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Practical Statistics for Physicists

This comprehensive guide explores practical statistics for physicists, focusing on error analysis, hypothesis testing, and the use of error matrices. It provides insight into comparing data with different hypotheses, fitting data to models, and estimating uncertainties. Key topics include 1-D and 2-D Gaussian distributions, correlation measurements, and the impact of theoretical inputs on precision. It also covers advanced methods like Δχ², logarithmic likelihood ratios, and Bayesian evidence. Learn to harness these statistical tools for effective interpretation and analysis in physics.

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Practical Statistics for Physicists

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

  2. PARADOX Histogram with 100 bins Fit 1 parameter Smin: χ2 with NDF = 99 (Expected χ2 = 99 ± 14) For our data, Smin(p0) = 90 Is p1 acceptable if S(p1) = 115? • YES. Very acceptable χ2 probability • NO. σp from S(p0 +σp) = Smin +1 = 91 But S(p1) – S(p0) = 25 So p1 is 5σ away from best value

  3. Comparing data with different hypotheses

  4. Choosing between 2 hypotheses Possible methods: Δχ2 lnL–ratio Bayesian evidence Minimise “cost”

  5. 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

  6. Element Eij - <(xi – xi) (xj – xj)> Diagonal Eij = variances Off-diagonal Eij = covariances

  7. N.B. Small errors

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

  9. 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”

  10. Small error xbest outside x1 x2 ybest outside y1  y2

  11. b y a x

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

  13. Tomorrow • Upper Limits • How Neural Networks work

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