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A probabilistic approach to teaching measurement and uncertainty

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					    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

				
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posted:3/9/2011
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