Mathematics and Statistics Vol. 12(1), pp. 80 - 98
DOI: 10.13189/ms.2024.120111
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Moments of Gaussian Distributions for Small and Large Sample Sizes Revisited

Florian Heiser ^1,2, E W Knapp ^1,*
1 Institute of Chemistry and Biochemistry, Free University of Berlin, D-14195 Berlin, Germany
² Leibniz-Institute for Molecular Pharmacology (FMP), D-13125 Berlin, Germany

ABSTRACT

Central moments of statistical samples provide coarse-grained information on width, symmetry and shape of the underlying probability distribution. They need appropriate corrections for fulfilling two conditions: (1) yielding correct limiting values for large samples; (2) yielding these values also, if averaged over many samples of the same size. We provide correct expressions of unbiased central moments up to the fourth and provide an unbiased expression for the kurtosis, which generally is available in a biased form only. We have verified the derived general expressions by applying them to the Gaussian probability distribution (GPD) and we show how unbiased central moments and kurtosis behave for finite samples. For this purpose, we evaluated precise distributions of all four moments for finite samples of the GPD. These distributions are based on up to 3.2*10⁸ randomly generated samples of specific sizes. For large samples, these moment distributions become Gaussians whose second moments decay with the inverse sample size. We parameterized the corresponding decay laws. Based on these moment distributions, we demonstrate how p-values can be computed to compare the values of mean and variance evaluated from a sample with the corresponding expected values. We also show how one can use p-values for the third moment to investigate the symmetry and for the fourth moment to investigate the shape of the underlying probability distribution, certifying or ruling out a Gaussian distribution. This all provides new power for the usage of statistical moments. Finally, we apply the evaluation of p-values for a dataset of the percent of people of age 65 and above in the 50 different states of the USA.

KEYWORDS
Statistical Moments, Finite Samples, Raw Moments, Central Moments, Correct Unbiased Moments, Kurtosis, Inferential Statistics, Distribution of Moments, Sample Size Dependence, p-values, Compare Mean and Variance of a Sample with Expected Values

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Florian Heiser , E W Knapp , "Moments of Gaussian Distributions for Small and Large Sample Sizes Revisited," Mathematics and Statistics, Vol. 12, No. 1, pp. 80 - 98, 2024. DOI: 10.13189/ms.2024.120111.

(b). APA Format:
Florian Heiser , E W Knapp (2024). Moments of Gaussian Distributions for Small and Large Sample Sizes Revisited. Mathematics and Statistics, 12(1), 80 - 98. DOI: 10.13189/ms.2024.120111.