统计学英文版课件：ch03 Descriptive Statistics Numerical Methods Part A

Chapter3

DescriptiveStatistics:NumericalMethods

PartAMeasuresofLocationMeasuresofVariabilityx%MeasuresofLocationMeanMedianModePercentilesQuartilesCountingaverage:mean、harmonicmean、geometricmean；Positionaverage:median、mode、percentiles,quartiles。Example:ApartmentRents Givenbelowisasampleofmonthlyrentvalues($)forone-bedroomapartments.Thedataisasampleof70apartmentsinaparticularcity.Thedataarepresentedinascendingorder.

MeanThemeanofadatasetistheaverageofallthedatavalues.Ifthedataarefromasample,themeanisdenotedby

Ifthedataarefromapopulation,themeanisdenotedbym(mu).Example:ApartmentRentsMeanSTAT[e.g.]agefortenpersons:15,16,16,17,17,17,18,18,18,18。Findtheaverageage.STATAttention:（1）weight：theamounttobalancedegreeforf/f（2）calculationforthemeanofclasswidthseries：usethemidpointtosubstitutevariablex,andcalculateitwithformula.STATThemeanofratioAnalysisofingredientsformean:STATHarmonicMeana、definition：thereciprocalofmeanwhichthevariables’reciprocalshave.b、formula:Simpleharmonicmean:Weightedharmonicmean:Meanwhile,m=xfisweightSTAT[e.g.]theinformationforfourcorporationsbelongingtoonebureauasfollows,trytocalculatetheaverageplanaccomplishedpercentagefortheindustrybureau.

Weightedharmonicmean：（itcanbeusedwhenvariableshasdifferentweight)STAT1、Basicformula：mean=symbolgross/populationgrossThepreconditionforthemeanandharmonicmean:A、whenthedenominatorisknown,meancanbeusedincalculation;(numeratorisunknown)B、whenthenumeratorisknown,harmonicmeancanbeusedincalculation;(denominatorisunknown)2、Weightedharmonicmean：（itcanbeusedwhenvariableshasdifferentweight)HarmonicmeanisthetransformationofmeanSTATa、definition:nhypo-squareofnvariables’product.b、precondition:It’ssuitabletocomputetheaverageofratioorspeed.c、formula:d、notice:whenthereisazerooranegativevalueintheobservation,itisnotsuitabletousegeometricmeanforcalculation.e、ifusethesamedatatocalculatethearithmeticmean、harmonicmeanandgeometricmeanseparately,therelationwillshowasfollow:GeometricMeanMedianThemedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.Foranoddnumberofobservations,themedianisthemiddlevalue.Foranevennumberofobservations,themedianistheaverageofthetwomiddlevalues.MedianThemedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.Afewextremelylargeincomesorpropertyvaluescaninflatethemean.[e.g.]theagesofnineofficersinsectionoffice:24,25,25,26,26,27,28,29,55Sequence：A1,A2,A3,A4,A5,A6,A7,A8,A9Example:ApartmentRentsMedian Median=50thpercentile

i=(p/100)n=(50/100)70=35.5 Averagingthe35thand36thdatavalues: Median=(475+475)/2=475STATEx:median=180/2=theninetieth，soMeoughttotheageoftheninetiethSo:Me=18。whenmaterialisgrouped,anditformsintomonomialvariablesequence,middlepoint=f/2STAT（3）Thedatahasalreadygroupingandformintotypeofclassintervaloffluentsequence

（A）Listhelowerwardlimitofthemean，theUistheupwardlimit（B）Iistheclassintervalofmeaninplaceset（C）Sm-1isthesumofsmallerthaneachnumberoftimesofmedian（D）Sm+1isthesumoflargerthaneachnumberoftimesofmedian（E）fmisthetimesofmeaninplaceset

STAT[EX]lowerwardformula：

upwardformula：

AndSTATdeduce：

506070（L）80（U）90100

xy103060110150180(Sm-1)TheninetiethpersonMe=L+x=U-ySupposethatthevariableofmediangroupsisaveragedistribution,thentakethemethodsofinterpolationbyproportionalpartsSTAT3、Attentionoftheproblems:（1）Notaffectedbytheextremevalue,moresteadiness.

（2）Themediantakesvalueonlybearononeortwonumeralvaluein

neutralposition,makeuseofinformationinsufficiency,ignoreothersizeofdata,andisnotsuitforalgebraicoperation.ModeThemodeofadatasetisthevaluethatoccurswithgreatestfrequency.Thegreatestfrequencycanoccurattwoormoredifferentvalues.Ifthedatahaveexactlytwomodes,thedataarebimodal.Ifthedatahavemorethantwomodes,thedataaremultimodal.Example:ApartmentRentsMode

450occurredmostfrequently(7times) Mode=450STAT1、definition：Themodeisthedatavaluethatoccurswithgreatestfrequency。ExpressedbytheMo。A、20,15,18,20,20,22,20,23；n=8Mo=20B、20,20,15,19,19,20,19,25；n=8Mo=20Mo=19C、10,11,13,16,15,25,8,12；n=8，nomode2、calculation（1）Ifthedataisthemonomialnumbersequence。

Firstidentifythemodegroups

thenidentifymode：Mo=18STAT2)IfthedataistypesofclassintervalofnumbersequenceMakesuremodalclassfirst；Thenusethefollowformulatocalculate：Signmeaning：（A）Listhelowerlimitofmodalclass,Uistheupward；（B）Iistheclassintervalofmodalclass；（C）1=fm－fm-1,isthedifferenceoforderofmodalclassandandex-numberoforder

2=fm－fm+1，isthedifferenceoforderofmodalclassandheelnumberoforderSTATThecharacteristicsofmodetakesvalue（1）Advantage：notaffectedbytheextremevalue（2）Disadvantage：DidnotmakeuseofallinformationLackthesensitivityandisnotsuitableforthealgebraoperation

PercentilesApercentileprovidesinformationabouthowthedataarespreadovertheintervalfromthesmallestvaluetothelargestvalue.Admissiontestscoresforcollegesanduniversitiesarefrequentlyreportedintermsofpercentiles.Thepthpercentileofadatasetisavaluesuchthatatleastppercentoftheitemstakeonthisvalueorlessandatleast(100-p)percentoftheitemstakeonthisvalueormore.Arrangethedatainascendingorder.Computeindexi,thepositionofthepthpercentile.

i=(p/100)nIfiisnotaninteger,roundup.Thep

thpercentileisthevalueinthei

thposition.Ifiisaninteger,thep

thpercentileistheaverageofthevaluesinpositionsiandi

+1.PercentilesExample:ApartmentRents90thPercentile

i=(p/100)n=(90/100)70=63 Averagingthe63rdand64thdatavalues: 90thPercentile=(580+590)/2=585QuartilesQuartilesarespecificpercentilesFirstQuartile=25thPercentileSecondQuartile=50thPercentile=MedianThirdQuartile=75thPercentileExample:ApartmentRentsThirdQuartile

Thirdquartile=75thpercentile

i=(p/100)n=(75/100)70=52.5=53 Thirdquartile=525STATTherelationshipamountofMean、MedianandMode（a）Therelationshipofthem:1、quantitativerelations:（1）symmetricdistribution：Thispointallequal35。

STAT（2）BiaseddistributionA、Divergeright(positive)：STATB、Divergeleft(negative)：BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-35ShapeofaDistributionDescribeshowdataaredistributedMeasuresofshapeSymmetricorskewedMean=Median

Mean<Median

Median<MeanRight-SkewedLeft-SkewedSymmetricMeasuresofVariabilityItisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-37MeasuringvariationSmallstandarddeviationLargestandarddeviationSTATFunction:（1）Measurethesizeofmeanvaluerepresentativeness。（2）Reflectthedispersityofvariablevaluedistribution。（3）ReflecttheproportionalityandstabilityofdevelopingphenomenaMeasuresofVariabilityRangeInterquartileRangeA.DVarianceStandardDeviationCoefficientofVariationRangeTherangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalues.Itisthesimplestmeasureofvariability.Itisverysensitivetothesmallestandlargestdatavalues.Example:ApartmentRentsRange

Range=largestvalue-smallestvalue Range=615-425=190InterquartileRangeTheinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.Itistherangeforthemiddle50%ofthedata.Itovercomesthesensitivitytoextremedatavalues.Example:ApartmentRentsInterquartileRange

3rdQuartile(Q3)=525 1stQuartile(Q1)=445 InterquartileRange=Q3-Q1=525-445=80STATA.D1、Definition:Theaveragedeviationofthevariableandthemean.2、Theformula:–5–2250522514STATExample:–5.61–2.611.394.395.612=11.222.615=13.051.398=11.224.393=13.1748.66VarianceThevarianceisameasureofvariabilitythatutilizesallthedata.Itisbasedonthedifferencebetweenthevalueofeachobservation(xi)andthemean(xforasample,mforapopulation).VarianceThevarianceistheaverageofthesquareddifferencesbetweeneachdatavalueandthemean.Ifthedatasetisasample,thevarianceisdenotedbys2.

Ifthedatasetisapopulation,thevarianceisdenotedby2.StandardDeviationThestandarddeviation

ofadatasetisthepositivesquarerootofthevariance.Itismeasuredinthesameunitsasthedata,makingitmoreeasilycomparable,thanthevariance,tothemean.Ifthedatasetisasample,thestandarddeviationisdenoteds.Ifthedatasetisapopulation,thestandarddeviationisdenoted(sigma).STATExample:Trytocalculatethevarianceandthestandarddeviationaboutthegradesofthese40studentsasfollowing:X5565758595Xf1105201200850380306-21.5-11.5-1.58.518.5462.25132.252.2572.25342.25924.5105836722.51369 4110S