Sampling Distribution Calculator

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Sampling Distribution Calculator

This sampling distribution calculator answers questions like “what is the probability that the mean of 36 randomly chosen adults is above 173 cm?” It finds the mean and standard error of the sampling distribution, converts your value to a z-score and returns the probability, for a sample mean x̄ or a sample proportion p̂.

For a sample mean the sampling distribution is centred on the population mean μ with standard error σ ÷ √n. By the central limit theorem it is approximately normal once n is about 30 or more, whatever the shape of the population, and exactly normal at any n if the population is normal. For a proportion it is centred on p with standard error √[p(1 − p) ÷ n], and the normal approximation is reasonable when np and n(1 − p) are both at least 10; the calculator warns you when they are not.

The example: μ = 170, σ = 12, n = 36 gives a standard error of 2, so 173 is 1.5 standard errors above the mean and P(x̄ > 173) = 0.0668. To find just the standard error, see the standard error calculator. For a probability about a single observation rather than a mean, use the normal distribution calculator.

Frequently asked questions

What is a sampling distribution?

The distribution of a statistic, such as the sample mean, across all possible samples of the same size from a population.

What is the standard deviation of the sampling distribution of the mean?

σ ÷ √n, called the standard error. With σ = 12 and n = 36 it is 2.

When can I use the normal approximation for p̂?

When np ≥ 10 and n(1 − p) ≥ 10. Some textbooks use 5 instead of 10.

Does the central limit theorem need a normal population?

No. For n of about 30 or more the sample mean is approximately normal for most populations. Strongly skewed populations may need larger samples.

Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.