Central Limit Theorem

The central limit theorem says that when you average many independent random samples, those averages follow a normal, bell-shaped distribution, even when the individual samples do not. The more samples you take, the tighter that distribution becomes around the true average.

Why it matters in simulation

A single simulation run is one random sample of how a process might unfold, and its results shift each time you run it. The central limit theorem is why running the model many times as replications and averaging the results gives a stable, trustworthy estimate. It is also what lets ProcessModel put a confidence interval around that average, so you know how precise it is.

LegacyHow this worked in the previous version

In probability theory, the central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (Rice 1995). The central limit theorem also requires the random variables to be identically distributed, unless certain conditions are met. Since real-world quantities are often the balanced sum of many unobserved random events, this theorem provides a partial explanation for the prevalence of the normal probability distribution. The CLT also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.