ASSESSING THE ACCURACY OF THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION IN LARGE SAMPLES: AN ANALYTICAL STUDY AND MONTE CARLO SIMULATION

Objective: This study aims to evaluate the accuracy of the natural approximation of the binomial distribution in large samples through an analytical study and Monte Carlo simulations, focusing on the effect of sample size n and probability parameter P, as well as the continuity correction, and certain properties related to the shape of the distribution. Method: The study included 143 scenarios resulting from the use of sample sizes ranging from 10 to 10,000 and different values of P ranging from 0.01 to 0.99, with the calculation of a set of accuracy metrics, including absolute error, relative error, mean absolute error (MAE), root mean square error (RMSE), and maximum cumulative error, along with an examination of the relationship between the error and each of (nP) and n(1−P), as well as the absolute skewness of the binomial distribution. Monte Carlo simulations with 10,000 iterations were also used to empirically verify the computational results. Results: It was also observed that there is a significant decrease in errors through continuity correction, where the mean absolute error reduced from (0.0362579686) without correction to (0.007330879) with the correction; the root mean square error also reduced from (0.0690117985) to (0.0162159958), which is an improvement in the maximum cumulative error of (79.539328%) on the set of correct values. Novelty: The study concludes that the assessment of the accuracy of the natural approximation should not depend solely on a large n, but should take into account P, (n P), n(1-P), and the shape parameters of the distribution, with the importance of using a continuity correction when applying the natural approximation to the binomial distribution.