Practical%20Statistics%20for%20Physicists

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Practical Statistics for Physicists
Louis Lyons Oxford
l.lyons_at_physics.ox.ac.uk

• LBL
• January 2008

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PARADOX

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

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Comparing data with different hypotheses
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Choosing between 2 hypotheses

• Possible methods
• ??2
• lnLratio
• Bayesian evidence
• Minimise cost

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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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Element Eij - lt(xi xi) (xj xj)gt Diagonal Eij
variances Off-diagonal Eij covariances
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N.B. Small errors
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Mnemonic (22) (24) (44) (42)
r c
r c 2 x_a, x_b 4 p_i, p_j

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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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Small error
xbest outside x1 ? x2 ybest outside y1 ? y2
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b
y
a
x
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Conclusion

• Error matrix formalism makes life easy when
  correlations are relevant

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Tomorrow

• Upper Limits
• How Neural Networks work