THE MATHEMATICAL FOUNDATIONS OF RELIABILITY: WHY KR-20 AND CRONBACH’S ALPHA CAN BE USED INTERCHANGEABLY

Abstract

The reliability of research instruments represents a critical foundation in quantitative research, with internal consistency reliability being the most widely employed approach. Two statistical measures are commonly utilized: the Kuder-Richardson Formula 20 (KR20), designed specifically for dichotomously scored data (0 or 1), and Cronbach's Alpha Coefficient, applicable to both rating scales and dichotomous data. However, many researchers routinely apply Cronbach's Alpha to analyze all data types, including dichotomous responses, often remaining uncertain whether this practice aligns with established psychometric principles.

This article aims to explicate the mathematical relationship between KR-20 and Cronbach's Alpha, demonstrating that when instruments employ binary scoring (0 and 1), reliability coefficients calculated using both formulas yield identical values. This equivalence exists because both measures are grounded in Classical Test Theory and employ identical computational principles based on item variance and total test variance. Specifically, item variance in the KR-20 formula, derived from the Bernoulli distribution (pq), equals item variance in Cronbach's Alpha when applied to dichotomous data. The article presents mathematical proofs, empirical calculation examples, and analytical demonstrations using Jamovi statistical software to confirm that these formulas are interchangeable for dichotomously scored instruments. By providing theoretical justification and practical application, this work clarifies appropriate selection and implementation of these reliability statistics, thereby enhancing methodological rigor and strengthening the validity of educational research findings. Researchers can confidently employ either measure for dichotomous data, understanding that their mathematical equivalence validates interchangeable use in psychometric assessment.