商務(wù)統(tǒng)計(jì)學(xué)課件英文版BSFC7e-CH03

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