Galton Board Experiment: Proof of Central Limit Theorem

Abstract

This paper investigates the application of the Central Limit Theorem (CLT) using a Galton board experiment. The Galton board, which produces a binomial distribution of ball positions, serves as a practical model to test the CLT’s assertion that the distribution of sample means approaches a normal distribution as the sample size increases. In this experiment, 30 independent trials were conducted, with 100 balls passing through 10 rows of pegs in each trial. The sample means for each trial were computed, and the resulting sampling distribution was analyzed. A normal distribution curve was fitted to the data, visually demonstrating alignment with CLT predictions. Additionally, statistical tests, including the Shapiro-Wilk test, were applied to rigorously evaluate the normality of the sample means, providing empirical validation for the theoretical model. The findings confirm the applicability of the CLT to the Galton board, as the sampling distribution closely follows a normal pattern, highlighting the theorem’s generalizability even when the original data follows a binomial distribution.