VIEWS: 7 PAGES: 23 POSTED ON: 3/9/2011 Public Domain
A probabilistic approach to teaching measurement and uncertainty Saalih Allie and Andy Buffler Department of Physics, University of Cape Town, South Africa Fred Lubben and Bob Campbell University of York, United Kingdom SAIP conference F13, Stellenbosch 2003 Some problems terms before concepts Terminology e.g. “errors” The lecturer’s readings are correct but the students are in error The experiment did not work because of human error Rules of thumb e.g. significant figures Formalism: large N and frequency interpretation of data Few readings in first year practicals inconsistencies Ad hoc approach for single reading No formal logical link between data {xi} and measurand Xbar and sigma tell us about the data not the measurand ISO / GUM - Guide to Reporting Uncertainty in Measurement What is a measurement? Purpose of a measurement is to update our state of knowledge about some physical property which we have conceptualised the measurand measurand voltage battery V reading 3.24 volts What can say about the value of the measurand (the battery voltage)? “say” infer Measurement result involves inference Conceptualise and define measurand X Observe readings {xi} (numbers) Probabilistic Inference Standard Inference “What can I infer from “What would happen if I these data?" did this a very large Incorporate {xi} into number of times?" Prob. Inference Model Statement about the Statement about data {xi} not the measurand X measurand X Bayesian approach Frequentist approach A probabilistic model for measurement all prior new information data Observations from apparatus inference Probability density engine functions best estimate final Inferences uncertainty result about the measurand coverage probability Probability density function P(M) describes our knowledge of the “true” value of mass M. P(M) is the probability that the mass lies within a given range P(M) [kg-1] ∫ P( M ) dM = 1 M [kg] If the shaded area = 0.2 say, then P(30 < M < 40) = 0.2 then we can make statements of the following type: “The probability that the value of M lies between 30 kg and 40 kg is 0.2.” Measurement involves seeking the final pdf which we summarise in terms of 3 quantities Best estimate Mbest = the expectation value E(M) Mbest = <M> = E(M) = ∫ M P(M) dM Standard Uncertainty = square root of the variance (2nd moment) variance = σ (M) = ∫ (M −M ) P(M) dM 2 best 2 Coverage probability (or level of confidence) area bounded by the curve and the uncertainty interval Standard uncertainties (u) and associated coverage probabilities 1/a u =a 2 3 p(x) Uniform (or rectangular) 0.58 pdf 0 a x a 2/a u =a 2 6 Triangular p(x) 0.65 pdf 0 x a Best estimate, uncertainty & coverage probability for the Gaussian pdf b u=σ 0.68 b e 1 ⎡ ( x − µ )2 ⎤ p (x ) = exp ⎢ − σ 2π ⎣ 2σ 2 ⎥ ⎦ 0 x µ− σ µ µ+ σ 1 N s (x ) = ∑ ( x i − x )2 N (N − 1) i =1 From N data we estimate µ by x [the mean] and σ by s(x ) [the experimental u = s(x) standard deviation of the mean Type A and type B evaluation of uncertainties In .the case of large enough numbers of readings the analysis is the same as the traditional approach for calculating the mean and the standard uncertainty. The calculation of uncertainties based on the usual statistical methods is called a Type A evaluation (uA) In the frequentist approach the single measurement poses a problem but follows naturally in the probabilistic framework by using an a priori pdf. Evaluation of uncertainties not using the usual statistical methods is known as a Type B evaluation (uB) Random and systematic “errors” do not correspond to Type A or Type B evaluations! If measurand X =x ± xcorrection where x is obtained from many readings and xcorrection is some correction term then the uncertainty (u1) associated with x will be estimated from a Type A evaluation while the uncertainty associated with the correction (u2) will come from a Type B evaluation. The combined uncertainty for X, u, is then calculated from u2 = (u1)2 + (u2)2 General procedure Identify all of sources uncertainty Uncertainty Assign a pdf to each source of uncertainty Budget Evaluate each contribution (Type A or Type B evaluation) Gaussian, Student-t, Triangular, Rectangular Several standard uncertainties. Model equation & quadrature GUM Combined standard workbench uncertainty uc Example Consider a digital voltmeter rated at ±1%, showing Volts What is the result of the measurement? In the absence of any other information we assume that the voltage could be anywhere between 2.855 and 2.865 with equal probability, otherwise the digit would be either a 5 or a 7. Thus, a suitable pdf is a uniform distribution over this interval P(Vtrue) [V-1] V(true) (Volts) 2.855 2.865 2.86 Example Consider a digital voltmeter rated at ±1%, showing Volts What is the result of the measurement? Best estimate of the voltage at this stage is of course 2.86 V Identify two sources of uncertainty (at least) (1) uncertainty due to the scale us and (2) uncertainty due to the rating ur The scale uncertainty us (type B evaluation) In the absence of any other information we assume that the voltage could be anywhere between 2.855 V and 2.865 V with equal probability, otherwise the digit would be either a 5 or a 7. Thus, a suitable pdf is a uniform distribution over this interval. P(Vtrue) [V-1] V(true) (Volts) 2.855 2.865 2.860 half the width of the rectangle 0.005 us = = = 0.0029 V 3 3 0.3 Relative frequency 0.2 (s-1) 0.1 0.0 0.4 0.6 0.8 1.0 1.2 1.4 tmeasured (s) Area between 30 1.015 ± 0.033 s = 0.68 P(ttrue) (s-1) Total area under P(ttrue) = 1.00 0.0 0.4 0.6 0.8 1.0 1.2 1.4 ttrue (s) tresult = 1.015 ± 0.033 s Conclusion The probabilistic formalism for metrology offers a logical and consistent framework for data analysis, naturally incorporating the limiting cases of only a single reading and a large number of dispersed data. The approach also offers the basis for a systematic teaching framework at first year level and beyond, for promoting a better understanding of the nature of experimental measurement and uncertainty. The new course on measurement and uncertainty Framework for development based on .... 1. Our point and set paradigmatic model of student reasoning 2. Expectations of laboratory work We have interviewed physics staff and students (from South Africa, USA, UK, France and Greece) concerning their views on the purpose of practical work in physics. Uncover or demonstrate Develop skills of physics phenomena experimentation (in a physics context) Main focus of our course (and new materials) 3. Philosophy and theory of measurement and data analysis - ISO (probabilistic) approach. The new course on measurement and uncertainty ... 2 An interactive student workbook has been written to introduce the new concepts. Students work through the activities in the workbook in small groups in a tutorial-type collaborative learning environment. On alternate weeks, the students are engaged in activities in the laboratory which are designed to support the new ideas about measurements and provide “hands-on” laboratory experiences. The course consists a 3 hour session per week for 16 weeks. The new course on measurement and uncertainty ... 3 The course has been piloted in the Physics Department at the University of Cape Town in 2002 and 2003. The evaluation of the new course involved the diagnostic testing of the students both before and after the course as well as interviews with individual students. The materials will be edited and published (hopefully for 2004, but more realistically 2005). Unit Description 1. Introduction to The relationship between science and experiment. measurement Designing an experiment. Tables and graphs. The laboratory report. 2. Basic concepts Probability and inference. Reading digital and analogue of measurement scales. The nature of uncertainty. A probabilistic model of measurement. 3. The single Probability density functions. Representing knowledge measurement graphically using a pdf. Evaluating standard uncertainties for a single reading. The result of a measurement. 4. The repeated Dispersion in data sets. Evaluating standard uncertainties measurement for multiple readings. Type A and Type B evaluation of uncertainties. 5. Working with Propagation of uncertainties. Combined standard uncertainties uncertainty. The uncertainty budget. Comparing different results. Repeatability and reproducibility 6. Modelling Principle of least squares. Least squares fitting of straight trends in data lines. Total area under P(Vtrue) P(Vtrue) = 1.00 (volts-1) 0 Vtrue (volts) 2.855 2.860 2.865