Sampling Distribution of the Mean Calculator
Calculate the properties of the sampling distribution of the sample mean. Given the population mean, population standard deviation, and sample size, this tool computes the standard error, a 95% confidence interval, and the probability of the sample mean falling within one standard error.
The sampling distribution of the sample mean describes how sample means vary from sample to sample. If you repeatedly draw random samples of size n from a population with mean mu and standard deviation sigma, the distribution of those sample means will have a mean equal to mu and a standard deviation (called the standard error) equal to sigma / sqrt(n).
The Central Limit Theorem (CLT) guarantees that for large enough sample sizes (typically n >= 30), the sampling distribution of the mean is approximately normal, regardless of the shape of the population distribution. This is one of the most important results in statistics because it justifies using normal-based confidence intervals and hypothesis tests even when the underlying data is not normal.
The standard error decreases as the sample size increases, which means larger samples produce more precise estimates of the population mean. Quadrupling the sample size halves the standard error. This relationship guides sample size planning in research studies.