P8130:BiostatisticalMethodsILecture2:DescriptiveStatistics

CodyChiuzan,PhDDepartmentofBiostatisticsMailmanSchoolofPublicHealth(MSPH)

Lecture1:Recap• IntrotoBiostatistics• TypesofData• StudyDesigns

DescriptiveStatistics

• Thecollectionandpresentationofthedatathroughgraphicalandnumericaldisplays

• Lookforpatternsinthedataandsummarizeinformation

• Measuresoflocation

• Measuresofdispersion

• Graphicaldisplay

MeasuresofLocation• Measuresoflocationorcentraltendency indicatethecenterofthedata

• Mean(average)

• Median(the50th percentile)

• Mode

MeasuresofLocation:MeanDefinition:thearithmeticmeanrepresentsthesumofallobservationsdividedbythenumberofobservations

Samplemeanforasampleofn observationsisgivenby:

𝑥=∑ 𝑥#/𝑛&#'(

Samplemeanisusedtoestimatethepopulationmeanμ whichistypicallyunknown

MeasuresofLocation:Mean• Themostcommonusedmeasureoflocation

• Overlysensitivetooutliers(unusualobservations),thusnotrecommendedifthedataareskewed

• Notappropriatefornominalorcategoricalvariables

MeasuresofLocation:MedianDefinition:Thesamplemedianiscomputedas:1. Ifnisodd,medianiscomputedas &)(

*𝑡ℎ largestiteminthesample

2. Ifniseven,medianiscomputedastheaveragebetween &*𝑎𝑛𝑑 &

*+ 1 th

largestitems

Example:Givenn=7(odd)totalsampleobservations,medianisthe1)(* = 4𝑡ℎ largestitemGivenn=10(even)totalsampleobservations,medianistheaverageofthe

(4* = 5𝑡ℎand (4* + 1 = 6𝑡ℎ largestitems

MeasuresofLocation:Median• Comparedtothemean,themedianisnotaffectedbyeveryvalueinthedatasetincludingoutliers

• Themedianisdefinedasthemiddlevalueorthe50th percentile• Thismeansthathalfofthedataarelessthanorequaltoit,andatleastaregreatertanorequaltoit

•Mediancalculationstartsbyfirstorderingthedata(increasingorder)• Appropriatemeasureforordinaldata

OtherMeasuresofLocationPercentiles:medianisthe50th percentile

• Ingeneral:thekth percentileisavaluesuchthatmostk%ofthedataaresmallerthanitand(100-k)%arelarger• Deciles:10th,20th,30th,…• Quartiles:25th (Q1),50th,75th (Q3)

• Question:whatdoesitmeanifyourGREscoreisinthe90thpercentile?

MeasuresofLocation:ModeDefinition:themostfrequentlyoccurringvalueinthedata

• Youcanhavemultiplemodesornone(really?)

• Problematicifthereisalargenumberofpossiblevalueswithinfrequentoccurrence

MeasuresofDispersionDescribethespreadofthedata:• Range

• Inter-quartilerange(IQR)

• Variance/Standarddeviation

• Coefficientofvariation(CV)

MeasuresofDispersionRange:Max– Min

Inter-quartilerange:IQR=75th (Q3)– 25th (Q1)

Sincetherangeonlydependsontheminimumandmaximumvalues,itcanbeinfluencedbytheextremes

Solution?UsetheIQR

MeasuresofDispersionPopulationVarianceistheaveragesquareddeviationfromthemean:

𝜎*=(<∑ (𝑥# − 𝜇)*<#'(

PopulationStandardDeviationisjustthesquarerootofthevariance:

𝜎 = 𝜎*�

Valuesoftenunknownandthenwereferbacktosample…

MeasuresofDispersionSampleVarianceistheaveragesquareddeviationfromthemean:

𝑠*= (&C(

∑ (𝑥# − 𝑥)*&#'(

PopulationStandardDeviationisjustthesquarerootofthevariance:

s= 𝑠*�

Lotsofchangesinnotationandalsoformula!!

MeasuresofDispersion

Meanandstandarddeviationsarethemostusedmeasuresoflocationandspread.Why?It’sallaboutthe…

Property:lineartransformationsdoaffectthesemeasures

Let𝑌 = 𝑐𝑋 + 𝑏 bealineartransformationavariableX

Meanof𝑌 = 𝑐𝑋 + 𝑏StandardDeviation𝑠H = 𝑐𝑠I

MeasuresofDispersion

CoefficientofVariation(CV)isameasurethatrelatesthemeanandthestandarddeviation.• Sometimesthevariancechangeswithitsmean

• Population:𝐶𝑉 = LM×100%

• Sample:𝐶𝑉 = QR×100%

• CVisunitless andcanbeinterpretedintermsofvariabilitytotheaverage

GraphicalDisplay

• Apictureisworthathousandwords(sometimes)

• Bargraphs

• Histograms

• Box-plots

• Scatterplots(laterinlinearregression)

BarGraph• Dataaredividedintogroupsandfrequenciesaredeterminedforeachgroup• Rectanglesareconstructedwiththebaseofconstantwidthandheightsproportionaltothefrequencies

Histogram• Numericalvaluesaregroupedintomeasurementsclasses,definedbyequal-lengthintervalsalongthenumericalscale• Eachvaluebelongstoonlyoneclass• Usually5-12classes• Likebargraph,thisplothasfrequenciesontheverticalaxis• Ifthemean>median:rightskew• Ifthemean<median:leftskew

Box-plot• ExtendsfromtheQ1(25th)totheQ3(75th)quartile– thebox• The‘whiskers’extendfromthesmallesttothelargestvalues• Ifoneofthewhiskersislong,itindicatesskewnessinthatdirection• IfadatavalueislessthanQ1–1.5(IQR)orgreaterthanQ3+1.5(IQR),thenitisconsideredanoutlierandgivenaseparatemarkontheboxplot

Readings

Rosner,FundamentalsofBiostatistics,Chapter2

• Sections:2.2– 2.6

• Sections:2.9– 2.10