1、Unit ThirteenMeasurement Errors and AccuracyBasic Concepts and TermsA measurable quantity is a property of phenomena,bodies,or substances that can be defined qualitatively and expressed quantitatively.Measurable quantities are also called physical quantities.Measurement is the process of determinati
2、ng the value of a physical quantity experimentally with the help of special technical means called measuring instruments.The value of a physical quantity is the product of a number and a unit adapted for these quantities.The true value of a measurand is the value of the measured physical quantity,wh
3、ich,being known,would ideally reflect,both qualitatively and quantitatively,the corresponding property of the object.We shall use the term uncertainty to characterize the inaccuracy of a measurement result,whereas the term error is used to characterize the components of the uncertainty.Basic Concept
4、s and TermsThe measurement error is the deviation of the result of measurement from the true value of the measurable quantity,expressed in absolute or relative form.If A is the true value of the measurable quantity and A is the result of measurement,then the absolute error of measurement is =A A.The
5、 absolute error is usually identified by the fact that it is expressed in the same units as the measurable quantity.Absolute error is a physical quantity,and its value may be positive,negative,or even given by an interval that contains that value.One should not confuse the absolute error with the ab
6、solute value of that error.For example,the absolute error 0.3 mm has the absolute value 0.3.The relative error is the error expressed as a fraction of the true value of the measurable quantity =(A A)/A.Relative errors are normally given as percent and sometimes per thousand(denoted by).Measurement E
7、rror测量误差测量误差Measurement error may be defined as the difference between the true value and the measured value of the quantity.Systematic errors 系统误差系统误差 Random errors 随机误差随机误差What Causes Measurement Errors?Now that we know the types of measurement errors that can occur,what factors lead to errors whe
8、n we take measurements?We can separate this category into 2 basic categories:instrument and operator errors.instrument errors仪器误差仪器误差operator errors操作误差操作误差Instrument ErrorsSome basic information that usually comes with an instrument is:accuracy range response time sensitivityaccuracy-this is simply
9、 a measurement of how accurate is a measurement likely to be when making that measurement within the range of the instrument.For instance a mercury thermometer that is only marked off in 10ths of a degree can really only be measured to that degree of accuracy.Instrument ErrorsSome basic information
10、that usually comes with an instrument is:accuracy range response time sensitivityrange-instruments are generally designed to measure values only within a certain range.This is usually a result of the physical properties of the instruments,such as instrument mass or the material used to make the inst
11、rument.For instance a cup anemometer that measures wind speed has a minimum rate that is can spin and thus puts a limit on the minimum wind speed it can measure.Instrument ErrorsSome basic information that usually comes with an instrument is:accuracy range response time sensitivityresponse time-if a
12、n instrument is making measurements in changing conditions every instrument will take time to detect that change.This again is often associated with the physical properties of the instrument.For instance a mercury thermometer taken from room temperature and put into boiling water will take some time
13、 before it gets to 100 oC.Reading the thermometer too early will give an inaccurate observation of the temperature of boiling water.Instrument ErrorsSome basic information that usually comes with an instrument is:accuracy range response time sensitivitysensitivity-many instruments are have a limited
14、 sensitivity when detecting changes in the parameter being measured.For instance some cup anemometers,because of their mass cannot detect small wind speeds.The problem gets the worse as the anemometer gets heavier.Operator ErrorsThese errors generally lead to systematic errors and sometimes cannot b
15、e traced and often can create quite large errors.ExampleMeasurement Location Errors Data often has errors because the instrument making the measurements was not placed in an optimal location for making this measurement.A good example of this,is again associated with measurements of temperature.Any t
16、emperature measurement will be inaccurate if it is directly exposed to the sun or is not properly ventilated.In addition,a temperature device place too close to a building will also be erroneous because it receives heat from the building through radiation.Quality Indicator:Precision of measurement P
17、recision is the degree of repeatability(or closeness)that repeated measurements of the same quantity display,and is therefore a means of describing the quality of the data with respect to random errors.Quality Indicator:Accuracy of measurement Accuracy is the degree of closeness(or conformity)of a m
18、easurement to its true value.Quality Indicator:Reliability of measurementreliability=precision+accuracymean percentage error(MPE)mean absolute percentage error(MAPE)mean bias error(MBE)mean absolute bias error(MABE)root mean square error(RMSE).kiimimicHHHkMPE11001kiimimicHHHkMAPE11001kiimicHHkMBE11k
19、iimicHHkMABE11kHHRMSEkiimic12where Him is the ith measured value,Hic is the ith calculated value and n is the total number of the observations.Linear associationCorrelation can be used to summarize the amount of linear association between two continuous variables x and yIf there is a strong linear a
20、ssociation between the two variables,then the points lie nearly in a straight line,like this:A positive association between the x and y variables(i.e.an increase in x is accompanied by an increase in y)is shown by the scatterplot having a positive slope.Similarly,a strong negative association(i.e.an
21、 increase in x is accompanied by a decrease in y)is shown by points with a negative slope.The strength of linear association,is summarized by the correlation coefficient,defined as:标准差标准差Example-Calculation of R for students heights and weights Engineers are increasingly being asked to monitor or ev
22、aluate the efficiency of a process or the performance of a device.1.Measurement Errors,Accuracy,and Precision three kinds of errors how to characterize measurements and instrumentation as being of high or low precision2.Estimating Measurement Uncertainty multi-sample experiments single-sample experi
23、ments In this case we refer to an uncertainty distribution rather than a frequency distribution.Frequently used words and phrases:uncertainty 不确定性不确定性 measurement error 测量误差测量误差true value 真值真值 measured value 测量值测量值recording errors 记录误差记录误差 systematic or fixed errors 系统误差系统误差accidental or random erro
24、rs 随机误差随机误差measurement system 测量系统测量系统mean value 平均值平均值 frequency distribution 频率分布频率分布probability density function 概率密度函数概率密度函数distribution function 分布函数分布函数discrete probability distribution 离散型概率分布离散型概率分布continuous probability densities 连续型概率密度连续型概率密度conditional probability 条件概率条件概率 Law of Large N
25、umbers 大数定律大数定律Central Limit Theorem 中心极限定律中心极限定律1.Results are often derived from the combination of values determined from a number of individual measurements.2.Unfortunately,every measurement is subject to error,and the degree to which this error is minimized is a compromise between the(overall)ac
26、curacy desired and the expense required to reduce the error in the component measurements to an acceptable value.3.Good engineering practice dictates that an indication of the error or uncertainty should be reported along with the derived results.be derived from:从从中得到中得到be subject to:受受支配支配 compromi
27、se:妥协妥协 折衷折衷dictate:要求要求 规定规定 indication:指标指标4.Implicit in this assumption is that the worst-case errors will occur simultaneously and in the most detrimental fashion.5.A more realistic estimate of error was presented by Kline and McClintock based on single-sample uncertainty analysis.6.The followin
28、g discussion is meant to provide an insight into measurement uncertainty rather than a rigorous treatment of the theoretical basis.implicit:暗示暗示 the worst-case error:最大误差最大误差 detrimental:有害的有害的insight into:对对的洞察力或深入的理解的洞察力或深入的理解 rigorous:严格的严格的single-sample uncertainty analysis:单样本不确定性分析单样本不确定性分析7.T
29、he errors that occur in an experiment are usually categorized as mistakes or recording errors,systematic or fixed errors,and accidental or random errors.8.Systematic errors may result from incorrect instrument calibrations and relate to instrument accuracy(the ability of the instrument to indicate t
30、he true value).9.Random errors cause readings to take random values on either side of some mean value.They may be due to the observer or the instrument and are revealed by repeated observations.be categorized as:可分为可分为 recording errors:记录误差记录误差systematic or fixed errors:系统误差系统误差 accidental or random
31、 errors:随机误差随机误差instrument calibration:仪表刻度仪表刻度 instrument accuracy:仪表精度仪表精度indicate:指示指示 显示显示mean value:平均值平均值 reveal:显现显现 显示显示10.In measurement systems,accuracy generally refers to the closeness of agreement of a measured value and the true value.11.All measurements are subject to both systematic(
32、bias)and random errors to differing degrees,and consequently the true value can only be estimated.12.To illustrate the above concepts,consider the case shown in Fig.13-1,where measurements of a fixed value are taken over a period of time.accuracy:准确性准确性 refer to:指的是指的是differing degree:不同程度不同程度 true
33、value:真值真值illustrate:举例说明举例说明 case:例子例子 案例案例13.If we further grouped the data into ranges of values,it would be possible to plot the frequency of occurrence in each range as a histogram.14.Figure 13.3 is often referred to as a plot of the probability density function,and the area under the curve rep
34、resents the probability that a particular value of x(the measured quantity)will occur.15.The total area under the curve has a value of 1,and the probability that a particular measurement will fall within a specified range(e.g.,between x1 and x2)is determined by the area under the curve bounded by th
35、ese values.grouped into:按按分类分类 frequency of occurrence:出现频率出现频率histogram:柱状图柱状图 直方图直方图probability density function:概率密度函数概率密度函数area:面积面积fall within:落入落入 bounded by:受受限制限制 以以为界为界16.Figure 13-3 indicates that there is a likelihood of individual measurements being close to xm,and that the likelihood of
36、 obtaining a particular value decreases for values farther away from the mean value,xm.17.The frequency distribution shown in Fig.13-3 corresponds to a Gaussian or normal distribution curve,the form generally assumed to represent random measurement uncertainty.18.There is no guarantee that this symm
37、etrical distribution,indicating an equal probability of measurements falling above or below the mean value,xm,will occur,but experience has shown that the normal distribution is generally suitable for most measurement applications.likelihood:可能性可能性guarantee:保证保证 证明证明 symmetrical distribution:对成分布对成分
38、布unsymmetrical:非对称的非对称的frequency distribution:频率分布频率分布 Gaussian or normal distribution:高斯分布高斯分布 正态分布正态分布19.In analyzing these results we may apply standard statistical tools to express our confidence in the determined value based on the probability of obtaining a particular result.20.If experimental
39、 errors follow a normal distribution,then a widely reported value is the standard deviation,.There is a 68%(68.27%)probability that an observed value x will fall within of xm(Fig.13-3).21.In reporting measurements,an indication of the probable error in the result is often stated based on an absolute
40、 error prediction e.g.,a temperature of 48.30.1(based on a 95%probability)or on a relative error basis e.g.,voltage of 9.0 V 2%(based on a 95%probability).statistical tool:统计工具统计工具 confidence:信心信心 信任信任indication:指标指标 state:表达表达 陈述陈述absolute error:绝对误差绝对误差 relative error:相对误差相对误差standard deviation:标准
41、偏差标准偏差 observed value:观测值观测值 测量值测量值22.Based on the previous discussions we may characterize measurements and instrumentation as being of high or low precision.23.The low-precision measurements have a wider distribution and are characterized by a greater standard deviation,lp,compared with the high-p
42、recision measurements,hp.24.Therefore,in the absence of bias or systematic error,the mean of a large sample of low-precision measurements theoretically indicates the true value.characterize as:描述为描述为in the absence of:缺少缺少 mean:平均值平均值a large sample:大量样本大量样本low-precision measurement:低精度测量低精度测量high-pre
43、cision measurement:高精度测量高精度测量25.The previous discussion,illustrating the concept of random measurement error,has not addressed the effects of systematic or fixed(bias)errors.26.In reality,even though a high probability exists that an individual measurement will be close to the mean value,there is no
44、 guarantee that the value of the mean of the large sample of measurements will be the true value(Fig.13-5).27.An example of a source of systematic error is a pressure gauge needle that is bent(i.e.,not zeroed).address：指出指出 pressure gauge needle:压力表指针压力表指针bent:弯曲的弯曲的 偏离的偏离的 zero:指零指零exist:存在存在28.Mult
45、i-sample experiments are those in which,for a given set of the independent experimental variables,the readings are taken many times.29.If we could repeat our tests many times,with many observers and a variety of instruments,we could apply statistics to determine the reliability of the results as in
46、the previously discussed methods.30.Single-sample experiments are those in which,for a given set of experimental conditions,the readings are taken only once.These are typical in engineering,where financial or time constraints limit the number of repetitions of a particular test.multi-sample experime
47、nt:多样本实验多样本实验 set:系列系列independent experimental variable:独立测量变量独立测量变量single-sample experiment:单样本实验单样本实验 statistics:统计学统计学 31.The experimenter also expresses the degree of confidence in the stated uncertainty based on odds,in a manner analogous to the standard deviation.odds:几率几率 set:系列系列analogous to:类似于类似于
