The Central Limit Theorem (CLT) does not have a single formula, but rather a set of conditions and consequences. However, one of the most common ways to express the CLT is in terms of the sample mean. If we take a random sample of size n from a population with mean μ and standard deviation σ, then the sample mean X̄ is approximately normally distributed with mean μ and standard deviation σ/√n, as n becomes large. Mathematically, we can express this as:
(X̄ – μ) / (σ/√n) ~ N(0,1)
where ~ denotes “is distributed as” and N(0,1) denotes the standard normal distribution with mean 0 and variance 1.
This formula tells us that as the sample size increases, the distribution of the sample mean becomes increasingly concentrated around the population mean, and the standard deviation of the sample mean decreases proportionally to the square root of the sample size. This means that larger samples are more likely to provide estimates that are close to the true population mean, and that the normal distribution provides a good approximation of the distribution of the sample mean, regardless of the underlying distribution of the population.