Probability for Physicists pp 177-201 | Cite as

# Statistical Inference from Samples

## Abstract

Any kind of empirical determination of probability distributions and their parameters amounts to statistical inference procedures based on samples randomly drawn from a population. The concepts of the statistic and the estimator are introduced, paying attention to their consistency and bias. Sample mean and sample variance are defined, and three most relevant sample distributions are investigated: distribution of sums and differences, distribution of variances, and distribution of variance ratios. Confidence intervals for the sample mean and sample variance are discussed. The problem of outliers is elucidated in the context of robust measures, and linear (Pearson) and non-parametric (Spearman) correlations are presented.

## Keywords

Sample Variance Population Variance Sample Distribution Unbiased Estimator Confidence Region## References

- 1.M.R. Spiegel, J. Schiller, R.A. Srinivasan,
*Theory and Problems of Probability and Statistics*, 4th edn. (McGraw-Hill, New York, 2012)Google Scholar - 2.R. Kandel,
*Our Changing Climate*(McGraw-Hill, New York, 1991)Google Scholar - 3.L. Davies, U. Gather, Robust statistics, Chap. III.9, in
*Handbook of computational statistics. Concepts and methods*, ed. by J.E. Gentle, W. Härdle, Y. Mori (Springer, Berlin, 2004), pp. 655–695Google Scholar - 4.Analytical Methods Committee. Robust statistics – how not to reject outliers, Part 1: basic concepts. Analyst
**114**, 1693 (1989), Part 2: Inter-laboratory trials. Analyst**114**, 1699 (1989)Google Scholar - 5.A.M. Walker, A note on the asymptotic distribution of sample quantiles. J. R. Stat. Soc. B
**30**, 570 (1968)MathSciNetzbMATHGoogle Scholar - 6.J.P. Romano, A.F. Siegel,
*Counterexamples in Probability and Statistics*(Wadsworth & Brooks/Cole, Monterey, 1986)zbMATHGoogle Scholar - 7.M. Hardy, An illuminating counterexample. Am. Math. Mon.
**110**, 234 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 8.T.R. Lange, H.E. Royals, L.L. Connor, Influence of water chemistry on mercury concentration in largemouth bass from Florida lakes. Trans. Am. Fish. Soc.
**122**, 74 (1993)CrossRefGoogle Scholar