1/73TheoryofError&DataProcessing误差理论与数据处理ReviewLesson(复习课)2/73期末考试时间:2015年1月6日周三14:00~15:40(100分钟)地点:闭卷,请携带计算器3/73Thepurposesoferrorstudying研究误差的意义Understandthepropertiesoferrorsandinvestigatethesourcesoferrors,soastoreduceorremovethem;认识误差的性质,研究误差的来源,以减小和消除误差;Properlyhandlemeasuringanddataprocessing,andcorrectlyanalyzetheresultsofmeasurement,inordertoobtainoutcomesasclosetotruevaluesaspossible;正确处理测量和实验数据,合理分析所得结果,以便得到更接近于真值的数据;Correctlyarrangetheexperiments,designorselectinstrumentsandapplymeasuringmethods,thusconstructtheoptimalsystemtoobtainthebestresult.正确组织实验,合理设计或选择仪器和测量方法,以便在合理优化的系统下得到理想结果。4/73DefinitionsAbsoluteError绝对误差L=L-L0C=L-L0=-LLLLLr0RelativeError相对误差mmmxrxFiducialError仪器引用误差测得值绝对误差真值绝对误差相对误差Correction修正值修正值=真值-测得值绝对误差=测得值-真值测量范围上限最大示值误差引用误差最大仪器)(5/73Thesourcesoferror(误差来源)Thesourcesoferrorareeverywhere,andalmosteverystepofthemeasuringprocedurebringsinerror.误差无处不在,几乎测量的每一个步骤都会导入误差。Majorsourcesoferror误差的主要来源Device(设备)Environ-ment(环境)Method(方法)People(人员)6/73ClassificationsClassificationsoferrors误差分类:ErrorAbsoluteErrorRelativeErrorAbnormalErrorSystemErrorRandomErrorIntermsofexpression按表示方式分类Intermsofproperty按性质分类绝对误差相对误差系统误差随机误差粗大误差7/73Correctness,Precision&AccuracyHighcorrectness高准确度Lowprecision低精密度Lowcorrectness低准确度Highprecision高精密度Lowcorrectness低准确度Lowprecision低精密度Highcorrectness高准确度Highprecision高精密度CorrectnessSystemErrorPrecisionRandomErrorAccuracySystem&RamdomError8/73SignificantDigitForanyapproximatenumber,theleftmostnon-zerodigitiscalledthefirstsignificantdigit.Fromthefirstsignificantdigittothelastdigit,alldigits,zeroornon-zero,arecalledsignificantdigits.对于任何近似数,从左边起的第一个非零的数字称为第一位有效数字,从第一位有效数字起到最末一位数字为止的所有数字,不论零或者非零,都是有效数字。SignificantDigit有效数字:9/73SignificantDigit35.6Vs0.03560.0027≠0.002702400???2.4x103or2.40x103or2.400x103Standardexpression:ax10n;1≤a10L=20.53±0.01=(2.053±0.001)x101L=20.531±0.01Forimportantmeasurements:L=15.214±0.04210/73Examplesofrounding-offoperation数字舍入运算示例Originaldata:3.141592.717294.510503.215506.3785017.6914995.43460Roundedoffdata:3.1422.7174.5103.2166.3797.6915.43511/73Rulesofdatacomputation数据运算规则Forsummationandsubtraction,thecomputingdatawiththefewestnumberofdecimaldigitsshouldbethereference,andotherdatamaykeepanextradecimaldigit,butthefinalresultshouldhavethefewestnumberofdecimaldigitsasthereference;在加减运算时,各运算数据以小数位数最少的数据位数为准,其余各数据可多保留一位小数,但最后结果应保留最少的小数位数;Formultiplicationanddivision,thecomputingdatawiththefewestnumberofsignificantdigitsshouldbethereference,andotherdatamaykeepanextrasignificantdigit,butthefinalresultshouldhavethefewestnumberofsignificantdigitsasthereference;在乘除运算时,各运算数据以有效位数最少的数据为准,其余各数据可以多保留一位有效位数,但最后的结果应象参考数据一样保留最少的有效位数;Forcomputationofsquareandsquareroot,thecomputingnumbersaretreatedthesamewayasinmultiplicationanddivision;在平方和开方运算时,按照乘除运算处理;12/73NormalDistribution正态分布:)2/(2221)()(ePfdeF)2(2221)()(f0)(dfEExpectation期望值:Variance方差:df)(2254)(||df326745.0Averageerror平均误差:Probableerror或然误差:13/73Thefigureshowsnormaldistributioncurveandrelatedparameters.正态分布密度曲线和相应的各种参数如图所示σistheXvalueofinflexionpointA;θistheXvalueofpointB,thegravitycenteroftherighthalfofthecurve;lineparalleltoY-axisatx=ρevenlysplitstheareaoftherighthalfofthecurve.σ是曲线拐点A的横坐标;θ是B点的横坐标,也就是右半部曲线下方面积的重心;在x=ρ处与Y轴平行的直线等分曲线右半部分下方面积.NormalDistribution正态分布14/73PropertiesofNormalDistribution正态分布的特性:01limniinThedistributionissymmetrictoY-axis,i.e.forerrorw/thesameabsolutevalue,positiveerrorandnegativeerrorappearw/sameprobability分布曲线关于Y-轴对称,就是说,绝对值相等的正负误差出现的概率相同;(对称性)Errorw/smallerabsolutevalueappearsw/higherprobability绝对值小的误差出现的概率大;(单峰性)Undercertaincondition,theabsolutevalueofrandomerrorsareboundedwithinacertainrange在一定的条件下,随机误差的绝对值不会查过一定界限;(有界性)Theaverageofrandomerrorsgoesto0asthenumberofmeasurementincreases随着测量次数增加,随机误差的平均值趋于0;(抵偿性)15/73letbetheresultsofnmeasurements,thenthearithmeticmeanisgivenasfollows:设为n次测量所得的值,则算术平均值按如下计算:ArithmeticMean算数平均值Whenmeasuringthesameobjectwithequalprecision,theresultsaredifferentduetorandomerrors,andthearithmeticmeanshouldbeusedasthefinalmeasurementresult.对某一量进行多次等精度测量,由于随机误差的存在,测量结果各不相同,应以全部测量值的算术平均值作为最后测量结果。thesignificanceofarithmeticmean算术平均值的意义:nlll,,,21niinlnnlllx1211nlll,,,2116/73ResidualError残余误差xlviininiiixnlv11niiv1017/73单次测量的标准差02Llwhereniiixlvwherenviii12Standarddeviationofasinglemeasurement单次测量的标准差Bessel’sformula:贝塞尔公式:Peter’sformula:别捷尔斯公式:)1(253.11nnvnii18/73StandardDeviationofArithmeticMean算数平均值的标准差nlllxn21Foraseriesofmultipleindependentmeasurements,arithmeticmeanisusedastheresultofthemeasurement.对于一系列重复的独立测量,我们把算数平均值作为测量结果。nx19/73)(2limtptxxyprobabilitconfidencewithNormaldistribution20/73ExtremeerrorsoxLxxtxlimTheerrorofarithmeticmeanWhenthenumberofmeasurementsislarge,thearithmeticmeanisalsonormallydistributed,sosimilarlywehave:Ifthenumberofmeasurementsnissmall,weneedtouse“StudentDistribution”,alsocalledtdistribution,tocalculatetheextremeerrorofarithmeticmean.xatxlimistheconfidencecoefficientofSt