﻿
Chapter26: Statistics
615
[FunctionFile]
gaminv
(
x
,
a
,
b
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
[FunctionFile]
geopdf
(
x
,
p
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
geometricdistributionwithparameterp.
Thegeometric distributionmodelsthenumberoffailures (x-1)ofaBernoullitrial
withprobabilitypbeforetheﬁrstsuccess(x).
[FunctionFile]
geocdf
(
x
,
p
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thegeometricdistributionwithparameterp.
Thegeometric distributionmodelsthenumberoffailures (x-1)ofaBernoullitrial
withprobabilitypbeforetheﬁrstsuccess(x).
[FunctionFile]
geoinv
(
x
,
p
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
geometricdistributionwithparameterp.
Thegeometric distributionmodelsthenumberoffailures (x-1)ofaBernoullitrial
withprobabilitypbeforetheﬁrstsuccess(x).
[FunctionFile]
hygepdf
(
x
,
t
,
m
,
n
)
Computetheprobabilitydensity function(PDF) at x ofthehypergeometricdistri-
butionwithparameterst,m,andn.
Thisistheprobabilityofobtainingxmarkeditemswhenrandomlydrawingasample
ofsizenwithoutreplacementfromapopulationoftotalsizetcontainingmmarked
items.
Theparameterst,m,andnmustbepositiveintegerswithmandnnotgreaterthan
t.
[FunctionFile]
hygecdf
(
x
,
t
,
m
,
n
)
Compute the cumulative distribution n function (CDF) at x x of f the e hypergeometric
distributionwithparameterst,m,andn.
Thisis theprobability ofobtainingnotmorethanx x markeditems s whenrandomly
drawing asample of size n without replacement from a population of totalsize t
containingmmarkeditems.
Theparameterst,m,andnmustbepositiveintegerswithmandnnotgreaterthan
t.
[FunctionFile]
hygeinv
(
x
,
t
,
m
,
n
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
hypergeometricdistributionwithparameterst,m,andn.
Thisistheprobabilityofobtainingxmarkeditemswhenrandomlydrawingasample
ofsizenwithoutreplacementfromapopulationoftotalsizetcontainingmmarked
items.
Theparameterst,m,andnmustbepositiveintegerswithmandnnotgreaterthan
t.
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616
GNUOctave
[FunctionFile]
kolmogorov_smirnov_cdf
(
x
,
tol
)
Returnthecumulativedistributionfunction(CDF)atxoftheKolmogorov-Smirnov
distribution.
Thisisdeﬁnedas
Q(x)=
X1
k= 1
( 1)
k
exp( 2k
2
x
2
)
forx>0.
Theoptionalparametertolspeciﬁes theprecisionuptowhichtheseries shouldbe
evaluated;thedefaultistol=eps.
[FunctionFile]
laplace_pdf
(
x
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
Laplacedistribution.
[FunctionFile]
laplace_cdf
(
x
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theLaplacedistribution.
[FunctionFile]
laplace_inv
(
x
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
Laplacedistribution.
[FunctionFile]
logistic_pdf
(
x
)
Foreachelementofx,computethePDFatx ofthelogisticdistribution.
[FunctionFile]
logistic_cdf
(
x
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thelogisticdistribution.
[FunctionFile]
logistic_inv
(
x
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
logisticdistribution.
[FunctionFile]
lognpdf
(
x
)
[FunctionFile]
lognpdf
(
x
,
mu
,
sigma
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
lognormaldistributionwithparametersmuandsigma.
Ifarandomvariablefollows thisdistribution,its logarithmis normally distributed
withmeanmuandstandarddeviationsigma.
Defaultvaluesaremu=0,sigma=1.
[FunctionFile]
logncdf
(
x
)
[FunctionFile]
logncdf
(
x
,
mu
,
sigma
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thelognormaldistributionwithparametersmuandsigma.
Ifarandomvariablefollows thisdistribution,its logarithmis normally distributed
withmeanmuandstandarddeviationsigma.
Defaultvaluesaremu=0,sigma=1.
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Chapter26: Statistics
617
[FunctionFile]
logninv
(
x
)
[FunctionFile]
logninv
(
x
,
mu
,
sigma
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
lognormaldistributionwithparametersmuandsigma.
Ifarandomvariablefollows thisdistribution,its logarithmis normally distributed
withmeanmuandstandarddeviationsigma.
Defaultvaluesaremu=0,sigma=1.
[FunctionFile]
nbinpdf
(
x
,
n
,
p
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
negativebinomialdistributionwithparametersnandp.
WhennisintegerthisisthePascaldistribution.Whennisextendedtorealnumbers
Thenumberoffailures inaBernoulliexperiment withsuccessprobability pbefore
then-thsuccessfollowsthisdistribution.
[FunctionFile]
nbincdf
(
x
,
n
,
p
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thenegativebinomialdistributionwithparametersnandp.
WhennisintegerthisisthePascaldistribution.Whennisextendedtorealnumbers
Thenumberoffailures inaBernoulliexperiment withsuccessprobability pbefore
then-thsuccessfollowsthisdistribution.
[FunctionFile]
nbininv
(
x
,
n
,
p
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
negativebinomialdistributionwithparametersnandp.
WhennisintegerthisisthePascaldistribution.Whennisextendedtorealnumbers
Thenumberoffailures inaBernoulliexperiment withsuccessprobability pbefore
then-thsuccessfollowsthisdistribution.
[FunctionFile]
normpdf
(
x
)
[FunctionFile]
normpdf
(
x
,
mu
,
sigma
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
normaldistributionwithmeanmuandstandarddeviationsigma.
Defaultvaluesaremu=0,sigma=1.
[FunctionFile]
normcdf
(
x
)
[FunctionFile]
normcdf
(
x
,
mu
,
sigma
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thenormaldistributionwithmeanmuandstandarddeviationsigma.
Defaultvaluesaremu=0,sigma=1.
[FunctionFile]
norminv
(
x
)
[FunctionFile]
norminv
(
x
,
mu
,
sigma
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
normaldistributionwithmeanmuandstandarddeviationsigma.
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618
GNUOctave
Defaultvaluesaremu=0,sigma=1.
[FunctionFile]
poisspdf
(
x
,
lambda
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
Poissondistributionwithparameterlambda.
[FunctionFile]
poisscdf
(
x
,
lambda
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thePoissondistributionwithparameterlambda.
[FunctionFile]
poissinv
(
x
,
lambda
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
Poissondistributionwithparameterlambda.
[FunctionFile]
stdnormal_pdf
(
x
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
standardnormaldistribution(mean=0,standarddeviation=1).
[FunctionFile]
stdnormal_cdf
(
x
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thestandardnormaldistribution(mean=0,standarddeviation=1).
[FunctionFile]
stdnormal_inv
(
x
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
standardnormaldistribution(mean=0,standarddeviation=1).
[FunctionFile]
tpdf
(
x
,
n
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)atxofthet
(Student)distributionwithndegreesoffreedom.
[FunctionFile]
tcdf
(
x
,
n
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thet(Student)distributionwithndegreesoffreedom.
[FunctionFile]
tinv
(
x
,
n
)
Foreachelementofx,computethequantile(theinverseoftheCDF)atx ofthet
(Student)distributionwithndegreesoffreedom.
This function is analogous to o looking in a a table e for r the e t-value of a a single-tailed
distribution.
[FunctionFile]
unidpdf
(
x
,
n
)
For each h element t of x, , compute the probability density y function n (PDF) ) at x x of f a
discreteuniformdistributionwhichassumestheintegervalues1–nwithequalprob-
ability.
Warning: The e underlying g implementation uses the e double class and will l only be
accuratefornbitmax(2
53
1onIEEE754compatiblesystems).
[FunctionFile]
unidcdf
(
x
,
n
)
For each h element of x, compute the cumulative distribution function (CDF) ) at t x
probability.
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Chapter26: Statistics
619
[FunctionFile]
unidinv
(
x
,
n
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
discreteuniformdistributionwhichassumestheintegervalues1–nwithequalprob-
ability.
[FunctionFile]
unifpdf
(
x
)
[FunctionFile]
unifpdf
(
x
,
a
,
b
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
uniformdistributionontheinterval[a,b].
Defaultvaluesarea=0,b=1.
[FunctionFile]
unifcdf
(
x
)
[FunctionFile]
unifcdf
(
x
,
a
,
b
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theuniformdistributionontheinterval[a,b].
Defaultvaluesarea=0,b=1.
[FunctionFile]
unifinv
(
x
)
[FunctionFile]
unifinv
(
x
,
a
,
b
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
uniformdistributionontheinterval[a,b].
Defaultvaluesarea=0,b=1.
[FunctionFile]
wblpdf
(
x
)
[FunctionFile]
wblpdf
(
x
,
scale
)
[FunctionFile]
wblpdf
(
x
,
scale
,
shape
)
Computetheprobabilitydensityfunction(PDF)atxoftheWeibulldistributionwith
scaleparameterscaleandshapeparametershape.
Thisisgivenby
shape
scale
shape
x
shape 1
e
(
x
scale
)
shape
forx0.
Defaultvaluesarescale =1,shape =1.
[FunctionFile]
wblcdf
(
x
)
[FunctionFile]
wblcdf
(
x
,
scale
)
[FunctionFile]
wblcdf
(
x
,
scale
,
shape
)
Computethecumulativedistributionfunction(CDF)atxoftheWeibulldistribution
withscaleparameterscale andshapeparametershape.
Thisisdeﬁnedas
1 e
x
scale
)
shape
forx0.
[FunctionFile]
wblinv
(
x
)
[FunctionFile]
wblinv
(
x
,
scale
)
[FunctionFile]
wblinv
(
x
,
scale
,
shape
)
Computethequantile(theinverseoftheCDF)atxoftheWeibulldistributionwith
scaleparameterscaleandshapeparametershape.
Defaultvaluesarescale =1,shape =1.
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620
GNUOctave
26.6 Tests
Octavecanperformmany diﬀerentstatisticaltests. . Thefollowingtable e summarizes the
availabletests.
Hypothesis
TestFunctions
Equalmeanvalues
anova,hotelling
test2,t
test
2,
welch
test,wilcoxon
test,z
test
2
Equalmedians
kruskal
wallis
test,sign
test
Equalvariances
bartlett
test,manova,var
test
Equaldistributions
chisquare
test
homogeneity,
kolmogorov
smirnov
test
2,u
test
Equalmarginalfrequencies
mcnemar
test
Equalsuccessprobabilities
prop
test
2
Independentobservations
chisquare
test
independence,
run
test
Uncorrelatedobservations
cor
test
Givenmeanvalue
hotelling
test,t
test,z
test
Observationsfromdistribution
kolmogorov
smirnov
test
Regression
f
test
regression,t
test
regression
The tests returnap-valuethat describestheoutcomeofthetest. . Assumingthatthe
testhypothesisistrue,thep-valueis theprobabilityofobtainingaworseresultthanthe
observedone. Solargep-valuescorrespondstoasuccessfultest.Usuallyatesthypothesis
isacceptedifthep-valueexceeds0.05.
[FunctionFile]
[pval, f, df_b, df_w] = anova
(
y
,
g
)
Performaone-wayanalysisofvariance(ANOVA).
Thegoalistotestwhetherthepopulationmeansofdatatakenfromkdiﬀerentgroups
areallequal.
Data may y be e given in a single vector r y with groups speciﬁedby y a corresponding
vectorofgrouplabelsg(e.g.,numbersfrom1tok). Thisisthegeneralformwhich
doesnotimposeanyrestrictiononthenumber ofdata ineachgrouporthegroup
labels.
Ify isamatrixandgisomitted,eachcolumnofy istreatedasagroup. . Thisform
isonlyappropriateforbalancedANOVAinwhichthenumbersofsamplesfromeach
groupareallequal.
Underthenullofconstantmeans,thestatisticf followsanFdistributionwithdf
b
anddf
wdegreesoffreedom.
Thep-value(1minustheCDFofthisdistributionatf)isreturnedinpval.
Ifnooutputargumentisgiven,thestandardone-wayANOVAtableisprinted.
Seealso: [manova],page623.
[FunctionFile]
[pval, chisq, df] = = bartlett_test
(
x1
,...)
PerformaBartletttestforthehomogeneityofvariancesinthedatavectors x1,x2,
...,xk,wherek >1.
Chapter26: Statistics
621
Under the nullof equal l variances, the test t statistic chisq q approximately y follows s a
chi-squaredistributionwithdf degreesoffreedom.
Thep-value(1minustheCDFofthisdistributionatchisq)isreturnedinpval.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
[pval, chisq, df] = = chisquare_test_homogeneity
(
x
,
y
,
c
)
Giventwo samples x x andy, performa a chisquare test for r homogeneity y of the null
hypothesis that t x x and y come e from the same distribution, based onthe partition
inducedbythe(strictlyincreasing)entriesofc.
Forlargesamples,theteststatisticchisqapproximatelyfollowsachisquaredistribu-
tionwithdf =length(c)degreesoffreedom.
Thep-value(1minustheCDFofthisdistributionatchisq)isreturnedinpval.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
[pval, chisq, df] = = chisquare_test_independence
(
x
)
Performachi-squaretestforindependencebasedonthecontingencytablex.
Under the null hypothesis of f independence, , chisq approximately y has s a a chi-square
distributionwithdf degreesoffreedom.
Thep-value(1minustheCDFofthisdistributionatchisq)ofthetestisreturnedin
pval.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
cor_test
(
x
,
y
,
alt
,
method
)
Testwhethertwosamplesx andy comefromuncorrelatedpopulations.
The optional argument stringalt describes s the e alternative hypothesis, and can be
"!="or"<>"(nonzero),">"(greaterthan0),or"<"(lessthan0).Thedefaultisthe
two-sidedcase.
The optional argument string method speciﬁes which correlationcoeﬃcient touse
fortesting. Ifmethodis"pearson"(default),the(usual)Pearson’sprodutmoment
correlationcoeﬃcientis used. . Inthis s case,thedata shouldcomefromabivariate
normaldistribution. Otherwise,theothertwomethodsoﬀernonparametricalterna-
tives.Ifmethodis"kendall",thenKendall’srankcorrelationtauisused.Ifmethod
is"spearman",thenSpearman’srankcorrelationrhoisused.Onlytheﬁrstcharacter
isnecessary.
Theoutputisastructurewiththefollowingelements:
pval
Thep-valueofthetest.
stat
Thevalueoftheteststatistic.
dist
Thedistributionoftheteststatistic.
params
Theparametersofthenulldistributionoftheteststatistic.
alternative
Thealternativehypothesis.
method
Themethodusedfortesting.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
622
GNUOctave
[FunctionFile]
[pval, f, df_num, df_den] = = f_test_regression
(
y
,
x
,
rr
,
r
)
PerformanFtestforthenullhypothesisrr*b=rinaclassicalnormalregression
modely=X*b+e.
Underthenull,theteststatisticf followsanFdistributionwithdf
numanddf
den
degreesoffreedom.
Thep-value(1minustheCDFofthisdistributionatf)isreturnedinpval.
Ifnotgivenexplicitly,r=0.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
[pval, tsq] ] = = hotelling_test
(
x
,
m
)
For a sample x x from a a multivariate normaldistributionwithunknown mean and
covariancematrix,testthenullhypothesisthatmean(x)==m.
Hotelling’s T
2
is returned d intsq. . Under r the e null, (n p)T
2
=(p(n 1)) ) has s anF
distributionwithpandn pdegreesoffreedom,wherenandparethenumbersof
samplesandvariables,respectively.
Thep-valueofthetestisreturnedinpval.
Ifnooutputargumentisgiven,thep-valueofthetestisdisplayed.
[FunctionFile]
[pval, tsq] ] = = hotelling_test_2
(
x
,
y
)
variables (columns), , unknownmeans s andunknown n equalcovariance matrices, test
thenullhypothesismean(x)==mean(y).
Hotelling’stwo-sampleT
2
isreturnedintsq.Underthenull,
(n
x
+n
y
p 1)T
2
p(n
x
+n
y
2)
hasanFdistributionwithpandn
x
+n
y
p 1degreesoffreedom,wheren
x
and
n
y
arethesamplesizesandpisthenumberofvariables.
Thep-valueofthetestisreturnedinpval.
Ifnooutputargumentisgiven,thep-valueofthetestisdisplayed.
[FunctionFile]
[pval, ks] = kolmogorov_smirnov_test
(
x
,
dist
,
params
,
alt
)
PerformaKolmogorov-Smirnovtestofthenullhypothesisthatthesamplex comes
fromthe(continuous)distributiondist.
ifFandGaretheCDFscorrespondingtothesampleanddist,respectively,thenthe
nullisthatF==G.
Theoptionalargumentparamscontainsalistofparametersofdist.Forexample,to
testwhetherasamplex comesfromauniformdistributionon[2,4],use
kolmogorov_smirnov_test (x, , "unif", 2, 4)
distcanbeanystringforwhichafunctiondistcdf thatcalculatestheCDFofdistri-
butiondistexists.
Chapter26: Statistics
623
Withtheoptionalargumentstringalt,thealternativeofinterestcanbeselected. If
altis "!="or "<>",thenullis testedagainstthetwo-sidedalternativeF!=G. . In
thiscase,theteststatisticks followsatwo-sidedKolmogorov-Smirnovdistribution.
Ifalt is">", , theone-sidedalternativeF > > Gis considered. . Similarly y for "<", , the
one-sided alternative F > > Gis considered. . Inthis s case,the test statistic ks has a
one-sidedKolmogorov-Smirnovdistribution.Thedefaultisthetwo-sidedcase.
Thep-valueofthetestisreturnedinpval.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
[pval, ks, d] = kolmogorov_smirnov_test_2
(
x
,
y
,
alt
)
Performa2-sampleKolmogorov-Smirnovtestofthenullhypothesisthatthesamples
x andy comefromthesame(continuous)distribution.
IfFandGaretheCDFscorrespondingtothex andy y samples,respectively,then
thenullisthatF==G.
Withtheoptionalargumentstringalt,thealternativeofinterestcanbeselected. If
altis "!="or "<>",thenullis testedagainstthetwo-sidedalternativeF!=G. . In
thiscase,theteststatisticks followsatwo-sidedKolmogorov-Smirnovdistribution.
Ifalt is">", , theone-sidedalternativeF > > Gis considered. . Similarly y for "<", , the
one-sided alternative F < < Gis considered. . Inthis s case,the test statistic ks has a
one-sidedKolmogorov-Smirnovdistribution.Thedefaultisthetwo-sidedcase.
Thep-valueofthetestisreturnedinpval.
The third d returned d value, d, is s the test statistic, , the maximum m vertical distance
betweenthetwocumulativedistributionfunctions.
Ifnooutputargumentisgiven,thep-valueisdisplayed.
[FunctionFile]
[pval, k, df] = kruskal_wallis_test
(
x1
,...)
PerformaKruskal-Wallisone-factoranalysisofvariance.
Supposeavariableisobservedfork >1diﬀerentgroups,andletx1,...,xk bethe
correspondingdatavectors.
Underthenullhypothesisthattheranksinthepooledsamplearenotaﬀectedbythe
groupmemberships,theteststatistick isapproximatelychi-squarewithdf f =k k -1
degreesoffreedom.
Ifthedatacontainsties(somevalueappearsmorethanonce)kisdividedby
1-sum
ties/(n^3-n)
wheresum
ties isthesumoft^2-tovereachgroupoftieswheret isthenumber
oftiesinthegroupandnisthetotalnumberofvaluesintheinputdata. Formore