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Chapter3,PartA

DescriptiveStatistics:NumericalMeasuresMeasuresofLocationMeasuresofVariabilityMeasuresofLocationIfthemeasuresarecomputedfordatafromasample,theyarecalledsamplestatistics.Ifthemeasuresarecomputedfordatafromapopulation,theyarecalledpopulationparameters.Asamplestatisticisreferredtoasthepointestimatorofthecorrespondingpopulationparameter.MeanMedianModePercentilesQuartilesMeanThemeanofadatasetistheaverageofallthedatavalues.Thesamplemeanisthepointestimatorofthepopulationmeanm.Perhapsthemostimportantmeasureoflocationisthemean.Themeanprovidesameasureofcentrallocation.SampleMeanNumberofobservationsinthesampleSumofthevaluesofthenobservationsPopulationMeanmNumberofobservationsinthepopulationSumofthevaluesoftheNobservationsSeventyefficiencyapartmentswererandomlysampledinasmallcollegetown.Themonthlyrentpricesfortheseapartmentsarelistedbelow.SampleMeanExample:ApartmentRentsSampleMeanExample:ApartmentRentsMedianWheneveradatasethasextremevalues,themedianisthepreferredmeasureofcentrallocation.Afewextremelylargeincomesorpropertyvaluescaninflatethemean.Themedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.Themedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.Median12141926271827Foranoddnumberofobservations:

inascendingorder261827121427197observationsthemedianisthemiddlevalue.Median=1912141926271827MedianForanevennumberofobservations:

inascendingorder261827121427308observationsthemedianistheaverageofthemiddletwovalues.Median=(19+26)/2=22.51930MedianAveragingthe35thand36thdatavalues:Median=(475+475)/2=475Note:Dataisinascendingorder.Example:ApartmentRentsTrimmedMeanItisobtainedbydeletingapercentageofthesmallestandlargestvaluesfromadatasetandthencomputingthemeanoftheremainingvalues.Forexample,the5%trimmedmeanisobtainedbyremovingthesmallest5%andthelargest5%ofthedatavaluesandthencomputingthemeanoftheremainingvalues.Anothermeasure,sometimesusedwhenextremevaluesarepresent,isthetrimmedmean.ModeThemodeofadatasetisthevaluethatoccurswithgreatestfrequency.Thegreatestfrequencycanoccurattwoormoredifferentvalues.Ifthedatahaveexactlytwomodes,thedataare

bimodal.Ifthedatahavemorethantwomodes,thedataare

multimodal.Caution:Ifthedataarebimodalormultimodal,Excel’sMODEfunctionwillincorrectlyidentifyasinglemode.Mode450occurredmostfrequently(7times)Mode=450Note:Dataisinascendingorder.Example:ApartmentRentsPercentilesApercentileprovidesinformationabouthowthedataarespreadovertheintervalfromthesmallestvaluetothelargestvalue.Admissiontestscoresforcollegesanduniversitiesarefrequentlyreportedintermsofpercentiles.Thepthpercentileofadatasetisavaluesuchthatatleastppercentoftheitemstakeonthisvalueorlessandatleast(100-p)percentoftheitemstakeonthisvalueormore.PercentilesArrangethedatainascendingorder.Computeindexi,thepositionofthepthpercentile.i=(p/100)nIfiisnotaninteger,roundup.Thep

thpercentileisthevalueinthei

thposition.Ifiisaninteger,thep

thpercentileistheaverageofthevaluesinpositionsiandi

+1.80thPercentilei=(p/100)n=(80/100)70=56Averagingthe56thand57thdatavalues:80thPercentile=(535+549)/2=542Note:Dataisinascendingorder.Example:ApartmentRents80thPercentile“Atleast80%oftheitemstakeonavalueof542orless.”“Atleast20%oftheitemstakeonavalueof542ormore.”56/70=.8or80%14/70=.2or20%Example:ApartmentRentsQuartilesQuartilesarespecificpercentiles.FirstQuartile=25thPercentileSecondQuartile=50thPercentile=MedianThirdQuartile=75thPercentileThirdQuartileThirdquartile=75thpercentilei=(p/100)n=(75/100)70=52.5=53Thirdquartile=525Note:Dataisinascendingorder.Example:ApartmentRentsMeasuresofVariabilityItisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.MeasuresofVariabilityRangeInterquartileRangeVarianceStandardDeviationCoefficientofVariationRangeTherangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalues.Itisthesimplestmeasureofvariability.Itisverysensitivetothesmallestandlargestdatavalues.RangeRange=largestvalue-smallestvalueRange=615-425=190Note:Dataisinascendingorder.Example:ApartmentRentsInterquartileRangeTheinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.Itistherangeforthemiddle50%ofthedata.Itovercomesthesensitivitytoextremedatavalues.InterquartileRange3rdQuartile(Q3)=5251stQuartile(Q1)=445InterquartileRange=Q3-Q1=525-445=80Note:Dataisinascendingorder.Example:ApartmentRentsThevarianceisameasureofvariabilitythatutilizesallthedata.VarianceItisbasedonthedifferencebetweenthevalueofeachobservation(xi)andthemean(forasample,

mforapopulation).Thevarianceisusefulincomparingthevariabilityoftwoormorevariables.VarianceThevarianceiscomputedasfollows:

Thevarianceistheaverageofthesquared

differencesbetweeneachdatavalueandthemean.forasampleforapopulationStandardDeviationThestandarddeviationofadatasetisthepositiv

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