Introduction to Data Analysis Data Reduction Example by gregoria

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									                                                                                     Data Reduction



                                                                  • Translate measured data into one or more physical
                                                                    variables of interest
                  Introduction to Data Analysis                   • Obtain
                                                                       – best estimate for physical variable
                                                                       – estimate for precision and accuracy of measurement
                               c.f. Bevington                            (systematic and statistical uncertainty)
                               Chapters 1-3                       • Example (from my experiment):




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        Example: Determination of particle
                                                                                         Histograms
            momentum distribution

              z
      Beam




                      By
                                            R = pzx/(q*By)             Binned representation of data in 1, 2 or 3 dimensional
                           2                                           variable space


                  1


  x
         10 cm




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                                                                                                                                1

              Statistical and Systematic Error                                                   Distributions
                                                                                                        Mean

                                                                          Sample
     • Systematic error                                                 Distribution
          – inherent to measurement, apparatus, methods                                                           Variance
          – estimate magnitude by comparing different approaches
          – limits accuracy
     • Statistical error
          – Measurements jitter around truth
          – Average many measurements to improve estimate (if they
            are independent)
          – limits precision, but lim(N-> inf) = truth                                Mean
                                                                                                                             for N -> inf :
                                                                                                                             Parent distribution
                                                                                     Variance
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                   Binominal Distribution                                                   Poisson Distribution




               Mean                                                                  Mean

               Std. Dev                                                              Std. Dev


          P(x:n,p): Probability to get ‘yes’ x times out of n tries,             Derives from binomial distribution for p << 1 with n*p = µ
          if probability of ‘yes’ for single try is p                            Important for counting experiments with low count rate



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                                                                                                                                                   2

                     Gaussian Distribution                                                  Statistical Error on Mean




                Mean

               Std. Dev                                                              • Repeated measurements increase precision
                                                                                         – but only as sqrt(N)
                                                                                         – ultimate limit may come from systematic uncertainty

                Derives from Poisson distribution for n*p >> 1

                Seen everywhere b/c of Central Limit Theorem


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                          Error propagation

              Interested in error on ‘x’, but measure ‘u’, with x = f(u)




               What if x = f(u,v):




               Errors add in quadrature, provided error in u and v
               is independent


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