Central limit theorem: the cornerstone of modern statistics

Area of Science:

• Statistics
• Biostatistics
• Medical Research Methodology

Background:

• Parametric statistical tests rely on assumptions about population distributions, offering higher statistical power than non-parametric alternatives.
• The central limit theorem (CLT) underpins the validity of many parametric tests by describing the normal distribution of sample means.
• Many medical researchers apply parametric tests without a full understanding of the CLT's foundational role.

Purpose of the Study:

• To elucidate the fundamental concepts of the central limit theorem.
• To demonstrate the CLT's significance in the context of binomial distributions and the Student's t-test.
• To provide practical insights into sampling distributions, including those from small populations.

Main Methods:

• Review of statistical theory, focusing on the central limit theorem.
• Explanation of mathematical concepts underpinning the CLT.
• Illustrative examples of sampling distributions and their relation to population parameters.

Main Results:

• The central limit theorem posits that sample means converge to a normal distribution regardless of the original population distribution.
• This convergence justifies the use of parametric tests, such as the Student's t-test, under specific conditions.
• The review details the mathematical proof and practical implications of the CLT.

Conclusions:

• A thorough understanding of the central limit theorem is essential for the appropriate application of parametric statistical methods in medical research.
• This review aims to bridge the knowledge gap regarding the CLT's contribution to common statistical tests.
• Accurate application of statistical tests enhances the reliability and precision of research findings.