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

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practical statistics for physicists

Practical Statistics for Physicists

Louis Lyons

Oxford

l.lyons@physics.ox.ac.uk

LBL

January 2008

paradox
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

choosing between 2 hypotheses
Choosing between 2 hypotheses

Possible methods:

Δχ2

lnL–ratio

Bayesian evidence

Minimise “cost”

learning to love the error matrix
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
slide11

Element Eij - <(xi – xi) (xj – xj)>

Diagonal Eij = variances

Off-diagonal Eij = covariances

slide19

Mnemonic: (2*2) = (2*4) (4*4) (4*2)

r c r c

2 = x_a, x_b

4 = p_i, p_j………

difference between averaging and adding
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”

slide25

Small error

xbest outside x1 x2

ybest outside y1  y2

slide26

b

y

a

x

conclusion
Conclusion

Error matrix formalism makes life easy when correlations are relevant

tomorrow
Tomorrow
  • Upper Limits
  • How Neural Networks work