Published in Q. Jl R. astr. Soc.
(1996), **37**, 519-563

# PRACTICAL STATISTICS FOR ASTRONOMERS:
II. CORRELATION, DATA-MODELLING AND SAMPLE COMPARISON

*J. V. Wall*

Royal Greenwich
Observatory, Madingley Road, Cambridge CB3 OEZ, UK

**SUMMARY.** Correlation is discussed in various
guises in which astronomers may
stumble across it, beginning with the pitfalls of searching for
correlations between two variables and the tests, both parametric and
non-parametric, for such correlations. This leads to the subject of
regression analysis, a particular form of data modelling. Some general
aspects and procedures in data modelling and parameter estimation are
then described, including least squares, maximum-likelihood, Bayesian
techniques and minimum chi-square. The final topic is sample
comparison, an area of hypothesis testing: here some seven tests in
all are described, three for the comparison of single samples with
prediction and four for inter-sample comparison.

Non-parametric methods are emphasized throughout as being of most
use to the astronomer, who frequently faces (i) very small samples and
(ii) lack of control over the Universe needed either to rerun the
experiments or to understand frequency distributions from which the
small samples were drawn.

**Table of Contents**

- INTRODUCTION

- CORRELATION BETWEEN TWO VARIABLES
- The Fishing Trip
- Correlation Testing

- DATA MODELLING: PARAMETER ESTIMATION
- The Method of Least Squares: Regression Analysis
- The Maximum-Likelihood (ML) Method
- Bayes Estimation
- The Minimum Chi-Square Method
- The Bootstrap

- HYPOTHESIS TESTING: COMPARISON OF SAMPLES
- Methodology
- Single-Sample Goodness-of-Fit Tests
- Tests for Comparison of Two Independent Samples
- Summary, One- and Two-Sample Tests

- CONCLUSION

- REFERENCES

- Table A I

- Table A II

- Table A III

- Table A IV

- Table A V

- Table A VI

- Table A VII

- Table A VIII

- Table A IX