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Probability distributionsDr. S. A. Rizwan, M.D.
PublicHealthSpecialistSBCM, JointProgram– Riyadh
MinistryofHealth,KingdomofSaudiArabia
Learningobjectives
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Defineprobabilitydistributions• Describethecommontypesofprobabilitydistributions• Describesamplingdistribution• Understandthecentrallimittheorem
Probabilitydistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Probabilitydistributionisamathematicalfunctionthatcanbethoughtofasprovidingtheprobabilityofoccurrenceofdifferentpossibleoutcomesinanexperiment.
• Thedistributionofastatisticaldataset(orapopulation)isalistingorfunctionshowingallthepossiblevalues(orintervals)ofthedataandhowoftentheyoccur.
Probabilitydistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Section1:Binomialdistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 5
Binomialdistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Followingconditionsneedtobesatisfiedforabinomialexperiment/distribution:• Thereisafixednumberofntrialscarriedout.• Theoutcomeofagiventrialiseithera“success”or“failure”.
• Theprobabilityofsuccess(p)remainsconstantfromtrialtotrial.
• Thetrialsareindependent, theoutcomeofatrialisnotaffectedbytheoutcomeofanyothertrial.
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)
Binomialdistribution– example
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Supposethatusingthenewtreatmentwefindthat16outofthe40patientssurviveatleast5yearspastdiagnosis.
• Q:Doesthisresultsuggestthatthenewtherapyhasabetter5-yearsurvivalratethanthecurrent,i.e.istheprobabilitythatapatientsurvivesatleast5yearsgreaterthan.20ora20%chancewhentreatedusingthenewtherapy?
Binomialdistribution– example
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Weessentiallyaskourselvesthefollowing:
• Ifweassumethatnewtherapyisnobetterthanthecurrentwhatistheprobabilitywewouldseetheseresultsbychancevariationalone?
• Morespecificallywhatistheprobabilityofseeing16ormoresuccessesoutof40ifthesuccessrateofthenewtherapyis.20or20%aswell?
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)
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”
Binomialdistribution– example
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Thechancethatwewouldsee16ormorepatientsoutof40survivingatleast5yearsifthenewmethodhasthesamechanceofsuccessasthecurrentmethods(20%)isVERYSMALL,0.0029.
Section2:Normaldistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 13
Normaldistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Thenormaldistributionisadescriptivemodelthatdescribesrealworldsituations.
• Itisdefinedasacontinuousfrequencydistributionofinfiniterange(cantakeanyvalue).
• Thisisthemostimportantprobabilitydistributioninstatisticsandimportanttoolinanalysisofepidemiologicaldata
Normaldistribution- properties
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Thenormaldistributionisdefinedbytwoparameters,μandσ.• Youcandrawanormaldistributionforanyμandσcombination.• Thereisonenormaldistribution,Z,thatisspecial.• Ithasμ=0andσ=1.• Alsocalledstandardnormaldistribution.
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
Normaldistribution- properties
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Normaldistribution- properties
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
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?
Normaldistribution– example
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Normaldistribution– example
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Section3:Samplingdistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 22
Samplingdistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Samplingdistributionofthemean– Atheoreticalprobabilitydistributionofsamplemeansthatwouldbeobtainedbydrawingfromthepopulationallpossiblesamplesofthesamesize.
Samplingdistribution
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
CentralLimitTheorem
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Nomatterwhatwearemeasuring,thedistributionofanymeasureacrossallpossiblesampleswecouldtakeapproximatesanormaldistribution,aslongasthenumberofcasesineachsampleisabout30orlarger.
• Ifwerepeatedlydrewsamplesfromapopulationandcalculatedthemeanofavariableorapercentageor,thosesamplemeansorpercentageswouldbenormallydistributed.
• ItenablesustocalculateStandarderrorfromasinglesample
Section4:Percentiles
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 26
Percentiles
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Valuebelowwhichapercentageofdatafalls.• Forexample:80%ofpeopleareshorterthanyou,Thatmeansyouareatthe80thpercentile.Ifyourheightis1.85mthen"1.85m"isthe80thpercentileheightinthatgroup.
Percentiles
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
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
Takehomemessages
Demystifying statistics! – Lecture 2 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Understandingthedistributionsletsusunderstandtheinferentialstatisticsbetter