The Beta Distribution: Applications and Insights for Social and Health Sciences

The objective of this methodological-academic chapter is to provide a detailed introduction to the Beta distribution, with examples applied to the social and health sciences using data randomly generated from Beta distributions. This continuous distribution maintains its status as a well-recognized statistical tool in mathematical statistics but practitioners in these fields rarely apply it despite its wide modeling capabilities for probabilities, proportions and bounded data.

The chapter begins with a historical overview to provide context, followed by an explanation of its support, parameters, and key functions, including density, cumulative distribution, quantile, moment-generating, and characteristic functions. Descriptive measures such as central tendency, variation, and shape are also discussed. The chapter then covers parameter estimation, focusing on the method of moments and maximum likelihood. The fit of a random sample to a Beta distribution with unknown parameters is tested using the G test and quantile-quantile plot. Additionally, the chapter presents the associated limiting distributions and generalizations. Four examples illustrate applications of the Beta distribution. The first example shows how to compute probabilities and descriptive measures when the two parameters of the distribution are known. The second example estimates the two shape parameters (alpha and beta) using the method of moments, calculates descriptive measures and performs probability calculations. The third example estimates alpha and beta, along with their standard errors and confidence intervals, using likelihood methods. The fourth example tests the goodness of fit of a sample to a Beta distribution using the G-test and the QQ plot. The work provides accessibility by including appendices containing R scripts for all calculations so researchers with different statistical abilities can benefit from this resource. The text closes with conclusions and suggestions for using the Beta distribution.

This chapter connects mathematical theory to practical examples along with computational tools to simplify the understanding of Beta distribution thereby advancing its adoption across social and health science data modeling and analysis. Researchers can now exploit this flexible distribution across policy evaluation, clinical trials and behavioral studies to enhance their data-driven decision-making capacity.