Central Limit Theorem Calculator

The central limit theorem explains why normal distributions appear throughout statistical inference. When independent observations are drawn from a population with a finite mean and variance, the distribution of their sample mean becomes approximately normal as the sample size grows. This can happen even when the underlying population is skewed or otherwise non-normal. The sampling distribution is centered at the population mean and has a standard deviation called the standard error. For a population mean μ, population standard deviation σ, and sample size n, the standard error of the mean is σ/√n. Increasing the sample size therefore makes sample means cluster more tightly around μ. To find the probability that a sample mean lies between lower and upper bounds, convert each bound to a z-score using z = (x̄ - μ)/(σ/√n). The desired probability is the area under the standard normal curve between those two z-scores. Suppose a population has mean 100 and standard deviation 15. For samples of 36 observations, the standard error is 15/6 = 2.5. Bounds of 95 and 105 correspond to z-scores of -2 and 2. The standard normal distribution places about 95.45 percent of its area between those points, so approximately 95.45 percent of repeated sample means would fall in that interval under the model. The theorem is asymptotic, not a universal rule that every sample of 30 is automatically normal. How quickly the approximation improves depends on the population's shape. A nearly normal population may need only a small sample, while a severely skewed or heavy-tailed population may require much more data. Dependence between observations, infinite variance, strong selection bias, or an unrepresentative sampling process can invalidate the simple calculation. If the source population itself is normal, however, the sample mean is exactly normal for every positive sample size. This calculator assumes the population standard deviation is known. In practical inference it is often estimated from sample data; confidence intervals and tests may then use a t distribution, especially for small samples. The displayed probability describes repeated random samples under the supplied population model, not the probability that a fixed unknown parameter lies in the entered range. Use the tool for sampling-distribution exercises, quality-control planning, survey estimates, and probability checks, while matching its assumptions to the design that produced the observations.