# Statistical methods and data analysis

## The purpose

Probability theory and mathematical statistics are an integral part of modern research. Calculation of errors, correct presentation of the result, risk assessment - all these are important components of the work of a scientist. At the same time, as practice shows, many scientists (and not just students) complain about the lack of practical skills in this area. This is due to the fact that the teaching of the theoretical aspect of probability theory is often well-placed in technical universities, but the purely practical aspect is completely overlooked.

In our course, we will try to analyze in detail the issues of the practical application of the statistical methods of in planning and processing the results of a physical experiment (using specific examples). Theoretical calculations will be mainly excluded from lectures and left for independent study.

## Course format

The course is planned in the optional format once a week, while lectures will be held every second week, and practical classes (seminars) will be held between the lectures, discussing examples and solving problems from modern experimental physics and everyday life (including laboratory work) .

Announcements of important events, as well as discussion of any issues related to the course, are available in the Telegram group (https://t.me/mipt_statmethods).

Materials

## Course structure (preliminary program)

1. Statistical decision-making theory.
2. Decisions in non-deterministic tasks, risk function.
3. Conditional probability, decision-making strategies.
2. Basic concepts of probability theory.
1. Definitions of probability.
2. Function of plausibility.
3. Point and interval estimates of distribution parameters.
4. Confidence intervals.
3. Errors in physical experiment.
1. Statistical and systematic errors.
2. Properties of distributions at replacement of variables.
3. Uncorrector stacking.
4. Adding results of various experiments.
4. Properties of distributions.
1. Poisson's binomial distribution and distribution.
2. Normal distribution and its properties.
3. Average values, moments of distributions.
5. Checking statistical hypotheses.
1. Functions of random variables.
2. Statistical criteria and their properties.
3. Methods of criteria construction.
4. Criteria of data agreement with the theory.
6. Evaluation of parameters.
1. Parameter criteria.
2. Maximum probability and chi-square method.
3. Using the probability function to construct the Chi-square maximum and Chi-square maximum. Interval estimates.
4. Interval estimates in the case of normal distribution.
7. Modern data analysis methods (optional).
1. Fitting of experimental curves. Criteria of phytate quality. Computer methods for solving optimization problems.
2. Multi-parameter analysis. Analysis of correlations.
3. Fisher Information and its Application. Maximum information and its application. the border between Rao and Kramer.
4. Two approaches to probability: frequency approach and subjective probability. The problem of unique events.
5. Using a computer to analyze experimental data.

## Reporting

The test takes place in the form of a presentation based on the materials of an individual project. Each student has the opportunity to prepare a report analyzing the results of a particular real or thought experiment (you can take laboratory work).

## Recommended literature

• The main textbook for the course - W. Idieu, D. Dryard, F. James, M. Ruth, B. Sadule. Statistical methods in experimental physics M.: Atomizdat, 1976. The Russian-language edition of the book is a bibliographical rarity, but the English version is republished every few years. In addition, an electronic version of the Russian-language edition is available (including the course materials on Google-drive).
• A lot of useful information is contained in the introductory chapters to the MIPT laboratory workshop for the 1st and 3rd courses.
• In concentrated form, information on probability theory and mathematical statistics can be found in the online version of the Particle Data Group (PDG) handbook of particle physics: http://pdg.lbl.gov/2014/reviews/rpp2014-rev-probability.pdf; http://pdg.lbl.gov/2014/reviews/rpp2014-rev-statistics.pdf.