Adapted from Chapter 4, *Techniques for Nuclear and
Particle Physics Experiments*, by W. R. Leo, Springer-Verlag 1992
For a PDF version of the article, click
here.

# 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.

**Table of Contents**

- 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
REFERENCES