应用统计学Lecture-3-a

1、Chapter3DescriptiveStatistics:NumericalMeasuresPartA,MeasuresofLocationMeasuresofVariability,MeasuresofLocation,Ifthemeasuresarecomputedfordatafromasample,theyarecalledsamplestatistics.,Ifthemeasuresarecomputedfordatafromapopulation,theyarecalledpopulationparameters.,Asamplestatisticisreferredtoasth

2、epointestimatorofthecorrespondingpopulationparameter.,Mean,Median,Mode,Percentiles,Quartiles,Mean(平均数）,Themeanofadatasetistheaverageofallthedatavalues.Thesamplemeanisthepointestimatorofthepopulationmeanm.,SampleMean,Numberofobservationsinthesample,Sumofthevaluesofthenobservations,PopulationMeanm,Num

3、berofobservationsinthepopulation,SumofthevaluesoftheNobservations,Seventyefficiencyapartmentswererandomlysampledinasmallcollegetown.Themonthlyrentpricesfortheseapartmentsarelistedinascendingorderonthenextslide.,SampleMean,Example:ApartmentRents,SampleMean,SampleMean,Median（中位数）,Wheneveradatasethasex

4、tremevalues,themedianisthepreferredmeasureofcentrallocation.,Afewextremelylargeincomesorpropertyvaluescaninflatethemean.,Themedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.,Themedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.,Median,12

5、,14,19,26,27,18,27,Foranoddnumberofobservations:,inascendingorder,26,18,27,12,14,27,19,7observations,themedianisthemiddlevalue.,Median=19,12,14,19,26,27,18,27,Median,Foranevennumberofobservations:,inascendingorder,26,18,27,12,14,27,30,8observations,themedianistheaverageofthemiddletwovalues.,Median=(

6、19+26)/2=22.5,19,30,Median,Averagingthe35thand36thdatavalues:,Median=(475+475)/2=475,Mode（众数）,Themodeofadatasetisthevaluethatoccurswithgreatestfrequency.,Thegreatestfrequencycanoccurattwoormoredifferentvalues.,Ifthedatahaveexactlytwomodes,thedataarebimodal.,Ifthedatahavemorethantwomodes,thedataaremu

7、ltimodal.,Mode,450occurredmostfrequently(7times),Mode=450,Percentiles（百分位数）,Apercentileprovidesinformationabouthowthedataarespreadovertheintervalfromthesmallestvaluetothelargestvalue.,Admissiontestscoresforcollegesanduniversitiesarefrequentlyreportedintermsofpercentiles.,百分位数提供了关于数据是如何在最小值与最大值之间分布的信

8、息。,Percentiles,Thepthpercentileofadatasetisavaluesuchthatatleastppercentoftheitemstakeonthisvalueorlessandatleast(100-p)percentoftheitemstakeonthisvalueormore.,第p百分位数是满足下列条件的一个数值：至少有p%的观察值小于或等于该值，且至少（100-p)%的观察值大于或等于该值。,Percentiles,Arrangethedatainascendingorder.,Computeindexi,thepositionofthepthper

9、centile.,i=(p/100)n,Ifiisnotaninteger,roundup.Thepthpercentileisthevalueintheithposition.,Ifiisaninteger,thepthpercentileistheaverageofthevaluesinpositionsiandi+1.,90thPercentile,i=(p/100)n=(90/100)70=63,Averagingthe63rdand64thdatavalues:,90thPercentile=(580+590)/2=585,90thPercentile,“Atleast90%ofth

10、eitemstakeonavalueof585orless.”,“Atleast10%oftheitemstakeonavalueof585ormore.”,63/70=.9or90%,7/70=.1or10%,Quartiles（四分位数）,Quartilesarespecificpercentiles.,FirstQuartile=25thPercentile,SecondQuartile=50thPercentile=Median,ThirdQuartile=75thPercentile,ThirdQuartile,Thirdquartile=75thpercentile,i=(p/10

11、0)n=(75/100)70=52.5=53,Thirdquartile=525,MeasuresofVariability,Itisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.,Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.,Measuresof

12、Variability,Range,InterquartileRange,Variance,StandardDeviation,CoefficientofVariation,Range,Therangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalues.,Itisthesimplestmeasureofvariability.,Itisverysensitivetothesmallestandlargestdatavalues.,极差是最大值和最小值之间的差值。,Range,Range=largestvalue-sma

13、llestvalue,Range=615-425=190,InterquartileRange,Theinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.,Itistherangeforthemiddle50%ofthedata.,Itovercomesthesensitivitytoextremedatavalues.,四分位数间距是第三四分位数和第一四分位数的差值。,InterquartileRange,3rdQuartile(Q3)=525,1stQuartile(Q1

14、)=445,InterquartileRange=Q3-Q1=525-445=80,Thevarianceisameasureofvariabilitythatutilizesallthedata.,Variance（方差）,Itisbasedonthedifferencebetweenthevalueofeachobservation(xi)andthemean(forasample,mforapopulation).,Variance,Thevarianceiscomputedasfollows:,Thevarianceistheaverageofthesquareddifferences

15、betweeneachdatavalueandthemean.,forasample,forapopulation,StandardDeviation,Thestandarddeviationofadatasetisthepositivesquarerootofthevariance.,Itismeasuredinthesameunitsasthedata,makingitmoreeasilyinterpretedthanthevariance.,Thestandarddeviationiscomputedasfollows:,forasample,forapopulation,StandardDeviation,Thecoefficientofvariationiscomputedasfollows:,CoefficientofVariation（标准差系数,Thecoefficientofvariationindicate