Finite Population Sampling as n to N: Empirical Evidence for the Transition from Inference to Accuracy

Abstract: The Central Limit Theorem provides a foundation for inferential statistics and hypothesis testing. It describes how standardized statistics behave under repeated sampling from large populations. However, if the size of the sample (n) becomes so large that it approaches the size of the population (N), sampling variability becomes very small, and standard errors and margins of error both approach zero. The purpose of this project was to investigate the behavior of estimators as the sampling fraction (f = n/N) approaches 1, motivated by modern data streams from administrative records, transaction logs, sensor systems, and institutional databases that capture large portions of finite populations. We constructed two finite populations with known parameters and drew repeated samples across a range of sampling fractions. We then examined the resulting randomization distributions of the sample mean to understand how sampling variability collapses. Additional experiments were conducted using various CPU- and GPU-based methods to evaluate the deviation of the sample mean from the defined population mean under different computational conditions. The results confirm that sampling variability diminishes as expected under finite population theory and becomes negligible well before full enumeration is reached. Once sampling variability is minimized, remaining deviations in estimators are primarily related to numerical precision and computational structure rather than random sampling. These findings support a reassessment of inferential assumptions in high-coverage, large-scale data settings.