生物统计学课件chapter2bel A measure of central tendency

生物统计学课件chapter2belAmeasureofcentraltendencyAsarepresentativeofthedata,comparedwithotherdata,pointoutthecentralpositionofthedata.DifferentAveragesaresuitablefordifferentdata.Example：Comparisonoftheheight,weightofthepeopleindifferentcountries,regions,racesetc.;Comparisonoftheproductionperformanceindifferentspeciesoflivestockandpoultry2.6AmeasureofcentraltendencyContent：1、Arithmeticaverage2、Medium、Mode、GeometricmeanandHarmonicMean3、Therelationshipamong5meansandthecomment2.6Ameasureofcentraltendency2.6.1Arithmeticaverage1、DefinitionSumofallobservationsdividedbythenumber,knownasthearithmeticmean.Themostcommonlyusedindicatorofameasureofcentraltendency.SampleaveragePopulationaverage2.6.1Arithmeticaverage：Observationorvariablein：NumberofObservationsorvariables∑：Summationsymbol（sigma）Formula：1、Directcalculate：Thelittersizesof10Landracesows：7、7、8、8、8、9、9、9、10、10，Seekingtheaveragelittersizeofthe10Landracesows,Theaveragelittersizeof10Landracesowswas8.5,soaverageisrepresentativeofthedata.suitabletothesmallsample,samplesizesmallerthan302.6.1Arithmeticaverage2、Weightmethod(1)分类资料：每个类别在某个指标上取相同的值。(2)计数资料和连续性资料：频率分布 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加权法，即计算时先将各个变数乘上它的权数，再经过总和，然后除以权数的总和，称为加权平均数。2.6.1ArithmeticaverageFormula：xi=Variablefi=FrequencyofVariablexi2.6.1ArithmeticaverageExample2.2：Apopulationsize1000，thefrequencyofAlleleAis0.6，Apopulationsize400,AlleleAis0.3，whatisthefrequencyofAlleleA,whenthe2populationmixed.Frequency：2.6.1ArithmeticaverageExample2.3：litterweightof1monthfrom200largewhitesows2.6.1Arithmeticaveragexi=groupvaluefi=frequencyofgroupvalue2.6.1Arithmeticaverage3、characteristic（1）Meandifferencefromtheaverageiszero：Thedifferenceofobservationfromtheaverage,referredtothemeandifference.2.6.1ArithmeticaverageExample2.2：Theweightoffivepigswere70,72,80,83,88kg,thearithmeticaverageof5pigswas78.6kg.2.6.1Arithmeticaverage（2）Thesumofsquaredeviationfromaveragewastheminimum:Thesumofsquaredifferencebetweentheobservationandtheaverageinasamplewassmallerthanthatbetweentheobservationandanyothervalue.Namely:therefore：Averageisclosesttothevalueofalltheobservationstherefore：Theaveragerepresentscentraltrendofthissample.2.6.1ArithmeticaverageDefinition:thenthrootoftheproductofnnon-negativeobservationswascalledgeometricmean(G).Fortheconvenienceofcalculation,First,logarithmicofthevariables,thenthesumoflogarithmicvariabledividedbyn,namely,lgG,atlasttakeAntilog,thatis,Gvalue.2.6.2Geometricmean2.6.2GeometricmeanThegeometricmeanoftheoriginaldataisfirstconvertedtologarithms;andthenseekthearithmeticaverageofthevalues;andfinallytaketheantilog.Appliedtothepercentageandtheproportionofdata,suchasgrowth,interestrates,drugpotency,antibodytitersandsoon.Weakenthedataovertoolargeobservationvalue.2.6.2GeometricmeanExample2.3：Adairyfarmin1995,therewere100dairycattle,andthenumberofcowsin1996,1997and1998respectively,werethepreviousyear's2,3and4.5times,whatwastheaverageannualgrowthrateofthedairycattle。Solution：Thenumberofdairycattleintheyear1998：100×2×3×4.5＝2700headsor100×33＝2700heads2.6.2GeometricmeanWeightingmethod:classificationdataorcountandcontinuousdataLogarithmic例2.4P152.6.2Geometricmean对某地一免疫鸡群测定其对鸡新城疫的血球凝集抑制滴度分布情况见下表，请计算该鸡群的平均血球凝集抑制滴度。Definition:Thereciprocaloftheaverageofthereciprocalofalltheobservation.2.6.3HarmonicMeanAppliedtotheextremeright-skewness,example2-62.6.3HarmonicMeanWeightingmethodSimplemethodthegeometricmeanwasfirstconvertedtheoriginaldatatoreciprocal;andthenseekthearithmeticaverageoftransformedvalue;andfinallytakethereciprocaloftheaverage.usedforspeed,orthedataofextremelylargevalue.2.6.3HarmonicMean2.6.3HarmonicMean某头肉猪在原体重基础上净增重90kg，经测定第一个30kg的日增重为0.4kg，第二与第三个30kg分别为0.5与0.7kg，求全期平均日增重。forthespeedcategoriesdata,calculatetheaveragespeed2.6.3HarmonicMean例2.4：用某药物救治12只中毒的小鼠，它们的存活天数记录如下：8，8，8，10，10，7，13，10，9，14，另外有两只未死亡，求平均存活天数。解：未死亡的存活天数记为∞，为极端右偏态，用算术平均数不合理。定义：将n个观察值从小到大依次排队，位于中间的那个观察值称为中位数。2.6.4Median（Md）Applicabletoskeweddistributiondata例：2.5现有一窝仔猪的出生重资料为：1.4，1.0，1.3，1.2，1.6kg，试求其中位数。解：首先将数据资料排序：1.0，1.2，1.3，1.4，1.6；然后计算中位数：2.6.4Median（Md）如果增加一头仔猪，出生重为1.8kg，计算中位数：2.6.4Median（Md）对于频数分布的资料，公式如下：Lmd：中位数所在组的组下限；fm：中位数所在组的频数；C：从第一组到中位数所在组的累计频数n：样本含量；i：组距；2.6.4Median（Md）Example：Table2－5992.6.4Median（Md）Definition:themostfrequencyvariableinthedata,Itcallsamode。1.Discretedata：Themostfrequencyvariable2.ContinuousData：Inthefrequencytable,thegroupvalueofthehighestfrequencygroup.！！Somedatamayhavemultiplemode,thatismorethanoneofthemosthigh-frequencyvalue;somedatamayhavenomode,ifallthevalueshavethesamefrequency。2.6.5mode（M0）2.6.6Therelationshipbetweenthevariouscentraltendencyandevaluationofthem1、Therelationshipbetweenthevariouscentraltendency1.Innormaldistribution，arithmeticaverage,medianandmodewereequal.2、EvaluationofthevariouscentraltendencyShouldmeetthefollowingconditions：(1)shouldbeastrictdefinitionandalgorithms,toavoidtheexistenceofasubjectiveelementduring；(2)Thecalculationshouldmakeuseofalltheobservations；(3)Simple,easytocomprehend,easytocalculate；(4)Littleaffectedbythesamplingmethod,thatisasmallsamplingerror;(5)canbedealwiththealgebraicmethods.2.6.6Therelationshipbetweenthevariouscentraltendencyandevaluationofthem（1）Arithmeticaveragemeetallthecondition，Fornormaldistributiondata.Mostlivestocktraitsarenormallydistributedquantitativetraits,sothearithmeticmeanisthemostcommon,andmostimportantone。However,whenthedistributionofasymmetrical,wasskewed,thearithmeticmeanisdifficulttoexpressthecentraltendency。2.6.6Therelationshipbetweenthevariouscentraltendencyandevaluationofthem（2）Mediancanmeetthefirst2and3conditions,shallapplytonon-parametrictests,suchasthechi-squaretestandsoon.（3）Geometricmeanandharmonicmeancanmeetthefirst1,2and5conditions,suitabletorightskeweddistribution.2.6.6Therelationshipbetweenthevariouscentraltendencyandevaluationofthem1、TherepresentativedegreeoftheaverageisrelatedtothedegreeofvariationAsamplehavealotofvariables,usingtheaverageasampleasarepresentative,anditsrepresentativedependsonthedegreeofvariationinallvariables。(1)Ifthevariablesweresameorrelativelysmallvariation,thentheaveragecanrepresenttheentiresample.(2)Ifthevariationofvariableswasrelativelylarge,thentheaverageastherepresentationwassmall.Therefore,theaveragealonecannotfullyandcorrectlydescribethedata,andcannotexpressthesamplevariation2.7Dispersiontendencyofthemeasurement例：即使两个样本的平均数相同，但是样本内变数的变异程度不一定相同。2.7DispersiontendencyofthemeasurementFromtheaverage：1.甲、乙两品种的平均产仔数相同，都是11头。从平均数来看，两个品种没有差异。2.进一步观察各个变数，两个样品的变异程度并不相同。甲：最小为4，最大为22；乙：最小为8，最大为14甲的变异程度大于乙甲的平均数的代表性小于乙的平均数所以，应该测定其变异程度从公式可以看出：标准差考虑了每个变数与平均数的离差。每个变数与平均数相差愈小，样本变异程度愈小，反之，愈大。因此，标准差是离散程度的度量2.7.1Standarddeviationandvariance标准差公式的来源1.离均差2.离均差之和3.离均差平方和虽然离均差可以衡量变异程度，但是离均差之和为0，所以不是理想的指标为了合理地计算平均差异，用平方和的办法来消除离均差的正负号，离均差平方相加，得到平方和（SS），但是由于不同样本的观察值个数不同，所以离均差平方和也不是理想指标2.7.1StandarddeviationandvarianceSamplevarianceandstandarddeviation将离均差平方和求平均数，称为样本方差，目的是消除观察值个数的影响样本方差开方，目的是使变异还原，即标准差。2.7.1StandarddeviationandvariancePopulationvarianceandstandarddeviationGenerallyunknown,thesamplestandarddeviationtoestimateandinferthepopulationstandarddeviation2.7.1Standarddeviationandvariancedegreeoffreedom（df），df=n-1如果一个样本含有n个变数，从理论上讲，n个变数都同样用以计算标准差，n个变数与平均数相减有n个离均差。表面上虽有n个比较，但实质上仅有n-1个可以自由变动，最后一个离均差受到离均差之和这个条件的限制，所以不能自由。2.7.1Standarddeviationandvariance例如：有5个变数。其4个离均差为-2、-1、1、2，则第5个离均差必等于0；如4个离均差为-1、0、1、2时，则第5个离均差必等于-2，这样才能使离均差的总和等于0；这5个离均差中，因受离均差之和等于0的限制，所以只有4个能自由变动。这时的自由度就是n-1。自由度等于样本变数的总个数减去计算过程中使用的条件数。2.7.1Standarddeviationandvariancetheactualformulaofvarianceandstandarddeviation2.7.1Standarddeviationandvariance例：20-21页，例2-810头考力代绵羊产毛量资料计算标准差表ProofoftheformulaonthesumofsquaresTheformulaofweightingmethod2.7.1StandarddeviationandvarianceDefinition:ThedifferencebetweenthemaximumandminimumvaluesofallvariablesR=Max(x)-Min(x)范围或全距可以反映变异程度的一部分，但是不能代表样本内各变数之间的变异程度。目前，被广泛使用的是以标准差来度量变异程度。2.7.2Range2.7.3Theaverageabsolutedeviation定义：各观测值离均差绝对值的平均数。分组资料，加权法未分组资料2.7.4Coefficientofvariation（C.V）用于比较度量单位不同以及不同时期的资料。例：现有100头成年母猪，体重180kg，s为10.250头育成母猪，体重90kg，s为8.6请比较它们的变异程度。计算如下：C.V.育>C.V.成，育成母猪的相对变异大于成年母猪2.7.4Coefficientofvariation（C.V）