概率论与统计学原理chapt2english

CH2MethodsforDescribingSetsofData概率论与统计学原理chapt2english1. DescribeDataGraphically通过图表2. DescribeDataNumerically用数表示地LearningObjects概率论与统计学原理chapt2englishExampleAPHASIASubjectTypeofAphasia12345678910111213141516171819202122Broca’sAnomicAnomicConductionBroca’sConductionConductionAnomicConductionAnomicConductionBroca’sAnomicBroca’sAnomicAnomicAnomicConductionBroca’sAnomicConductionAnomicTheresearcherswanttodeterminewhetheronetypeofaphasiaoccursmoreoftenthananyother,and,ifso,howoften.概率论与统计学原理chapt2englishDescribingQualitativedataQualitativedataarenonnumericalinnature,thusthevalueofaqualitativevariablecanonlybeclassifiedintocategoriescalledclasses.Wecansummarisesuchdatanumericallyintwoways:(1)bycountingtheclass组frequency频数–thenumberofobservationsinthedatasetthatfallintoeachclass,or(2)bycalculatingtheclassrelativefrequency相对频数–theproportionofthetotalnumberofobservationsfallingintoeachclass.概率论与统计学原理chapt2englishDescribingQualitativedataDEF2.1Aclassisoneofthecategories分类intowhichqualitativedatacanbeclassified.DEF2.2Theclassfrequencyisthenumberofobservationsinthedatasetfallingintoaparticularclass.DEF2.3Theclassrelativefrequencyistheclassfrequencydividedbythetotalnumber(denoteasn,thesizeofthedataset)ofobservationsinthedataset,i.e., classrelativefrequency=classfrequency/n.概率论与统计学原理chapt2englishExampleAPHASIAClassTypeofAphasiaFrequencyNumberofSubjectsRelativeFrequencyProportionBroca’sConductionAnomic5710.227.318.455Totals221.000概率论与统计学原理chapt2englishBarGraphandPieChartThemostwidelyusedgraphicalmethodsforsummarizingqualitativedataarebargraphsandpiechart.BargraphshowstheamountofdatathatbelongstoeachclassasproportionallysizedrectangularareasPiechartshowstheamountofdatathatbelongstoeachclassasaproportionalpartofacircle概率论与统计学原理chapt2englishBarGraphBarLengthShowsFrequencyor%EqualBarWidthsTypeFrequency概率论与统计学原理chapt2englishBarGraph概率论与统计学原理chapt2englishPieChart饼图1. ShowsBreakdownofTotalQuantity

intoCategories2. UsefulforShowingRelativeDifferences3. AngleSize(360°)(Percent)概率论与统计学原理chapt2englishGraphicalMethodsfordescribingQuantitativedataQuantitativedatasetsconsistofdatathatarerecordedonameaningfulnumericalscale.Fordescribing,summarizing,suchdatasets,weintroduceherethreegraphicalmethods:dotplots,stem-and-leafdisplays,andhistograms.概率论与统计学原理chapt2englishExampleEPAGAS36.332.740.536.238.536.341.037.037.139.941.037.336.537.939.036.831.837.240.336.936.941.237.636.035.532.537.340.736.732.937.136.633.937.934.836.433.137.437.033.844.932.940.235.938.640.537.037.133.939.836.836.536.438.239.436.637.637.840.134.030.033.237.738.335.336.137.035.938.036.837.237.437.735.734.438.238.735.635.235.042.137.540.035.638.838.439.036.734.838.136.733.634.235.139.739.335.834.539.536.9EPAMileageRatingson100Cars概率论与统计学原理chapt2englishDotPlots点图Dotplotcondensesthedatabygropingallvaluesthatarethesametogetherintheplot.Inthedotplot,thehorizontalaxisisascaleforthequantitativevariableandthenumericalvalueofeachmeasurementinthedatasetislocatedonthehorizontalscalebyadot.Whendatavaluesrepeat,thedotsareplacedaboveoneanother.Seethefigureintheexamplebelow.概率论与统计学原理chapt2englishDotPlot概率论与统计学原理chapt2englishStem-and-LeafDisplay茎叶图

Stem-and-leafdisplaycombinesgraphictechniqueandsortingtechnique.Itisverypopularforsummarizingnumericaldata.概率论与统计学原理chapt2english1. DivideEachObservationintoStemValueandLeafValuetheleadingdigit(s)becomesthestemthetrailingdigit(s)becomestheleaf

2.Data:21,24,24,26,27,27,30,32,38,4126Stem-and-LeafDisplay概率论与统计学原理chapt2englishStem-and-LeafDisplayExampleConstructastem-and-leafdisplayofthefollowingsetof20testscores.8274886658747884967662687292867652768278概率论与统计学原理chapt2englishStem-and-LeafDisplayFigure1: 20ExamScores 5 28 6 268 7 24466688 8 22468 9 26概率论与统计学原理chapt2englishStem-and-LeafDisplayStem-and-leafofMPGN=100LeafUnit=0.10300318325799331268993402458835012356678993601233445566778993700001112233445667789938012234567839003457894001235574100242143449概率论与统计学原理chapt2englishHistogramsDividethedatasetintoclassintervalsofequalsizeCounttheclassfrequencyorcalculatetheclassrelativefrequencyConductthehistogram概率论与统计学原理chapt2english1. DetermineRange2.ComputeClassIntervals(Width)3. SelectNumberofClassesUsuallyBetween5&15Inclusive 4. DetermineClassBoundaries(Limits)5. CountObservations&AssigntoClassesClassify分类theQuantitativeData概率论与统计学原理chapt2english概率论与统计学原理chapt2englishHistogramMPGfrequency概率论与统计学原理chapt2englishHistogramTheEffectoftheSizeofadataSetontheOutlineofaHistogram概率论与统计学原理chapt2englishExerciseCalculatethenumberofthe500measurementsfallingintoeachofthemeasurementclasses.Thengraphafrequencyhistogramforthesedata.MeasurementClassRelativeFrequency0.5-2.52.5-4.54.5-6.56.5-8.58.5-10.510.5-12.512.5-14.514.5-16.50.100.150.250.200.050.100.100.05概率论与统计学原理chapt2englishSummationNotation总和符号Summationissomethingthatisdonequiteofteninmathematics,andthereisasymbolthatmeanssummation.ThatsymbolisthecapitalGreeklettersigma∑,andsothenotationissometimescalledSigmaNotationinsteadofSummationNotation.概率论与统计学原理chapt2englishNumerical数值MethodsfordescribingQuantitativedataTwomostimportantdatacharacteristics:Centraltendency:thetendencyofthedatatocluster,orcentre.Variability:thedispersionorspreadofthedata.概率论与统计学原理chapt2englishCentralTendency集中趋势

MeanMedianMode概率论与统计学原理chapt2englishDEF2.4Themean(arithmeticmean)ofasetofdataisthesumofthemeasurementsdividedbythenumberofmeasurementscontainedinthedataset.Themean(average)ofasetofdata(asample),x1,x2,...,xn,isdefinedbyMean平均数概率论与统计学原理chapt2englishMeanExampleRawData: 10.3 4.9 8.9 11.7 6.3 7.7XnXXXXXXiin11234566103498911763776830.......概率论与统计学原理chapt2englishMeanExample(EPAGAS)Themeangasmileageforthe100carsis概率论与统计学原理chapt2englishMedian中位数DEF2.5Themedianofaquantitativedatasetisthemiddlevalueofthedatarankedinascending(ordescending)order.PositionofMedianinSequencePositioningPointn122.Median概率论与统计学原理chapt2englishRawData: 24.1 22.6 21.5 23.7 22.6Ordered: 21.5 22.6 22.6 23.7 24.1Position: 1 2 3 4 5PositioningPointMediann1251230226..MedianExampleOdd-SizedSample概率论与统计学原理chapt2englishRawData: 10.3 4.9 8.9 11.7 6.3 7.7Ordered: 4.9 6.3 7.7

8.9 10.3 11.7Position: 1 2 3

4 5 6PositioningPointMediann126123577892830....MedianExampleEven-SizedSample概率论与统计学原理chapt2englishMedianExample(EPAGAS)Themedianis37.0.Thesamplesizen=100isanevennumberandtherearetwomiddlevalueslocatedat50thand51stpositionsafterordering.Thesetwomiddlevaluesare37.0and37.0andso,themedianis37.0.Thisvalueimpliesthatabouthalfofthe100mileagesinthedatasetfallbelow37.0andhalflieabove37.0.概率论与统计学原理chapt2englishComparingtheMeanandMedianAdatasetsaidtobeskewed斜的

ifonetailofthedistributionhasmoreextremeobservationsthanothertail.Ifthedatasetisskewedtotheright,thenthemedianislessthanthemean.Ifthedatasetissymmetric对称,thenthemedianisequaltothemean.Ifthedatasetisskewedtotheleft,thenthemedianislargerthanthemean.概率论与统计学原理chapt2englishDEF2.6Themodeisthemeasurementthatoccursmostfrequentlyinthedataset.1. MayBeNoModeorSeveralModes2. MayBeUsedforQuantitative&QualitativeDataMode重数概率论与统计学原理chapt2english

RawData: 10.3 4.9 8.9 11.7 6.3 7.7RawData: 6.3 4.9 8.9 6.3 4.9 4.9RawData: 21 28 28 41 43 43ModeExample概率论与统计学原理chapt2englishModeExamplee.g.considerthegradesof18students,B,C,B,A,F,D,B,C,BC,B,A,F,C,B,D,C,AGrade: ABCDFFrequency:36522Themodeisthegrade"B".概率论与统计学原理chapt2englishModeExample(EPAGAS)Modeofthemileageratingis37.0(occursmostoften)概率论与统计学原理chapt2englishExerciseCalculatethemean,median,andmodeforeachofthefollowingsamples:7,-2,3,3,0,42,3,5,3,2,3,4,3,5,1,2,3,451,50,47,50,48,41,59,68,45,37概率论与统计学原理chapt2englishVariationvariability离散程度RangeVarianceStandarddeviation概率论与统计学原理chapt2englishRange极差DEF2.8Therangeofaquantitativedatasetisequaltothelargestmeasurementminusthesmallestmeasurement.Itisthesimplestmeasureofdispersion.Range=Max-MinItisverysensitivetoextremevalues.概率论与统计学原理chapt2englishRangeExampleExampleForthedataset{11,12,13,13,13,14,15},Forthedataset{9,10,11,13,15,16,17}

=13,andrange=15-11=4=13,andrange=17-9=8概率论与统计学原理chapt2englishVariance方差&StandardDeviation标准差Thevarianceandstandarddeviationmeasurethespreadofthedatasetaroundthemean.Itistheaverageofthesquaresofthedistanceeachmeasurementinthedatasetisfromthemeanofallthemeasurementsinthedataset.Notethatthevariance,σ2,ofapopulationisdefinedas概率论与统计学原理chapt2englishSampleVarianceDEF2.8Thesamplevariance,

s2forasampleofnmeasurementsisequaltothesumofthesquareddistancesfromthemeandividedby(n-1).概率论与统计学原理chapt2englishSampleVarianceFormulan-1indenominator!(UseNifPopulationVariance)概率论与统计学原理chapt2englishVarianceExamplee.g.(1)Forthesample{11,12,13,13,13,14,15},=13

(2)Forthesample{9,10,11,13,15,16,17}=13概率论与统计学原理chapt2englishSampleStandardDeviationDEF2.9Thesamplestandarddeviationiss,thepositivesquarerootofthevariance.概率论与统计学原理chapt2englishInterpretingStandardDeviationTounderstandinghowthestandarddeviationprovidesameasureofvariabilityofadataset,consideraspecificdatasetandanswerthefollowingquestions.Howmanymeasurementsarewithin1standarddeviationofthemean?Howmanymeasurementsarewithin2standarddeviationsofthemean?概率论与统计学原理chapt2english1.Chebyshev'sRule

Letkbeanypositivenumbergreaterthan1.Foranydataset(regardlessoftheshapeofthefrequencydistributionofthedata),theproportionofobservationsthatliewithinkstandarddeviationsofthemeanisatleast1-1/k2.概率论与统计学原理chapt2english1.Chebyshev'sRuleThisrulesaysthatwithintwostandarddeviationsofthemean(k=2),(-2s,+2s),youwillalwaysfindatleast75%ofthedata(since1-1/k2=1-1/4=0.75)andthatwithinthreestandarddeviationsofthemean(k=3),(-3s,+3s),youwillalwaysfindatleast89%ofthedata(since1-1/k2=1-1/9=0.89).Remarks:1.Chebyshev’sruleappliestoanydataset,regardlessoftheshapeofthefrequencydistributionofthedata.2.Chebyshev’srulecanapplytosampleandpopulationaswell.概率论与统计学原理chapt2english2.EmpiricalRule

经验法则Theempiricalruleisaruleofthumbthatappliestodatasetswithfrequencydistributionsthataremound-shaped(bell-shaped)andsymmetric.Approximately68%ofthemeasurementswillfallwithin1standarddeviationofthemean,i.e.,withintheinterval(–s,+s)forsampleand(μ–σ,μ+σ)forpopulation.Approximately95%ofthemeasurementswillfallwithin2standarddeviationsofthemean,i.e.,withintheinterval(–2s,+2s)forsampleand(μ–2σ,μ+2σ)forpopulation.Approximately99.7%ofthemeasurementswillfallwithin3standarddeviationsofthemean,i.e.,withintheinterval(–3s,+3s)forsampleand(μ–3σ,μ+3σ)forpopulation.概率论与统计学原理chapt2englishExampleExample2.12(pp.64-65).AmanufactureofautomobilebatteriesclaimsthattheaveragelengthoflifeforitsgradeAbatteryis60months.However,theguaranteeonthisbrandisforjust36months.Supposethestandarddeviationofthelifelengthisknowntobe10months,andthefrequencydistributionofthelife-lengthdataisknowntobemound-shaped.a.Approximatelywhatpercentageofthemanufacturer’sgradeAbatterieswilllastmorethan50months,assumingthemanufacturer’sclaimistrue?b.Approximatelywhatpercentageofthemanufacturer’sgradeAbatterieswilllastlessthan40months,assumingthemanufacturer’sclaimistrue?c.Supposeyourbatterylasts37months.Whatcouldyouinferaboutthemanufacturer’sclaim?概率论与统计学原理chapt2englishSolutionThepercentageofbatterieslastingmorethan50monthsisapproximately84%ofthebatteriesshouldhavelifelengthexceeding50months.Approximately2.5%ofthebatteriesshouldfailpriorto40months.IfyouaresounfortunatethatyourgradeAbatteryfailsat37months,youcanmaketwoinferences:Eitheryourbatterywasoneoftheapproximately2.5%thatfailpriorto40months,orsomethingaboutthemanufacturer’sclaimisnottrue.Becausethechancesaresosmallthatabatteryfailsbefore40months.概率论与统计学原理chapt2englishNumericalMeasuresofRelativeStanding相对位置Measuresofrelativestandingarenumberswhichindicatewhereaparticularvalueliesinrelationtotherestofvaluesinadataset.PercentilesZ-Score概率论与统计学原理chapt2englishPercentiles百分位DEF2.11Supposethemeasurementsx1,x2,...,xnhavebeenrankedinascendingorder.Thepthpercentileisanumbersuchthatp%ofthemeasurementsfallbelowthepthpercentileand(100-p)%fallaboveit.Lowerquartile(firstquartile),Q1isthe25thpercentile.Upperquartile(thirdquartile),Q3isthe75thpercentile.Themedianisthe50thpercentile(orsecondquartile).概率论与统计学原理chapt2englishProcedureforcalculatingpercentiles

Arrangethenobservationsinascendingorder,x(1),x(2),...,x(n).Calculatethepositionindexk=(pn)/100,wherepisthepercentileofinterestandnisthesamplesize.3a.Ifkisaninteger,thepthpercentileis(x(k)+x(k+1))/2.3b.Ifkisnotaninteger,thenextintegervaluegreaterthankisthepthpercentile.概率论与统计学原理chapt2englishcalculatingpercentiles