Adapted from Chapter 4, Techniques for Nuclear and Particle Physics Experiments, by W. R. Leo, Springer-Verlag 1992

# STATISTICS AND THE TREATMENT OF EXPERIMENTAL DATA

### W. R. Leo

Statistics plays an essential part in all the sciences as it is the tool which allows the scientist to treat the uncertainties inherent in all measured data and to eventually draw conclusions from the results. For the experimentalist, it is also a design and planning tool. Indeed, before performing any measurement, one must consider the tolerances required of the apparatus, the measuring times involved, etc., as a function of the desired precision on the result. Such an analysis is essential in order to determine its feasibility in material, time and cost.

Statistics, of course, is a subject unto itself and it is neither fitting nor possible to cover all the principles and techniques in a book of this type. We have therefore limited ourselves to those topics most relevant for experimental nuclear and particle physics. Nevertheless, given the (often underestimated) importance of statistics we shall try to give some view of the general underlying principles along with examples, rather than simple "recipes" or "rules of thumb". This hopefully will be more useful to the physicist in the long run, if only because it stimulates him to look further. We assume here an elementary knowledge of probability and combinatorial theory.

CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS
Cumulative Distributions
Expectation Values
Distribution Moments. The Mean and Variance
The Covariance

SOME COMMON PROBABILITY DISTRIBUTIONS
The Binomial Distribution
The Poisson Distribution
The Gaussian or Normal Distribution
The Chi-Square Distribution

MEASUREMENT ERRORS AND THE MEASUREMENT PROCESS
Systematic Errors
Random Errors

SAMPLING AND PARAMETER ESTIMATION. THE MAXIMUM LIKELIHOOD METHOD
Sample Moments
The Maximum Likelihood Method
Estimator for the Poisson Distribution
Estimators for the Gaussian Distribution
The Weighted Mean

EXAMPLES OF APPLICATIONS
Mean and Error from a Series of Measurements
Combining Data with Different Errors
Determination of Count Rates and Their Errors
Null Experiments. Setting Confidence Limits When No Counts Are Observed
Distribution of Time Intervals Between Counts

PROPAGATION OF ERRORS
Examples

CURVE FITTING
The Least Squares Method
Linear Fits. The Straight Line
Linear Fits When Both Variables Have Errors
Nonlinear Fits

SOME GENERAL RULES FOR ROUNDING-OFF NUMBERS FOR FINAL PRESENTATION