1 Practical Statistics for Physicists SLUO Lectures February 2007 Louis Lyons Oxford

2 Topics 1) Introduction Learning to love the Error Matrix 2) Do’s and Dont’s with L ikelihoods 3) χ 2 and Goodness of Fit 4) Discovery and p-values

3 Books Statistics for Nuclear and Particle Physicists Cambridge University Press, 1986 Available from CUP Errata in these lectures

4 Other Books CDF Statistics Committee BaBar Statistics Working Group

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10 Difference between averaging and adding Isolated island with conservative inhabitants How many married people ? Number of married men = 100 ± 5 K Number of married women = 80 ± 30 K Total = 180 ± 30 K Weighted average = 99 ± 5 K CONTRAST Total = 198 ± 10 K GENERAL POINT: Adding (uncontroversial) theoretical input can improve precision of answer Compare “kinematic fitting”

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12 Relation between Poisson and Binomial PeopleMaleFemale PatientsCuredRemain ill Decaying nuclei ForwardsBackwards Cosmic raysProtonsOther particles N people in lecture, m males and f females (N = m + f ) Assume these are representative of basic rates : ν people νp males ν(1-p) females Probability of observing N people = P Poisson = e –ν ν N /N! Prob of given male/female division = P Binom = p m (1-p) f Prob of N people, m male and f female = P Poisson P Binom = e –νp ν m p m * e -ν(1-p) ν f (1-p) f m! f ! = Poisson prob for males * Poisson prob for females

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14 Relevant for Goodness of Fit

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18 Gaussian = N (r, 0, 1) Breit Wigner = 1/{π * (r 2 + 1)}

19 Learning to love the Error Matrix Introduction via 2-D Gaussian Understanding covariance Using the error matrix Combining correlated measurements Estimating the error matrix

20 Element E ij - Diagonal E ij = variances Off-diagonal E ij = covariances

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32 Small error Example: Lecture 3 x best outside x 1  x 2 y best outside y 1  y 2

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36 Conclusion Error matrix formalism makes life easy when correlations are relevant

37 Next time: L ikelihoods What it is How it works: Resonance Error estimates Detailed example: Lifetime Several Parameters Extended maximum L Do’s and Dont’s with L