1 Practical Statistics for Physicists LBL January 2008 Louis Lyons Oxford [email protected].
Practical Statistics for Physicists
description
Transcript of Practical Statistics for Physicists
2
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
3
5
6
7
Comparing data with different hypotheses
8
Choosing between 2 hypotheses
Possible methods:
Δχ2
lnL–ratio
Bayesian evidence
Minimise “cost”
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
10
11
12
Element Eij - <(xi – xi) (xj – xj)>
Diagonal Eij = variances
Off-diagonal Eij = covariances
13
14
15
16
17
18
N.B. Small errors
19
20
Mnemonic: (2*2) = (2*4) (4*4) (4*2)
r c r c
2 = x_a, x_b
4 = p_i, p_j………
21
22
23
24
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”
25
26
Small error xbest outside x1 x2
ybest outside y1 y2
27
a
b
x
y
28
29
30
31
Conclusion
Error matrix formalism makes life easy when correlations are relevant
32
Tomorrow
• Upper Limits
• How Neural Networks work