1. 1 Practical Statistics for Physicists LBL January 2008 Louis Lyons Oxford l.lyons@physics.ox.ac.uk

2. 2 PARADOX Histogram with 100 bins Fit 1 parameter S min : χ 2 with NDF = 99 (Expected χ 2 = 99 ± 14) For our data, S min (p 0 ) = 90 Is p 1 acceptable if S(p 1 ) = 115? 1)YES. Very acceptable χ 2 probability 2) NO. σ p from S(p 0 +σ p ) = S min +1 = 91 But S(p 1 ) – S(p 0 ) = 25 So p 1 is 5σ away from best value

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7. 7 Comparing data with different hypotheses

8. 8 Choosing between 2 hypotheses Possible methods: Δχ 2 ln L –ratio Bayesian evidence Minimise “cost”

9. 9 Learning to love the Error Matrix Resume of 1-D Gaussian Extend to 2-D Gaussian Understanding covariance Using the error matrix Combining correlated measurements Estimating the error matrix

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12. 12 Element E ij - Diagonal E ij = variances Off-diagonal E ij = covariances

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18. 18 N.B. Small errors

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20. 20 Mnemonic: (2*2) = (2*4) (4*4) (4*2) r c r c 2 = x_a, x_b 4 = p_i, p_j………

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24. 24 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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26. 26 Small error x best outside x 1  x 2 y best outside y 1  y 2

27. 27 a b x y

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31. 31 Conclusion Error matrix formalism makes life easy when correlations are relevant

32. 32 Tomorrow Upper Limits How Neural Networks work