Sampling Distribution Calculator

When you take many random samples of size n from a population, the sample means form their own distribution. This is the sampling distribution of the mean, and the Central Limit Theorem describes its shape.

The mean of the sampling distribution equals the population mean mu. No surprise there. The standard deviation of the sampling distribution, called the standard error, equals sigma divided by the square root of n. Larger samples produce smaller standard errors because with more data, the sample mean is less variable.

The Central Limit Theorem says that for large enough n, the sampling distribution of the mean is approximately normal, regardless of the shape of the original population. For normal populations, this holds exactly for any n. For moderately skewed populations, n = 30 is often enough. For heavily skewed or discrete distributions, you may need more.

This has a practical consequence: even if individual observations are not normally distributed, the average of many of them is. That is why the normal distribution appears so often in statistics, even when the underlying data is not normal.

Given a target x-bar, this calculator finds P(X-bar <= x-bar) using the z-score z = (x-bar - mu) / SE. A z-score of 2 means the target is 2 standard errors above the mean, which puts it at roughly the 97.7th percentile of the sampling distribution.

The chart plots the normal sampling distribution centered at mu with standard deviation SE. The shaded area under the curve up to x-bar shows the probability P(X-bar <= x-bar) visually.

A common use: you are running a study and want to know how likely you are to see a sample mean at least this extreme by chance. That is the foundation of hypothesis testing for means.