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Practical Statistics for Physicists

LBL

January 2008

Louis Lyons

Oxford

l.lyons@physics.ox.ac.uk

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PARADOXHistogram with 100 binsFit 1 parameter

Smin: χ2 with NDF = 99 (Expected χ2 = 99 ± 14)

For our data, Smin(p0) = 90

Is p1 acceptable if S(p1) = 115?

1) YES. Very acceptable χ2 probability

2) NO. σp from S(p0 +σp) = Smin +1 = 91

But S(p1) – S(p0) = 25

So p1 is 5σ away from best value

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Comparing data with different hypotheses

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Choosing between 2 hypotheses

Possible methods:

Δχ2

lnL–ratio

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 - <(xi – xi) (xj – xj)>

Diagonal Eij = variances

Off-diagonal Eij = covariances

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N.B. Small errors

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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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Difference between averaging and adding

Isolated island with conservative inhabitantsHow many married people ?

Number of married men = 100 ± 5 KNumber of married women = 80 ± 30 K

Total = 180 ± 30 KWeighted 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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a

b

x

y

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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