Sample Size Determination in Applied Research: A Critical Review of Cochran’s, Yamane’s and Power-Analysis Approaches

Abstract

Few questions are asked more often by research students than how large a sample should be, and few are answered more mechanically. Many theses and articles in India justify their sample size with a single formula, most often Yamane’s or the Krejcie and Morgan table, without explaining whether the formula suits the study’s purpose. This review examines the three approaches most commonly used in applied research: Cochran’s formula for estimating a proportion, Yamane’s simplified formula, and statistical power analysis for testing hypotheses. It sets out the assumptions behind each, shows with worked calculations how they compare across population sizes and effect sizes, and discusses common errors. The comparison shows that Cochran’s formula and the Krejcie and Morgan table give practically identical results, that Yamane’s formula is equivalent to Cochran’s at a confidence level of about 95.5 per cent and therefore gives slightly larger samples, and that none of these formulas is appropriate when the aim is to compare groups or test a relationship, where power analysis is needed. A study comparing two groups for a small effect needs nearly 400 participants per group at 80 per cent power, far more than survey formulas suggest. The review recommends that researchers state the purpose of the sample, the assumptions used and adjustments for non-response and design effects, and that they justify sample size in the way the analysis will actually be carried out.