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NumericalDescriptiveMeasuresChapter3NumericalDescriptiveMeasuresInthischapter,youlearnto:
Describethepropertiesofcentraltendency,variation,andshapeinnumericaldataConstructandinterpretaboxplotComputedescriptivesummarymeasuresforapopulationCalculatethecovarianceandthecoefficientofcorrelationObjectivesInthischapter,youlearnto:SummaryDefinitionsThecentraltendencyistheextenttowhichthevaluesofanumericalvariablegrouparoundatypicalorcentralvalue.Thevariationistheamountofdispersionorscatteringawayfromacentralvaluethatthevaluesofanumericalvariableshow.Theshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue.DCOVASummaryDefinitionsThecentralMeasuresofCentralTendency:
TheMeanThearithmeticmean(oftenjustcalledthe“mean”)isthemostcommonmeasureofcentraltendencyForasampleofsizen:SamplesizeObservedvaluesTheithvaluePronouncedx-barDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheMean(con’t)ThemostcommonmeasureofcentraltendencyMean=sumofvaluesdividedbythenumberofvaluesAffectedbyextremevalues(outliers)11121314151617181920Mean=1311121314151617181920Mean=14DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheMedianInanorderedarray,themedianisthe“middle”number(50%above,50%below)
LesssensitivethanthemeantoextremevaluesMedian=13Median=131112131415161718192011121314151617181920DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
LocatingtheMedianThelocationofthemedianwhenthevaluesareinnumericalorder(smallesttolargest):Ifthenumberofvaluesisodd,themedianisthemiddlenumberIfthenumberofvaluesiseven,themedianistheaverageofthetwomiddlenumbers Notethatisnotthevalueofthemedian,onlythepositionofthemedianintherankeddataDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheModeValuethatoccursmostoftenNotaffectedbyextremevaluesUsedforeithernumericalorcategoricaldataTheremaybenomodeTheremaybeseveralmodes01234567891011121314
Mode=90123456NoModeDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
ReviewExampleHousePrices:
$2,000,000$500,000
$300,000
$100,000
$100,000Sum$3,000,000Mean:($3,000,000/5) =$600,000Median:middlevalueofrankeddata
=$300,000Mode:mostfrequentvalue
=$100,000DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
WhichMeasuretoChoose?Themeanisgenerallyused,unlessextremevalues(outliers)exist.Themedianisoftenused,sincethemedianisnotsensitivetoextremevalues.Forexample,medianhomepricesmaybereportedforaregion;itislesssensitivetooutliers.Insomesituationsitmakessensetoreportboththemeanandthemedian.DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
SummaryCentralTendencyArithmeticMeanMedianModeMiddlevalueintheorderedarrayMostfrequentlyobservedvalueDCOVAMeasuresofCentralTendency:
Samecenter,differentvariationMeasuresofVariationMeasuresofvariationgiveinformationonthespreadorvariabilityordispersionofthedatavalues.
VariationStandardDeviationCoefficientofVariationRangeVarianceDCOVASamecenter,MeasuresofVariaMeasuresofVariation:
TheRangeSimplestmeasureofvariationDifferencebetweenthelargestandthesmallestvalues:Range=Xlargest–Xsmallest01234567891011121314Range=13-1=12Example:DCOVAMeasuresofVariation:
TheRanMeasuresofVariation:
WhyTheRangeCanBeMisleadingDoesnotaccountforhowthedataaredistributedSensitivetooutliers789101112Range=12-7=5789101112Range=12-7=5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range=5-1=4Range=120-1=119DCOVAMeasuresofVariation:
WhyTheAverage(approximately)ofsquareddeviationsofvaluesfromthemeanSample
variance:MeasuresofVariation:
TheSampleVarianceWhere
=arithmeticmeann=samplesizeXi=ithvalueofthevariableXDCOVAAverage(approximately)ofsquMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthevarianceHasthesameunitsastheoriginaldataSample
standarddeviation:MeasuresofVariation:
TheSampleStandardDeviationDCOVAMostcommonlyusedmeasureofMeasuresofVariation:
TheStandardDeviationStepsforComputingStandardDeviation1. Computethedifferencebetweeneachvalueandthemean.2. Squareeachdifference.3. Addthesquareddifferences.4. Dividethistotalbyn-1togetthesamplevariance.5. Takethesquarerootofthesamplevariancetogetthesamplestandarddeviation.DCOVAMeasuresofVariation:
TheStaMeasuresofVariation:
SampleStandardDeviation:
CalculationExampleSample
Data(Xi):1012141517181824n=8Mean=X=16Ameasureofthe“average”scatteraroundthemeanDCOVAMeasuresofVariation:
SampleMeasuresofVariation:
ComparingStandardDeviationsMean=15.5S=3.338
11121314151617181920211112131415161718192021DataBDataAMean=15.5S=0.9261112131415161718192021Mean=15.5S=4.567DataCDCOVAMeasuresofVariation:
CompariMeasuresofVariation:
ComparingStandardDeviationsSmallerstandarddeviationLargerstandarddeviationDCOVAMeasuresofVariation:
CompariMeasuresofVariation:
SummaryCharacteristicsThemorethedataarespreadout,thegreatertherange,variance,andstandarddeviation.Themorethedataareconcentrated,thesmallertherange,variance,andstandarddeviation.Ifthevaluesareallthesame(novariation),allthesemeasureswillbezero.Noneofthesemeasuresareevernegative.DCOVAMeasuresofVariation:
SummaryMeasuresofVariation:
TheCoefficientofVariationMeasuresrelativevariationAlwaysinpercentage(%)ShowsvariationrelativetomeanCanbeusedtocomparethevariabilityoftwoormoresetsofdatameasuredindifferentunitsDCOVAMeasuresofVariation:
TheCoeMeasuresofVariation:
ComparingCoefficientsofVariationStockA:Averagepricelastyear=$50Standarddeviation=$5StockB:Averagepricelastyear=$100Standarddeviation=$5Bothstockshavethesamestandarddeviation,butstockBislessvariablerelativetoitspriceDCOVAMeasuresofVariation:
CompariMeasuresofVariation:
ComparingCoefficientsofVariation(con’t)StockA:Averagepricelastyear=$50Standarddeviation=$5StockC:Averagepricelastyear=$8Standarddeviation=$2StockChasamuchsmallerstandarddeviationbutamuchhighercoefficientofvariationDCOVAMeasuresofVariation:
CompariLocatingExtremeOutliers:
Z-ScoreTocomputetheZ-scoreofadatavalue,subtractthemeananddividebythestandarddeviation.TheZ-scoreisthenumberofstandarddeviationsadatavalueisfromthemean.AdatavalueisconsideredanextremeoutlierifitsZ-scoreislessthan-3.0orgreaterthan+3.0.ThelargertheabsolutevalueoftheZ-score,thefartherthedatavalueisfromthemean.DCOVALocatingExtremeOutliers:
Z-SLocatingExtremeOutliers:
Z-ScorewhereXrepresentsthedatavalue Xisthesamplemean SisthesamplestandarddeviationDCOVALocatingExtremeOutliers:
Z-SLocatingExtremeOutliers:
Z-ScoreSupposethemeanmathSATscoreis490,withastandarddeviationof100.ComputetheZ-scoreforatestscoreof620.Ascoreof620is1.3standarddeviationsabovethemeanandwouldnotbeconsideredanoutlier.DCOVALocatingExtremeOutliers:
Z-SShapeofaDistributionDescribeshowdataaredistributedTwousefulshaperelatedstatisticsare:SkewnessMeasurestheextenttowhichdatavaluesarenotsymmetricalKurtosisKurtosisaffectsthepeakednessofthecurveofthedistribution—thatis,howsharplythecurverisesapproachingthecenterofthedistributionDCOVAShapeofaDistributionDescribShapeofaDistribution(Skewness)MeasurestheextenttowhichdataisnotsymmetricalMean=Median
Mean<Median
Median<MeanRight-SkewedLeft-SkewedSymmetricDCOVASkewnessStatistic<0 0 >0ShapeofaDistribution(SkewnShapeofaDistribution--Kurtosismeasureshowsharplythecurverisesapproachingthecenterofthedistribution
SharperPeakThanBell-Shaped(Kurtosis>0)FlatterThanBell-Shaped(Kurtosis<0)Bell-Shaped(Kurtosis=0)DCOVAShapeofaDistribution--KGeneralDescriptiveStatsUsingMicrosoftExcelFunctionsDCOVAGeneralDescriptiveStatsUsinGeneralDescriptiveStatsUsingMicrosoftExcelDataAnalysisToolSelectData.SelectDataAnalysis.SelectDescriptiveStatisticsandclickOK.DCOVAGeneralDescriptiveStatsUsinGeneralDescriptiveStatsUsingMicrosoftExcel4.Enterthecellrange.5.ChecktheSummaryStatisticsbox.6.ClickOKDCOVAGeneralDescriptiveStatsUsinExceloutputMicrosoftExceldescriptivestatisticsoutput,usingthehousepricedata:HousePrices:
$2,000,000500,000
300,000
100,000
100,000DCOVAExceloutputMicrosoftExcelHoMinitabOutputMinitabdescriptivestatisticsoutputusingthehousepricedata:HousePrices:
$2,000,000500,000
300,000
100,000
100,000DCOVADescriptiveStatistics:HousePriceTotalVariableCountMeanSEMeanStDevVarianceSumMinimumHousePrice56000003577718000006.40000E+113000000100000 NforVariableMedianMaximumRangeModeMode SkewnessKurtosisHousePrice300000200000019000001000002 2.014.13MinitabOutputMinitabdescriptQuartileMeasuresQuartilessplittherankeddatainto4segmentswithanequalnumberofvaluespersegment25%Thefirstquartile,Q1,isthevalueforwhich25%oftheobservationsaresmallerand75%arelargerQ2isthesameasthemedian(50%oftheobservationsaresmallerand50%arelarger)Only25%oftheobservationsaregreaterthanthethirdquartileQ1Q2Q325%25%25%DCOVAQuartileMeasuresQuartilessplQuartileMeasures:
LocatingQuartilesFindaquartilebydeterminingthevalueintheappropriatepositionintherankeddata,whereFirstquartileposition:
Q1=(n+1)/4rankedvalue
Secondquartileposition:
Q2=(n+1)/2
rankedvalue
Thirdquartileposition:
Q3=3(n+1)/4rankedvalue
where
n
isthenumberofobservedvaluesDCOVAQuartileMeasures:
LocatingQuQuartileMeasures:
CalculationRulesWhencalculatingtherankedpositionusethefollowingrulesIftheresultisawholenumberthenitistherankedpositiontouseIftheresultisafractionalhalf(e.g.2.5,7.5,8.5,etc.)thenaveragethetwocorrespondingdatavalues.Iftheresultisnotawholenumberorafractionalhalfthenroundtheresulttothenearestintegertofindtherankedposition.DCOVAQuartileMeasures:
Calculation(n=9)Q1isinthe
(9+1)/4=2.5positionoftherankeddata
sousethevaluehalfwaybetweenthe2ndand3rdvalues,
soQ1=12.5QuartileMeasures:
LocatingQuartilesSampleDatainOrderedArray:111213161617182122
Q1andQ3aremeasuresofnon-centrallocationQ2=median,isameasureofcentraltendencyDCOVA(n=9)QuartileMeasures:
(n=9)Q1isinthe
(9+1)/4=2.5positionoftherankeddata,
soQ1=(12+13)/2=12.5Q2isinthe
(9+1)/2=5thpositionoftherankeddata,
soQ2=median=16Q3isinthe
3(9+1)/4=7.5positionoftherankeddata,
soQ3=(18+21)/2=19.5QuartileMeasures
CalculatingTheQuartiles:ExampleSampleDatainOrderedArray:111213161617182122
Q1andQ3aremeasuresofnon-centrallocationQ2=median,isameasureofcentraltendencyDCOVA(n=9)QuartileMeasures
CQuartileMeasures:
TheInterquartileRange(IQR)TheIQRisQ3–Q1andmeasuresthespreadinthemiddle50%ofthedataTheIQRisalsocalledthemidspreadbecauseitcoversthemiddle50%ofthedataTheIQRisameasureofvariabilitythatisnotinfluencedbyoutliersorextremevaluesMeasureslikeQ1,Q3,andIQRthatarenotinfluencedbyoutliersarecalledresistantmeasuresDCOVAQuartileMeasures:
TheInterquCalculatingTheInterquartileRangeMedian(Q2)XmaximumXminimumQ1Q3Example:25%25%25%25%1230455770Interquartilerange=57–30=27DCOVACalculatingTheInterquartileTheFiveNumberSummaryThefivenumbersthathelpdescribethecenter,spreadandshapeofdataare:XsmallestFirstQuartile(Q1)Median(Q2)ThirdQuartile(Q3)XlargestDCOVATheFiveNumberSummaryThefivRelationshipsamongthefive-numbersummaryanddistributionshapeLeft-SkewedSymmetricRight-SkewedMedian–Xsmallest>Xlargest–MedianMedian–Xsmallest≈Xlargest–MedianMedian–Xsmallest<Xlargest–MedianQ1–Xsmallest>Xlargest–Q3Q1–Xsmallest≈Xlargest–Q3Q1–Xsmallest<Xlargest–Q3Median–Q1>Q3–MedianMedian–Q1≈Q3–MedianMedian–Q1<Q3–MedianDCOVARelationshipsamongthefive-nFiveNumberSummaryand
TheBoxplotTheBoxplot:AGraphicaldisplayofthedatabasedonthefive-numbersummary:Example:Xsmallest
--Q1
--Median--Q3
--Xlargest25%ofdata25%25%25%ofdata ofdataofdata Xsmallest Q1 MedianQ3 XlargestDCOVAFiveNumberSummaryand
TheBoFiveNumberSummary:
ShapeofBoxplotsIfdataaresymmetricaroundthemedianthentheboxandcentrallinearecenteredbetweentheendpointsABoxplotcanbeshownineitheraverticalorhorizontalorientationXsmallestQ1MedianQ3XlargestDCOVAFiveNumberSummary:
ShapeofDistributionShapeand
TheBoxplotRight-SkewedLeft-SkewedSymmetricQ1Q2Q3Q1Q2Q3Q1Q2Q3DCOVADistributionShapeand
TheBoBoxplotExampleBelowisaBoxplotforthefollowingdata:
022233455927Thedataarerightskewed,astheplotdepicts023527XsmallestQ1Q2/MedianQ3XlargestDCOVABoxplotExampleBelowisaBoxpNumericalDescriptiveMeasuresforaPopulationDescriptivestatisticsdiscussedpreviouslydescribedasample,notthepopulation.Summarymeasuresdescribingapopulation,calledparameters,aredenotedwithGreekletters.Importantpopulationparametersarethepopulationmean,variance,andstandarddeviation.DCOVANumericalDescriptiveMeasuresNumericalDescriptiveMeasures
foraPopulation:ThemeanµThepopulationmeanisthesumofthevaluesinthepopulationdividedbythepopulationsize,Nμ=populationmeanN=populationsizeXi=ithvalueofthevariableXWhere
DCOVANumericalDescriptiveMeasuresAverageofsquareddeviationsofvaluesfromthemeanPopulation
variance:NumericalDescriptiveMeasuresForAPopulation:TheVarianceσ2Where
μ=populationmeanN=populationsizeXi=ithvalueofthevariableXDCOVAAverageofsquareddeviationsNumericalDescriptiveMeasuresForAPopulation:TheStandardDeviationσMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthepopulationvarianceHasthesameunitsastheoriginaldataPopulation
standarddeviation:DCOVANumericalDescriptiveMeasuresSamplestatisticsversuspopulationparametersMeasurePopulationParameterSampleStatisticMeanVarianceStandardDeviationDCOVASamplestatisticsversuspopulTheempiricalruleapproximatesthevariationofdatainabell-shapeddistributionApproximately68%ofthedatainabellshapeddistributioniswithin1standarddeviationofthemeanorTheEmpiricalRule68%DCOVATheempiricalruleapproximateApproximately95%ofthedatainabell-shapeddistributionlieswithintwostandarddeviationsofthemean,orµ±2σApproximately99.7%ofthedatainabell-shapeddistributionlieswithinthreestandarddeviationsofthemean,orµ±3σTheEmpiricalRule99.7%95%DCOVAApproximately95%ofthedataUsingtheEmpiricalRuleSupposethatthevariableMathSATscoresisbell-shapedwithameanof500andastandarddeviationof90.Then,Approximately68%ofalltesttakersscoredbetween410and590,(500±90).Approximately95%ofalltesttakersscoredbetween320and680,(500±180).Approximately99.7%ofalltesttakersscoredbetween230and770,(500±270).DCOVAUsingtheEmpiricalRuleSupposRegardlessofhowthedataaredistributed,atleast(1-1/k2)x100%ofthevalueswillfallwithinkstandarddeviationsofthemean(fork>1)
Examples:
(1-1/22)x100%=75%…..............k=2(μ±2σ) (1-1/32)x100%=88.89%………..k=3(μ±3σ)ChebyshevRuleWithinAtleastDCOVARegardlessofhowthedataareWeDiscussTwoMeasuresOfTheRelationshipBetweenTwoNumericalVariablesScatterplotsallowyoutovisuallyexaminetherelationshipbetweentwonumericalvariablesandnowwewilldiscusstwoquantitativemeasuresofsuchrelationships.TheCovarianceTheCoefficientofCorrelationWeDiscussTwoMeasuresOfTheTheCovarianceThecovariancemeasuresthestrengthofthelinearrelationshipbetweentwonumericalvariables(X&Y)Thesamplecovariance:OnlyconcernedwiththestrengthoftherelationshipNocausaleffectisimpliedDCOVATheCovarianceThecovariancemCovariancebetweentwovariables:cov(X,Y)>0XandYtendtomoveinthesamedirectioncov(X,Y)<0XandYtendtomoveinoppositedirectionscov(X,Y)=0XandYareindependentThecovariancehasamajorflaw:ItisnotpossibletodeterminetherelativestrengthoftherelationshipfromthesizeofthecovarianceInterpretingCovarianceDCOVACovariancebetweentwovariablCoefficientofCorrelationMeasurestherelativestrengthofthelinearrelationshipbetweentwonumericalvariablesSamplecoefficientofcorrelation:
whereDCOVACoefficientofCorrelationMeasFeaturesofthe
CoefficientofCorrelationThepopulationcoefficientofcorrelationisreferredasρ.Thesamplecoefficientofcorrelationisreferredtoasr.Eitherρorrhavethefollowingfeatures:UnitfreeRangebetween–1and1Thecloserto–1,thestrongerthenegativelinearrelationshipThecloserto1,thestrongerthepositivelinearrelationshipThecloserto0,theweakerthelinearrelationshipDCOVAFeaturesofthe
CoefficientofScatterPlotsofSampleDatawithVariousCoefficientsofCorrelationYXYXYXYXr=-1r=-.6r=+.3r=+1YXr=0DCOVAScatterPlotsofSampleDataTheCoefficientofCorrelationUsingMicrosoftExcelFunctionDCOVATheCoefficientofCorrelationTheCoefficientofCorrelationUsingMicrosoftExcelDataAnalysisToolSelectDataChooseDataAnalysisChooseCorrelation&ClickOKDCOVATheCoefficientofCorrelationTheCoefficientofCorrelation
UsingMicrosoftExcelInputdatarangeandselectappropriateoptionsClickOKtogetoutputDCOVATheCoefficientofCorrelationInterpretingtheCoefficientofCorrelation
UsingMicrosoftExcelr=.733Thereisarelativelystrongpositivelinearrelationshipbetweentestscore#1andtestscore#2.Studentswhoscoredhighonthefirsttesttendedtoscorehighonsecondtest.DCOVAInterpretingtheCoefficientoPitfallsinNumerical
DescriptiveMeasuresDataanalysisisobjectiveShouldreportthesummarymeasuresthatbestdescribeandcommunicatetheimportantaspectsofthedatasetDatainterpretationissubjectiveShouldbedoneinfair,neutralandclearmannerDCOVAPitfallsinNumerical
DescripEthicalConsiderationsNumericaldescriptivemeasures:ShoulddocumentbothgoodandbadresultsShouldbepresentedinafair,objectiveandneutralmannerShouldnotuseinappropriatesummarymeasurestodistortfactsDCOVAEthicalConsiderationsNumericaInthischapterwehavediscussed:
Describingthepropertiesofcentraltendency,variation,andshapeinnumericaldataConstructingandinterpretingaboxplotComputingdescriptivesummarymeasuresforapopulationCalculatingthecovarianceandthecoefficientofcorrelationChapterSummaryInthischapterwehavediscusNumericalDescriptiveMeasuresChapter3NumericalDescriptiveMeasuresInthischapter,youlearnto:
Describethepropertiesofcentraltendency,variation,andshapeinnumericaldataConstructandinterpretaboxplotComputedescriptivesummarymeasuresforapopulationCalculatethecovarianceandthecoefficientofcorrelationObjectivesInthischapter,youlearnto:SummaryDefinitionsThecentraltendencyistheextenttowhichthevaluesofanumericalvariablegrouparoundatypicalorcentralvalue.Thevariationistheamountofdispersionorscatteringawayfromacentralvaluethatthevaluesofanumericalvariableshow.Theshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue.DCOVASummaryDefinitionsThecentralMeasuresofCentralTendency:
TheMeanThearithmeticmean(oftenjustcalledthe“mean”)isthemostcommonmeasureofcentraltendencyForasampleofsizen:SamplesizeObservedvaluesTheithvaluePronouncedx-barDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheMean(con’t)ThemostcommonmeasureofcentraltendencyMean=sumofvaluesdividedbythenumberofvaluesAffectedbyextremevalues(outliers)11121314151617181920Mean=1311121314151617181920Mean=14DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheMedianInanorderedarray,themedianisthe“middle”number(50%above,50%below)
LesssensitivethanthemeantoextremevaluesMedian=13Median=131112131415161718192011121314151617181920DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
LocatingtheMedianThelocationofthemedianwhenthevaluesareinnumericalorder(smallesttolargest):Ifthenumberofvaluesisodd,themedianisthemiddlenumberIfthenumberofvaluesiseven,themedianistheaverageofthetwomiddlenumbers Notethatisnotthevalueofthemedian,onlythepositionofthemedianintherankeddataDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
TheModeValuethatoccursmostoftenNotaffectedbyextremevaluesUsedforeithernumericalorcategoricaldataTheremaybenomodeTheremaybeseveralmodes01234567891011121314
Mode=90123456NoModeDCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
ReviewExampleHousePrices:
$2,000,000$500,000
$300,000
$100,000
$100,000Sum$3,000,000Mean:($3,000,000/5) =$600,000Median:middlevalueofrankeddata
=$300,000Mode:mostfrequentvalue
=$100,000DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
WhichMeasuretoChoose?Themeanisgenerallyused,unlessextremevalues(outliers)exist.Themedianisoftenused,sincethemedianisnotsensitivetoextremevalues.Forexample,medianhomepricesmaybereportedforaregion;itislesssensitivetooutliers.Insomesituationsitmakessensetoreportboththemeanandthemedian.DCOVAMeasuresofCentralTendency:
MeasuresofCentralTendency:
SummaryCentralTendencyArithmeticMeanMedianModeMiddlevalueintheorderedarrayMostfrequentlyobservedvalueDCOVAMeasuresofCentralTendency:
Samecenter,differentvariationMeasuresofVariationMeasuresofvariationgiveinformationonthespreadorvariabilityordispersionofthedatavalues.
VariationStandardDeviationCoefficientofVariationRangeVarianceDCOVASamecenter,MeasuresofVariaMeasuresofVariation:
TheRangeSimplestmeasureofvariationDifferencebetweenthelargestandthesmallestvalues:Range=Xlargest–Xsmallest01234567891011121314Range=13-1=12Example:DCOVAMeasuresofVariation:
TheRanMeasuresofVariation:
WhyTheRangeCanBeMisleadingDoesnotaccountforhowthedataaredistributedSensitivetooutliers789101112Range=12-7=5789101112Range=12-7=5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range=5-1=4Range=120-1=119DCOVAMeasuresofVariation:
WhyTheAverage(approximately)ofsquareddeviationsofvaluesfromthemeanSample
variance:MeasuresofVariation:
TheSampleVarianceWhere
=arithmeticmeann=samplesizeXi=ithvalueofthevariableXDCOVAAverage(approximately)ofsquMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthevarianceHasthesameunitsastheoriginaldataSample
standarddeviation:MeasuresofVariation:
TheSampleStandardDeviationDCOVAMostcommonlyusedmeasureofMeasuresofVariation:
TheStandardDeviationStepsforComputingStandardDeviation1. Computethedifferencebetweeneachvalueandthemean.2. Squareeachdifference.3. Addthesquareddifferences.4. Dividethistotalbyn-1togetthesamplevariance.5. Takethesquarerootofthesamplevariancetogetthesamplestand
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