商務(wù)統(tǒng)計(jì)學(xué)課件-統(tǒng)計(jì)ch3-學(xué)生版

ChapterChapterNumericalDescriptive Chap3-LearningInthischapter,youTodescribethepropertiesofcentraltendency集中趨勢(shì),variation變異程度,andshape分布形狀innumericaldataToconstructandinterpretaboxplotTocomputedescriptivesummarymeasuresforaTocomputethecovarianceandthecoefficientofcorrelation相關(guān)系數(shù) Chap3-Summary

Thecentraltendencyistheextenttowhichallthedatavaluesgrouparoundatypicalorcentralvalue.ThevariationistheamountofdispersionorscatteringofvaluesTheshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue. Chap3-ChapterMean均值median中位數(shù)modeRange全距quartile四分位數(shù)，interquartilerange四分位間距,variance方差andstandarddeviation標(biāo)準(zhǔn)差,coefficientofvariation變異系數(shù),Z-scoresshapeofdistributionSkewness&Kurtosis5-numbersummary五值概括 Chap3-Chapter

DiscussedcovarianceandcorrelationAddressedpitfallsinnumericaldescriptivemeasuresandethicalconsiderations描述性數(shù)值 Chap3-MeasuresofCentralTendency:TheMean算術(shù)平均數(shù)Thearithmeticmean(oftenjustcalledthe“mean”)isthemostcommonmeasureofcentralTheithForasampleTheithPronouncedPronouncedx-nXin

XXnObservedX nObserved SampleSample Chap3-MeasuresofCentralTendency:TheMean

ThemostcommonmeasureofcentralMean=sumofvaluesdividedbythenumberofAffectedbyextremevalues(outliers) 111213141516171819 111213141516171819MeanMean=Mean=111213141565

111213142070 Chap3-111213141516171819 Chap3-111213141516171819 111213141516171819MeanMean=Mean=111213141565

111213142070 Chap3-MeasuresofCentralTendency:TheMedian中位數(shù)Inanorderedarray有序數(shù)組themedianisthe“middle”number(50%above,50%below)111213141516171819 111213141516171819Median=Median=Median=Notaffectedbyextremevalues Chap3-MeasuresofCentralTendency:LocatingtheMedian中位數(shù)的位置Thelocationofthemedianwhenthevaluesareinnumerical(smallesttoMedianMedianpositionn2positionintheorderedIfthenumberofvaluesisodd奇數(shù)themedianistheIfthenumberofvaluesiseven偶數(shù)themedianistheaverageofthetwomiddlenumbers兩個(gè)中間的數(shù)的和NoteNoten2isnotthevalueofthemedianonlytheposition位ofthemedianintheranked Chap3-MeasuresofCentralTendency:TheMode眾數(shù)ValuethatoccursmostoftenNotaffectedbyextremevaluesUsedforeithernumericalorcategoricalTheremaybenomodeTheremaybeseveralmodes Mode=

NoNoMeasuresofCentralTendency:ReviewExampleHouseHouse Sum$= rankedddata=$300,000 = Chap3-MeasuresofCentralTendency: easuretoChoose?Themeanisgenerallyused,unlessextremevaluesThemedianisoftenused,sincethemedianisnotsensitivetoextremevalues.Forexample,medianhomepricesmaybereportedforaregion;itislesssensitivetoInsomesituationsitmakessensetoreportboththemeanandthemedian.必要的情況下，可以同 Chap3-MeasuresofCentralTendency:nnX n

Middlevalueintheordered

Chap3-

Thecentraltendencyistheextenttowhichallthedatavaluesgrouparoundatypicalorcentralvalue.ThevariationistheamountofdispersionorscatteringofvaluesTheshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue.MeasuresofVariationofof變異方全Standard標(biāo)準(zhǔn)Samecenter,differentvariationMeasuresofvariationgiveinformationonthespreadorvariabilityordispersion散布o(jì)fthedataSamecenter,differentvariation Chap3-MeasuresofVariation:TheRangeSimplestmeasureof

DifferencebetweenthelargestandthesmallestRange=Xlargest– Range=13-1= Chap3-MeasuresofWhyTheRangeCanBeIgnoresthewayinwhichdataare

Range=12-7=

Range=12-7= Sensitivetooutliers Range=5-1= Range=120-1= Chap3- Chap3-n(Xn(X 2i n-TheSampleVarianceAverage(approxima yofsquareddeviationsofvaluesfromthemeanSample X=arithmeticn=sampleXi=ithvalueofthevariable Chap3-MeasuresofTheSampleStandardMostcommonlyusedmeasureofShowsvariationabouttheIsthesquarerootoftheHasthesameunitsastheoriginalnn(X 2iS n-Samplestandard

Chap3-MeasuresofVariation:TheStandardDeviationStepsforComputingStandard計(jì)算步

ComputethedifferencebetweeneachvalueandtheSquareeachAddthesquaredDividethistotalbyn-1togetthesampleTakethesquarerootofthesamplevariancetothesamplestandard Chap3-MeasuresofVariation:SampleStandardDeviationCalculationExample

nn= Mean=X= Chap3-MeasuresofVariation:SampleStandardDeviationCalculationExample

nn= Mean=X=(10(10X)2(12X)2(14X)2(24X)2n((1016)2(1216)2(1416)2(2416)813074.3095Ameasure13074.3095 scatteraroundthe Chap3-MeasuresofVariation:ComparingStandardDeviationsDataDataMean=S= Mean=S=DataDataMean=S= Mean=S=MeanMean=S=D Chap3-MeasuresofVariation:ComparingStandardDeviationsSmallerstandardLargerstandard Chap3-MeasuresofVariation:SummaryCharacteristicsThemorethedataarespreadout,thegreatertherange,variance,andstandarddeviation.Themorethedataareconcentrated,thesmallertherange,variance,andstandarddeviation.Ifthevaluesareallthesame(novariation),allthesemeasureswillbezero.Noneofthesemeasuresareever Chap3-MeasuresofTheCoefficientof

MeasuresrelativevariationAlwaysinpercentageShowsvariationrelativetomean標(biāo)準(zhǔn)差/Canbeusedtocomparethevariabilityoftwoormoresetsofdatameasuredindifferentunits便于比較兩組或多組不同單位度量的數(shù)CVCVS100%X Chap3-MeasuresofComparingCoefficientsofStockAveragepricelastyear=Bothstockshavethesamedeviation,butstockBislessvariablerelativetoitspriceStandarddeviationBothstockshavethesamedeviation,butstockBislessvariablerelativetoitspriceSCVA100%X

100%StockAveragepricelastyear=Standarddeviation=S X

100%

100% Chap3-MeasuresofComparingCoefficientsofStockAveragepricelastyear=Standarddeviation=

SCVA X

StockAveragepricelastyear=Standarddeviation=S StockChasamuchsmallerdeviationbutamuchhighercoefficientofStockChasamuchsmallerdeviationbutamuchhighercoefficientofX

100%

100% Chap3-如果StockABC均呈正太分布，試下畫出他們 Chap3-LocatingExtreme TocomputetheZ-scoreofadatavalue,subtractthemeananddividebythestandarddeviation.TheZ-scoreisthenumberofstandarddeviationsadatavalueisfromthemean.Adatavalueisconsideredanextremeoutlierifitsscoreislessthan-3.0orgreaterthanZ值大于3或小于-3ThelargertheabsolutevalueoftheZ-score,thefartherthedatavalueisfromthemean. Chap3-LocatingExtremeOutliers:ZXSwhereXrepresentsthedatavalueXisthesamplemeanSisthesamplestandard Chap3-LocatingExtremeOutliers:SupposethemeanmathSATscoreis490,withastandarddeviationof100.ZXX620490130 Ascoreof620is1.3standarddeviationsabovethemeanandwouldnotbeconsideredanoutlier. Chap3-ShapeofaDistributionDescribeshowdataareTwousefulshaperelatedstatisticsMeasurestheamountofasymmetryinadistribution對(duì)稱程度Measurestherelativeconcentrationofvaluesinthecenterofadistributionascomparedwiththetails峰的平緩程度 Chap3-

Thecentraltendencyistheextenttowhichallthedatavaluesgrouparoundatypicalorcentralvalue.ThevariationistheamountofdispersionorscatteringofvaluesTheshapeisthepatternofthedistributionofvaluesfromthelowestvaluetothehighestvalue.ShapeofaDistribution

DescribestheamountofasymmetryinSymmetricorMeanMean<Mean=Mean>Skewness<0 Chap3-ShapeofaDistribution

DescribesrelativeconcentrationofvaluesinthecenterascomparedtothetailsFlatterThanFlatterThanBell-SharperPeakThanBell-Shaped<0 Chap3-

Quartilessplittherankeddatainto4segmentswithanequalnumberofvaluespersegment把數(shù)據(jù)集 quartile,Q1,isthevalueforwhich25%ofobservationsaresmallerand75%are75%的數(shù)Q125%Q2isthesameasthemedian(50%ofthearesmallerand50arelarger)Only25%oftheobservationsaregreaterthanthe Chap3-QuartileMeasures:LocatingQuartilesFindaquartilebydeterminingthevalueintheappropriatepositionintherankeddata,where

quartileQ1= rankedQ2= rankedQ3=3(n+1)/4rankedSecondquartileposition:Thirdquartileposition:wherenisthenumberofobserved Chap3-QuartileMeasures:CalculationRulesWhencalculatingtherankedpositionusethefollowingrulesIftheresultisawholenumberthenitisthepositiontouseIftheresultisafractionalhalf(e.g.2.5,7.5,8.5,etc.) agethetwocorrespondingdatavalues.Iftheresultisnotawholenumberorafractionalhalfthenroundtheresulttothenearestintegertofindtherankedposition.分?jǐn)?shù)，就近取整 Chap3-QuartileMeasures:LocatingQuartiles

SampleSampleDatainOrderedArray: Q1 Chap3-QuartileMeasures:LocatingQuartiles

SampleSampleDatainOrderedArray: (n=Q1isin

(9+1)/4=2.5of(9+1)/4=2.5sousethevaluehalfwaybetweenthe2ndand3rdQ1Q1=QQ1andQ3aremeasuresofnon-centralQ2=median,isameasureofcentral Chap3-QuartileMeasures:LocatingQuartiles

SampleSampleDatainOrderedArray: Q2，Q3 Chap3-QuartileQuartileCalculatingTheQuartiles:SampleDatainOrderedArray: (n=Q1isinthe(9+1)/4=2.5positionoftheranked Q1=(12+13)/2=Q2isinthe(9+1)/2=5thpositionoftheranked Q2=median=Q3isinthe3(9+1)/4=7.5positionoftheranked Q3=(18+21)/2=QQ1andQ3aremeasuresofnon-centralQ2=median,isameasureofcentral Chap3-QuartileTheInterquartileRange

TheIQRisQ3–Q1andmeasuresthespreadinmiddle50%oftheTheIQRisalsocalledthemidspreadbecauseitcoversthemiddle50%ofthedata居中的一半 influencedbyoutliersorextremevaluesMeasureslikeQ1,Q3,andIQRthatarenotinfluencedbyoutliersarecalledresistantmeasures有抵抗力的度 Chap3-CalculatingTheInterquartileExampleboxplotX X

Interquartile=57–30= Chap3-TheFive-Number

Thefivenumbersthathelpdescribethecenter,spreadandshapeofdataare:QuartileMedianThirdQuartile Chap3-Five-NumberSum TheBoxplot盒須圖

----TheBoxplot:AGraphicaldisplayofthedatabasedonthefive-numbersummary:25%ofofof25%of Chap3-Relationshipsamongthefive-number nddistributionshape

左中位數(shù)>Right-右中位數(shù)<Median–>Xlargest–Median–≈Xlargest–Median–<Xlargest–Q1–>Xlargest–Q1–≈Xlargest–Q1–<Xlargest–Median–>Q3–Median–≈Q3–Median–<Q3– Chap3-Five-NumberSummary:ShapeofBoxplots

Ifdataaresymmetricaroundthemedianthentheboxandcentrallinearecenteredbetween ABoxplotcanbeshownineitheraverticalororientation Chap3-DistributionShapeandTheBoxplot

Left- Right- Q2

Q1Q2 Chap3-BoxplotBelowisaBoxplotforthefollowing

0235 0235 2 Thedataarerightskewed,astheplot Chap3-NumericalDescriptiveMeasuresforaPopulationDescriptivestatisticsdiscussedpreviouslydescribedasamplenotthepopulation.Summarymeasuresdescribingapopulation,parametersaredenotedwithGreekImportantpopulationparametersarethepopulationmean,variance,andstandarddeviation. Chap3-NumericalDescriptiveMeasuresforaPopulation:Themeanμ

Thepopulationmeanisthesumofthevaluesinthepopulationdividedbythepopulationsize,N XiNXXNN μ=populationN=populationXi=ithvalueofthevariable Chap3-NumericalDescriptiveMeasuresForAPopulation:TheVarianceσ2N(X2iN(X2i NPopulation μ=populationN=populationXi=ithvalueofthevariable Chap3-NumericalDescriptiveMeasuresForAPopulation:TheStandardDeviationσMostcommonlyusedmeasureofShowsvariationaboutthemeanIsthesquareroot平方根ofthepopulationHasthesameunitsastheoriginalN(XN(X2iσ N Chap3-SamplestatisticsversuspopulationparametersXSS Chap3-TheEmpiricalRuleTheempiricalruleapproximatesthevariationofdatainabell-shapeddistribution經(jīng)驗(yàn)法則利用鐘形分 y68%ofthedatainabelldistributioniswithin±onestandarddeviationofthemeanorμ1σ68%觀測(cè)值落在均值兩邊

μ

TheEmpirical y95%ofthedatainabell-

distributionlieswithin±twostandarddeviationsofmean,orμ± y99.7%ofthedatainabell-shapeddistributionlieswithin±threestandarddeviationsofthemean,orμ±3σμ

μUsingtheEmpirical

SupposethatthevariableMathSATscoresisbell-shapedwithameanof500andastandarddeviationof90.Then,68%ofalltesttakersscoredbetween410and(500±95%ofalltesttakersscoredbetween320and680(500±180).99.7%ofalltesttakersscoredbetween230and(500±Regardlessofhowthedataaredistributed,atleast(1-1/k2)x100%ofthevalueswillfallwithinkstandarddeviationsofthemean(fork>1)AtAt(1-1/22)x100%=75%…........k=2(μ±(1-1/32)x100%=89%……….k=3(μ± Chap3- Chap3-

Thecovariancemeasuresthestrengthoftherelationshipbetweentwonumericalvariables(X&Thesamplecovariance:nn cov(X,Y) (XiX)(YinOnlyconcernedwiththestrengthoftheNocausaleffectisimplied Chap3-InterpretingCovariancebetweentwo

cov(X,Y)>cov(X,Y)<

XandYtendtomoveinthesameXandYtendtomoveinoppositecov(X,Y)= XandYareThecovariancehasamajorflaw主要劣勢(shì)Itisnotpossibletodeterminetherelativestrengthoftherelationshipfromthesizeofthecovariance協(xié)