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Statistics in Particle Physics. 1. 20-29 November 2006 Tatsuo Kawamoto ICEPP, University of Tokyo. Outline. Introduction Probability Distributions Fitting and extracting parameters Combination of measurements Errors, limits and confidence intervals Likelihood, ANN, and sort of things.

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Statistics in Particle Physics

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Statistics in Particle Physics


20-29 November 2006

Tatsuo Kawamoto

ICEPP, University of Tokyo






Fitting and extracting parameters

Combination of measurements

Errors, limits and confidence intervals

Likelihood, ANN, and sort of things



  • Textbooks of statistics in HEP
  • PDG review (Probability, Statistics)
  • Relevant scientific papers

Why bother statistics ?

  • It’s not fundamental.
  • As soon as we come to the point to present results of
  • an experiment, we face to a few questions like:
  • What is the size of uncertainty?
  • How to combine results from different runs?
  • Discovered something new?
  • If not discovery, what we can say from the experiment?
  • Prescriptions to these problems often involve considerations
  • based on statistics.

Particle Physics

Study of elementary particles that have been discovered

- Quarks

- leptons

- Gauge bosons

- Hadrons

And anything that has not been discovered

- Higgs

- Supersymmetry

- Extradimensions


Goals of experiments

For each particle we want to know, eg.

  • What are its properties ?

- mass, lifetime, spin, ….

  • What are its decay modes ?
  • How it interacts with other particles ?
  • Does it exist at all ?

Observation is a result of fundamental rules of the nature

these are random, quantum mechanical, processes


Also, the detector effects (resolution, efficiency, …) are

often of random nature

Systematic uncertainty is a subtle subject, but we have to do

our best to say something about it, and treat it reasonably.


Template for an experiment

  • To study X
  • Arrange for X to occur
  • e.g colliding beams
  • Record events that might be X
  • trigger, data acquisition,
  • Reconstruct momentum, energy, … of visible particles
  • Select events that could be X by applying CUTS

Efficiency < 100%, Background > 0

  • Study distributions of interesting variables
  • Compare with/ fit to Theoretical distributions
  • Infer the value of parameter q and its uncertainty


  • Essentially counting numbers
  • Uncertainties of measurements are understood
  • Distributions are reproduced to reasonable accuracy

We don’t use:

  • Student’s t
  • F test
  • Markov chains


  • Monte Carlo simulation
  • Know in principle → Know in practice
  • Simple beautiful underlying physics
  • Unbeautiful effects (higher order, fragmentation,..)
  • Ugly detector imperfections (resolution, efficiency)
  • Likelihood
  • Fundamental tool to handle probability
  • Fitting
  • c2, Likelihood, Goodness of fit
  • Toy Monte Carlo
  • Handle complicated likelihood

Extracting parameters


mZ = 91.1853±0.0029 GeV

GZ = 2.4947 ±0.0041 GeV

shad= 41.82 ±0.044 nb


Likelihood, Artificial Neural Net

Use as much

Information as



W+W- → qqqq


There are other important things

which we don’t cover

  • Blind analysis
  • Unfolding
  • ….

2. Probability

What is it?



P(A) is a number obeying the rules:

Kolmogorov axioms

Ai are disjoint events




And, that’s almost it.



Laplace, …

From considerations of games of chances

Given by symmetry for equally-likely outcomes, for which

we are equally undecided.

Classify things into certain number of equally-likely cases,

And count the number of such favorable cases.

P(A) = number of equally-likely favorable cases / total number

Tossing a coin P(H)=1/2, Throwing a dice P(1)=1/6

How to handle continuous variables ?



Probability is the limit of frequency (taken over some ensemble)

The event A either occur or not. Relative frequency of occurence

Law of large numbers


Frequency definition is associated to some ensemble of ‘events’

Can’t say things like:

  • It will probably rain tomorrow
  • Probability of LHC collision in November 2007
  • Probability of existence of SUSY

But one can say:

  • The statement ‘It will rain tomorrow’ is probably true

Comeback later in the discussion of confidence level


Bayesian or Subjective probability

P(A) is the degree of belief in A

A can be anything:

Rain, LHC completion, SUSY, ….

You bet depending on odds P vs 1-P


Bayes theorem

Often used in subjective probability discussions

Conditional probability P(A|B)

Thomas Bayes 1702-1761


Bayes theorem

How it works?

Initial belief P(Theory) is modified by experimental results

If Result is negative, P(Result|Theory)=0, the Theory is killed


It’s an extreme case. Will comeback later in the discussion of

confidence level


Fun with Bayes theorem - 1

Monty Hall problem

  • There are 3 doors
  • Behind one of these, there is a prize (a car, etc)
  • Behind each of the other two, there is a goat (you lost)
  • you choose 1 door whatever you like (you bet), say, Nr 1.
  • A door will be opened to reveal a goat, either of Nr 2 or Nr 3,
  • chosen randomly if goat is behind the both.
  • Then you are asked if you stay Nr 1, or, switch to Nr 2.

You should stay or switch?


One would say:

you don’t know anyway if there is the prize

behind Nr 1 or Nr 2. They are equally probable.

To stay or to switch give equal chance.


But the correct strategy is to switch

A ‘classical’ reasoning (count the number of cases)

Before the door is opened

After the door is opened

Odds to win : stay 1/3

switch 2/3


Using Bayes theorem

P(Ci) : Prize is behind door i = 1/3

P(Ok) : Door k is opened

We want to know P(C1| O3) vs P(C2| O3)



A disease X (maybe AIDS, SARS, ….)

P(X) = 0.001 Prior probability

P(no X) = 0.999

Consider a test of X

P(+ | X) = 0.998

P(+ | no X) = 0.03

If the test result were +, how worried you should be ?

ie. What is P(X | +) ?