The historical development of sample size estimation: from Huygens and Bernoulli to the present.

Abstract

This paper traces the historical development of frequentist sample size estimation from its philosophical origins to its present-day complexity. Preliminary concepts were identified by Christiaan Huygens' work on expected value and Jacob Bernoulli's law of large numbers, which first linked sample size and estimation accuracy. The 18th and 19th centuries brought major advances in probability theory through the work of Pierre-Simon de Laplace, Carl Friedrich Gauss and Siméon-Denis Poisson, yet explicit sample size planning remained uncommon. The early 20th century saw the emergence of methods for sample size calculations based on Jerzy Neyman and Egon Pearson's hypothesis testing framework and Sir Ronald Aylmer Fisher's experimental design principles. While Donald Mainland and Austin Bradford Hill referred indirectly to these as early as the 1930s, it took many decades before their explicit use became common. After the Second World War, contributions from figures such as Abraham Wald further embedded sample size planning with sequential methodologies. From the 1970s onward, standardised formulas, regulatory requirements, reporting standards such as CONSORT and statistical software consolidated frequentist sample size estimation as a routine component in applied research. In the 21st century, simulation-based, adaptive and Bayesian approaches, together with open-source computational ecosystems, have expanded the scope and accessibility of sample size methods. In contemporary research, sample size estimation has evolved into a multifaceted discipline; its methodological sophistication is contingent upon the underlying objective, illustrating the persistent divergence between explanatory inference and decision-oriented design.