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StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter3,PartA

DescriptiveStatistics:NumericalMeasuresMeasuresofLocationMeasuresofVariabilityChapter3,PartA

DescriptiveMeasuresofLocationIfthemeasuresarecomputedfordatafromasample,theyarecalledsamplestatistics.Ifthemeasuresarecomputedfordatafromapopulation,theyarecalledpopulationparameters.Asamplestatisticisreferredtoasthepointestimatorofthecorrespondingpopulationparameter.MeanMedianModePercentilesQuartilesMeasuresofLocationIfthemeaMeanThemeanofadatasetistheaverageofallthedatavalues.Thesamplemeanisthepointestimatorofthepopulationmeanm.Perhapsthemostimportantmeasureoflocationisthemean.Themeanprovidesameasureofcentrallocation.MeanThemeanofadatasetisSampleMeanNumberofobservationsinthesampleSumofthevaluesofthenobservationsSampleMeanNumberofSumofthPopulationMeanmNumberofobservationsinthepopulationSumofthevaluesoftheNobservationsPopulationMeanmNumberofSumSeventyefficiencyapartmentswererandomlysampledinasmallcollegetown.Themonthlyrentpricesfortheseapartmentsarelistedbelow.SampleMeanExample:ApartmentRentsSeventyefficiencyapartmSampleMeanExample:ApartmentRentsSampleMeanExample:ApartmMedianWheneveradatasethasextremevalues,themedianisthepreferredmeasureofcentrallocation.Afewextremelylargeincomesorpropertyvaluescaninflatethemean.Themedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.Themedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.MedianWheneveradatasethMedian12141926271827Foranoddnumberofobservations:

inascendingorder261827121427197observationsthemedianisthemiddlevalue.Median=19Median12141926271827Foran12141926271827MedianForanevennumberofobservations:

inascendingorder261827121427308observationsthemedianistheaverageofthemiddletwovalues.Median=(19+26)/2=22.5193012141926271827MedianForanMedianAveragingthe35thand36thdatavalues:Median=(475+475)/2=475Note:Dataisinascendingorder.Example:ApartmentRentsMedianAveragingthe35thand3TrimmedMeanItisobtainedbydeletingapercentageofthesmallestandlargestvaluesfromadatasetandthencomputingthemeanoftheremainingvalues.Forexample,the5%trimmedmeanisobtainedbyremovingthesmallest5%andthelargest5%ofthedatavaluesandthencomputingthemeanoftheremainingvalues.Anothermeasure,sometimesusedwhenextremevaluesarepresent,isthetrimmedmean.TrimmedMeanItisobtainedModeThemodeofadatasetisthevaluethatoccurswithgreatestfrequency.Thegreatestfrequencycanoccurattwoormoredifferentvalues.Ifthedatahaveexactlytwomodes,thedataare

bimodal.Ifthedatahavemorethantwomodes,thedataare

multimodal.Caution:Ifthedataarebimodalormultimodal,Excel’sMODEfunctionwillincorrectlyidentifyasinglemode.ModeThemodeofadatasetMode450occurredmostfrequently(7times)Mode=450Note:Dataisinascendingorder.Example:ApartmentRentsMode450occurredmostfrequentPercentilesApercentileprovidesinformationabouthowthedataarespreadovertheintervalfromthesmallestvaluetothelargestvalue.Admissiontestscoresforcollegesanduniversitiesarefrequentlyreportedintermsofpercentiles.Thepthpercentileofadatasetisavaluesuchthatatleastppercentoftheitemstakeonthisvalueorlessandatleast(100-p)percentoftheitemstakeonthisvalueormore.PercentilesApercentileprovPercentilesArrangethedatainascendingorder.Computeindexi,thepositionofthepthpercentile.i=(p/100)nIfiisnotaninteger,roundup.Thep

thpercentileisthevalueinthei

thposition.Ifiisaninteger,thep

thpercentileistheaverageofthevaluesinpositionsiandi

+1.PercentilesArrangethedatai80thPercentilei=(p/100)n=(80/100)70=56Averagingthe56thand57thdatavalues:80thPercentile=(535+549)/2=542Note:Dataisinascendingorder.Example:ApartmentRents80thPercentilei=(p/100)n=80thPercentile“Atleast80%oftheitemstakeonavalueof542orless.”“Atleast20%oftheitemstakeonavalueof542ormore.”56/70=.8or80%14/70=.2or20%Example:ApartmentRents80thPercentile“Atleast80%oQuartilesQuartilesarespecificpercentiles.FirstQuartile=25thPercentileSecondQuartile=50thPercentile=MedianThirdQuartile=75thPercentileQuartilesQuartilesarespecThirdQuartileThirdquartile=75thpercentilei=(p/100)n=(75/100)70=52.5=53Thirdquartile=525Note:Dataisinascendingorder.Example:ApartmentRentsThirdQuartileThirdquartile=MeasuresofVariabilityItisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.MeasuresofVariabilityItiMeasuresofVariabilityRangeInterquartileRangeVarianceStandardDeviationCoefficientofVariationMeasuresofVariabilityRangeInRangeTherangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalues.Itisthesimplestmeasureofvariability.Itisverysensitivetothesmallestandlargestdatavalues.RangeTherangeofadataseRangeRange=largestvalue-smallestvalueRange=615-425=190Note:Dataisinascendingorder.Example:ApartmentRentsRangeRange=largestvalue-sInterquartileRangeTheinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.Itistherangeforthemiddle50%ofthedata.Itovercomesthesensitivitytoextremedatavalues.InterquartileRangeTheinteInterquartileRange3rdQuartile(Q3)=5251stQuartile(Q1)=445InterquartileRange=Q3-Q1=525-445=80Note:Dataisinascendingorder.Example:ApartmentRentsInterquartileRange3rdQuartilThevarianceisameasureofvariabilitythatutilizesallthedata.VarianceItisbasedonthedifferencebetweenthevalueofeachobservation(xi)andthemean(forasample,

mforapopulation).Thevarianceisusefulincomparingthevariabilityoftwoormorevariables.ThevarianceisameasureofVarianceThevarianceiscomputedasfollows:

Thevarianceistheaverageofthesquared

differencesbetweeneachdatavalueandthemean.forasampleforapopulationVarianceThevarianceiscompuStandardDeviationThestandarddeviationofadatasetisthepositivesquarerootofthevariance.Itismeasuredinthesameunitsasthedata,makingitmoreeasilyinterpretedthanthevariance.StandardDeviationThestandarThestandarddeviationiscomputedasfollows:

forasampleforapopulationStandardDeviationThestandarddeviationiscomThecoefficientofvariationiscomputedasfollows:

CoefficientofVariationThecoefficientofvariationindicateshowlargethestandarddeviationisinrelationtothemean.forasampleforapopulationThecoefficientofvariationthestandarddeviationisabout11%ofthemeanVarianceStandardDeviationCoefficientofVariationSampleVariance,StandardDeviation,AndCoefficientofVariationExample:ApartmentRentsthestandardVarianceStandardDEndofChapter3,PartAEndofChapter3,PartAStatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter3,PartA

DescriptiveStatistics:NumericalMeasuresMeasuresofLocationMeasuresofVariabilityChapter3,PartA

DescriptiveMeasuresofLocationIfthemeasuresarecomputedfordatafromasample,theyarecalledsamplestatistics.Ifthemeasuresarecomputedfordatafromapopulation,theyarecalledpopulationparameters.Asamplestatisticisreferredtoasthepointestimatorofthecorrespondingpopulationparameter.MeanMedianModePercentilesQuartilesMeasuresofLocationIfthemeaMeanThemeanofadatasetistheaverageofallthedatavalues.Thesamplemeanisthepointestimatorofthepopulationmeanm.Perhapsthemostimportantmeasureoflocationisthemean.Themeanprovidesameasureofcentrallocation.MeanThemeanofadatasetisSampleMeanNumberofobservationsinthesampleSumofthevaluesofthenobservationsSampleMeanNumberofSumofthPopulationMeanmNumberofobservationsinthepopulationSumofthevaluesoftheNobservationsPopulationMeanmNumberofSumSeventyefficiencyapartmentswererandomlysampledinasmallcollegetown.Themonthlyrentpricesfortheseapartmentsarelistedbelow.SampleMeanExample:ApartmentRentsSeventyefficiencyapartmSampleMeanExample:ApartmentRentsSampleMeanExample:ApartmMedianWheneveradatasethasextremevalues,themedianisthepreferredmeasureofcentrallocation.Afewextremelylargeincomesorpropertyvaluescaninflatethemean.Themedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.Themedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.MedianWheneveradatasethMedian12141926271827Foranoddnumberofobservations:

inascendingorder261827121427197observationsthemedianisthemiddlevalue.Median=19Median12141926271827Foran12141926271827MedianForanevennumberofobservations:

inascendingorder261827121427308observationsthemedianistheaverageofthemiddletwovalues.Median=(19+26)/2=22.5193012141926271827MedianForanMedianAveragingthe35thand36thdatavalues:Median=(475+475)/2=475Note:Dataisinascendingorder.Example:ApartmentRentsMedianAveragingthe35thand3TrimmedMeanItisobtainedbydeletingapercentageofthesmallestandlargestvaluesfromadatasetandthencomputingthemeanoftheremainingvalues.Forexample,the5%trimmedmeanisobtainedbyremovingthesmallest5%andthelargest5%ofthedatavaluesandthencomputingthemeanoftheremainingvalues.Anothermeasure,sometimesusedwhenextremevaluesarepresent,isthetrimmedmean.TrimmedMeanItisobtainedModeThemodeofadatasetisthevaluethatoccurswithgreatestfrequency.Thegreatestfrequencycanoccurattwoormoredifferentvalues.Ifthedatahaveexactlytwomodes,thedataare

bimodal.Ifthedatahavemorethantwomodes,thedataare

multimodal.Caution:Ifthedataarebimodalormultimodal,Excel’sMODEfunctionwillincorrectlyidentifyasinglemode.ModeThemodeofadatasetMode450occurredmostfrequently(7times)Mode=450Note:Dataisinascendingorder.Example:ApartmentRentsMode450occurredmostfrequentPercentilesApercentileprovidesinformationabouthowthedataarespreadovertheintervalfromthesmallestvaluetothelargestvalue.Admissiontestscoresforcollegesanduniversitiesarefrequentlyreportedintermsofpercentiles.Thepthpercentileofadatasetisavaluesuchthatatleastppercentoftheitemstakeonthisvalueorlessandatleast(100-p)percentoftheitemstakeonthisvalueormore.PercentilesApercentileprovPercentilesArrangethedatainascendingorder.Computeindexi,thepositionofthepthpercentile.i=(p/100)nIfiisnotaninteger,roundup.Thep

thpercentileisthevalueinthei

thposition.Ifiisaninteger,thep

thpercentileistheaverageofthevaluesinpositionsiandi

+1.PercentilesArrangethedatai80thPercentilei=(p/100)n=(80/100)70=56Averagingthe56thand57thdatavalues:80thPercentile=(535+549)/2=542Note:Dataisinascendingorder.Example:ApartmentRents80thPercentilei=(p/100)n=80thPercentile“Atleast80%oftheitemstakeonavalueof542orless.”“Atleast20%oftheitemstakeonavalueof542ormore.”56/70=.8or80%14/70=.2or20%Example:ApartmentRents80thPercentile“Atleast80%oQuartilesQuartilesarespecificpercentiles.FirstQuartile=25thPercentileSecondQuartile=50thPercentile=MedianThirdQuartile=75thPercentileQuartilesQuartilesarespecThirdQuartileThirdquartile=75thpercentilei=(p/100)n=(75/100)70=52.5=53Thirdquartile=525Note:Dataisinascendingorder.Example:ApartmentRentsThirdQuartileThirdquartile=MeasuresofVariabilityItisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.MeasuresofVariabilityItiMeasuresofVariabilityRangeInterquartileRangeVarianceStandardDeviationCoefficientofVariationMeasuresofVariabilityRangeInRangeTherangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalues.Itisthesimplestmeasureofvariability.Itisverysensitivetothesmallestandlargestdatavalues.RangeTherangeofadataseRangeRange=largestvalue-smallestvalueRange=615-425=190Note:Dataisinascendingorder.Example:ApartmentRentsRangeRange=largestvalue-sInterquartileRangeTheinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.Itistherangeforthemiddle50%ofthedata.Itovercomesthesensitivitytoextremedatavalues.InterquartileRangeTheinteInterquartileRange3rdQuartile(Q3)=5251stQuartile(

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