theformulaisadaptedfrompage89ofDuncanCramer’sBasicStatisticsforSocialResearch(Routledge,1997).(Someauthorssuggest√(24/n),butforsmallsamplesthat’sapoorapproximation.Andanyway,we’veallgotcalculators,soyoumayaswelldoitright.)

ThecriticalvalueofZg2isapproximately2.(Thisisatwo-tailedtestofexcesskurtosis≠0atapproximatelythe0.05signi icancelevel.)

IfZg2<−2,thepopulationverylikelyhasnegativeexcesskurtosis(kurtosis<3,platykurtic),thoughyoudon’tknowhowmuch.IfZg2isbetween−2and+2,youcan’treachanyconclusionaboutthekurtosis:excesskurtosismightbepositive,negative,orzero.IfZg2>+2,thepopulationverylikelyhaspositiveexcesskurtosis(kurtosis>3,leptokurtic),thoughyoudon’tknowhowmuch.

Forthesamplecollegemen’sheights(n=100),youfoundexcesskurtosisofG2=−0.2091.Thesampleisplatykurtic,butisthisenoughtoletyousaythatthewholepopulationisplatykurtic(haslowerkurtosisthanthebellcurve)?

Firstcomputethestandarderrorofkurtosis:SEK=2×SES×√[(n²−1)/((n−3)(n+5))]

n=100,andtheSESwaspreviouslycomputedas0.2414.SEK=2×0.2414×√[(100²−1)/(97×105)]=0.4784

TheteststatisticisZg2=G2/SEK=−0.2091/0.4784=−0.44

Youcan’tsaywhetherthekurtosisofthepopulationisthesameasordifferentfromthekurtosisofanormaldistribution.

Assessing Normality

Therearemanywaystoassessnormality,andunfortunatelynoneofthemarewithoutproblems.Graphicalmethodsareagoodstart,suchasplottingahistogramandmakingaquantileplot.(YoucanindaTI-83programtodothoseatMATH200AProgram—StatisticsUtilitiesforTI-83/84.)

TheUniversityofSurreyhasagoodsurveyorproblemswithnormalitytests,atHowdoItestthenormalityofavariable’sdistribution?Thatpagerecommendsusingtheteststatisticsindividually.


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OnetestistheD'Agostino-Pearsonomnibustest,socalledbecauseitusestheteststatisticsforbothskewnessandkurtosistocomeupwithasinglep-value.Theteststatisticis

DP=Zg1²+Zg2²followsχ²withdf=2Youcanlookupthep-valueinatable,oruseχ²cdfonaTI-83orTI-84.

Caution:TheD’Agostino-Pearsontesthasatendencytoerronthesideofrejectingnormality,particularlywithsmallsamplesizes.DavidMoriarty,inhisStatCatutility,recommendsthatyoudon’tuseD’Agostino-Pearsonforsamplesizesbelow20.

Forcollegestudents’heightsyouhadteststatisticsZg1=−0.45forskewnessandZg2=0.44forkurtosis.Theomnibusteststatisticis

DP=Zg1²+Zg2²=0.45²+0.44²=0.3961andthep-valueforχ²(2df)>0.3961,fromatableorastatisticscalculator,is0.8203.Youcannotrejecttheassumptionofnormality.(Remember,youneveracceptthenullhypothesis,soyoucan’tsayfromthistestthatthedistributionisnormal.)Thehistogramsuggestsnormality,andthistestgivesyounoreasontorejectthatimpression.

Example 2: Size of Rat Litters

Forasecondillustrationofinferencesaboutskewnessandkurtosisofapopulation,I’lluseanexamplefromBulmer’sPrinciplesofStatistics:

Frequencydistributionoflittersizeinrats,n=815

Littersize 1 2 3 4 5 6 7 8 9 10 11 12

Frequency 7 33 58 116 125 126 121 107 56 37 25 4

I’llspareyouthedetailedcalculations,butyoushouldbeabletoverifythembyfollowingequation(1)andequation(2):

n=815,x̄=6.1252,m2=5.1721,m3=2.0316skewnessg1=0.1727andsampleskewnessG1=0.1730

Thesampleisroughlysymmetricbutslightlyskewedright,whichlooksaboutrightfromthehistogram.Thestandarderrorofskewnessis

SES=√[(6×815×814)/(813×816×818)]=0.0856DividingtheskewnessbytheSES,yougettheteststatistic

Zg1=0.1730/0.0856=2.02Sincethisisgreaterthan2,youcansaythatthereissomepositiveskewnessinthepopulation.Again,“somepositiveskewness”justmeansa iguregreaterthanzero;itdoesn’ttellusanythingmoreaboutthemagnitudeoftheskewness.

Ifyougoontocomputea95%con idenceintervalofskewnessfromequation(4),youget0.1730±2×0.0856=0.00to0.34.

Whataboutthekurtosis?Youshouldbeabletofollowequation(5)andcomputeafourthmomentofm4=67.3948.Youalreadyhavem2=5.1721,andtherefore


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kurtosisa4=m4/m2²=67.3948/5.1721²=2.5194excesskurtosisg2=2.5194−3=−0.4806sampleexcesskurtosisG2=[814/(813×812)][816×(−0.4806+6)=−0.4762

Sothesampleismoderatelylesspeakedthananormaldistribution.Again,thismatchesthehistogram,whereyoucanseethehigher“shoulders”.

Whatifanythingcanyousayaboutthepopulation?Forthisyouneedequation(7).Beginbycomputingthestandarderrorofkurtosis,usingn=815andthepreviouslycomputedSESof0.0.0856:

SEK=2×SES×√[(n²−1)/((n−3)(n+5))]SEK=2×0.0856×√[(815²−1)/(812×820)]=0.1711

anddivide:Zg2=G2/SEK=−0.4762/0.1711=−2.78

SinceZg2iscomfortablybelow−2,youcansaythatthedistributionofalllittersizesisplatykurtic,lesssharplypeakedthanthenormaldistribution.Butbecareful:youknowthatitisplatykurtic,butyoudon’tknowbyhowmuch.

Youalreadyknowthepopulationisnotnormal,butlet’sapplytheD’Agostino-Pearsontestanyway:DP=2.02²+2.78²=11.8088p-value=P(χ²(2)>11.8088)=0.0027

Thetestagreeswiththeseparatetestsofskewnessandkurtosis:sizesofratlitters,fortheentirepopulationofrats,isnotnormallydistributed.

What’s New

26Apr2011:identifythet(4)distributionandthebetadistributionsintheircaptions

20Dec2010:updatecitationstotextbooks

23Oct2010:restoreamissingminussign,thankstoEdwardB.Taylor

(interveningchangessuppressed)

13Dec2008:newdocumentThispageusessomematerialfromtheoldSkewnessandKurtosisontheTI-83/84,whichwas irstcreated12Jan2008andreplaced7Dec2008byMATH200BProgrampart1;buttherearenewexamplesandpicturesandconsiderableneworrewrittenmaterial.

ThispageisusedininstructionatTompkinsCortlandCommunityCollegeinDryden,NewYork;it’snotanof icialstatementoftheCollege.Pleasevisitwww.tc3.edu/instruct/sbrown/toreporterrorsorasktocopyit.

Forupdatesandnewinfo,gotohttp://www.tc3.edu/instruct/sbrown/stat/


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measures of shape_ skewness and kurtosis — math200 (tc3, brown)

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