Probability distributions, sampling distributions and central limit theorem

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Probability distributions Dr. S. A. Rizwan, M.D. Public Health Specialist SBCM, Joint Program – Riyadh Ministry of Health, Kingdom of Saudi Arabia

Transcript of Probability distributions, sampling distributions and central limit theorem

Page 1: Probability distributions, sampling distributions and central limit theorem

Probability distributionsDr. S. A. Rizwan, M.D.

PublicHealthSpecialistSBCM, JointProgram– Riyadh

MinistryofHealth,KingdomofSaudiArabia

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Learningobjectives

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Defineprobabilitydistributions• Describethecommontypesofprobabilitydistributions• Describesamplingdistribution• Understandthecentrallimittheorem

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Probabilitydistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Probabilitydistributionisamathematicalfunctionthatcanbethoughtofasprovidingtheprobabilityofoccurrenceofdifferentpossibleoutcomesinanexperiment.

• Thedistributionofastatisticaldataset(orapopulation)isalistingorfunctionshowingallthepossiblevalues(orintervals)ofthedataandhowoftentheyoccur.

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Probabilitydistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Section1:Binomialdistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 5

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Binomialdistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Followingconditionsneedtobesatisfiedforabinomialexperiment/distribution:• Thereisafixednumberofntrialscarriedout.• Theoutcomeofagiventrialiseithera“success”or“failure”.

• Theprobabilityofsuccess(p)remainsconstantfromtrialtotrial.

• Thetrialsareindependent, theoutcomeofatrialisnotaffectedbytheoutcomeofanyothertrial.

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Supposewehaven=40patientswhowillbereceivinganexperimentaltherapywhichisbelievedtobebetterthancurrenttreatmentswhichhistoricallyhavehada5-yearsurvivalrateof20%,i.e.theprobabilityof5-yearsurvivalisp=0.20

• Thusthenumberofpatientsoutof40inourstudysurvivingatleast5yearshasabinomialdistribution,i.e.X~BIN(40,0.20)

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Supposethatusingthenewtreatmentwefindthat16outofthe40patientssurviveatleast5yearspastdiagnosis.

• Q:Doesthisresultsuggestthatthenewtherapyhasabetter5-yearsurvivalratethanthecurrent,i.e.istheprobabilitythatapatientsurvivesatleast5yearsgreaterthan.20ora20%chancewhentreatedusingthenewtherapy?

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Weessentiallyaskourselvesthefollowing:

• Ifweassumethatnewtherapyisnobetterthanthecurrentwhatistheprobabilitywewouldseetheseresultsbychancevariationalone?

• Morespecificallywhatistheprobabilityofseeing16ormoresuccessesoutof40ifthesuccessrateofthenewtherapyis.20or20%aswell?

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Thisisabinomialexperimentsituation

• Therearen=40patientsandwearecountingthenumberofpatientsthatsurvive5ormoreyears.TheindividualpatientoutcomesareindependentandIFWEASSUMEthenewmethodisNOTbetter,thentheprobabilityofsuccessisp=.20or20%forallpatients.

• SoX=#of“successes”intheclinicaltrialisbinomialwithn=40andp=0.20,i.e.X~BIN(40,0.20)

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• X~BIN(40,.20),findtheprobabilitythat16ormorepatientssurviveatleast5years. probabilities are computed

automatically for greater than or equal to and less than or equal to x.

Enter n = sample sizex = observed # of “successes”p = probability of “success”

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Binomialdistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Thechancethatwewouldsee16ormorepatientsoutof40survivingatleast5yearsifthenewmethodhasthesamechanceofsuccessasthecurrentmethods(20%)isVERYSMALL,0.0029.

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Section2:Normaldistribution

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Normaldistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Thenormaldistributionisadescriptivemodelthatdescribesrealworldsituations.

• Itisdefinedasacontinuousfrequencydistributionofinfiniterange(cantakeanyvalue).

• Thisisthemostimportantprobabilitydistributioninstatisticsandimportanttoolinanalysisofepidemiologicaldata

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Normaldistribution- properties

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Thenormaldistributionisdefinedbytwoparameters,μandσ.• Youcandrawanormaldistributionforanyμandσcombination.• Thereisonenormaldistribution,Z,thatisspecial.• Ithasμ=0andσ=1.• Alsocalledstandardnormaldistribution.

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Normaldistribution- properties

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Mean=Median=Mode• Spread determinedbySD• Bell-shaped• Symmetryaboutthecenter• 50%ofvalueslessthanthemeanand50%greaterthanthemean

• Itapproacheshorizontalaxisasymptotically:- ∞<X<+∞

• Areaunderthecurveis1

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Normaldistribution- properties

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Normaldistribution- properties

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Normaldistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Assumingthenormalheartrate(H.R)innormalhealthyindividualsisnormallydistributedwithMean=70andStandardDeviation=10

• Q1.Whatareaunderthecurveisabove80beats/min?• Q2.Whatareaofthecurveisabove90beats/min?• Q3.Whatareaofthecurveisbetween50-90beats/min?• Q4.Whatareaofthecurveisabove100beats/min?• Q5.Whatareaofthecurveisbelow40beatsperminorabove100beatspermin?

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Normaldistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Normaldistribution– example

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Section3:Samplingdistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 22

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Samplingdistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Samplingdistributionofthemean– Atheoreticalprobabilitydistributionofsamplemeansthatwouldbeobtainedbydrawingfromthepopulationallpossiblesamplesofthesamesize.

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Samplingdistribution

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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CentralLimitTheorem

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Nomatterwhatwearemeasuring,thedistributionofanymeasureacrossallpossiblesampleswecouldtakeapproximatesanormaldistribution,aslongasthenumberofcasesineachsampleisabout30orlarger.

• Ifwerepeatedlydrewsamplesfromapopulationandcalculatedthemeanofavariableorapercentageor,thosesamplemeansorpercentageswouldbenormallydistributed.

• ItenablesustocalculateStandarderrorfromasinglesample

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Section4:Percentiles

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 26

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Percentiles

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Valuebelowwhichapercentageofdatafalls.• Forexample:80%ofpeopleareshorterthanyou,Thatmeansyouareatthe80thpercentile.Ifyourheightis1.85mthen"1.85m"isthe80thpercentileheightinthatgroup.

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Percentiles

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

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Percentiles

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Quantilesarecutpoints dividingtherangeofaprobabilitydistributionintocontiguousintervalswithequalprobabilities

• Median,tertiles,quartiles,quintiles,sextiles,septiles,octiles,deciles,percentilesorcentiles

• Inter-quartilerange

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Takehomemessages

Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Understandingthedistributionsletsusunderstandtheinferentialstatisticsbetter

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