Normal probability calculator for sampling distributions

How this calculator works

First calculate the sampling centre and standard deviation. Standardise the cutoff or bounds using that sampling SD, then evaluate a standard normal area. The result is exact for independent normal observations with known population parameters and otherwise is an approximation. Enter bounds in units of the selected statistic; a sum can be much larger than a mean.

Formula

For means: z = (x̄-μ)/(σ/√n). For sums: z = (S-nμ)/(σ√n). P(a<T<b) = Φ(z_b)-Φ(z_a).

Worked example

For μ = 100, σ = 15 and n = 25, the mean has SE 3. The chance that it lies between 97 and 103 is about 0.6826895.

Frequently asked questions

Should I enter the sample SD?

This model uses the population SD. Estimating it adds uncertainty that a confidence interval handles differently.

Does a larger sample change the mean?

The centre of sample means stays at μ, while their spread shrinks.

When can I trust the normal shape?

For normal independent observations it is exact. Otherwise assess sample size, skew, tails and dependence.

Sources

Written byAmira MansourCalculator writer. Builds and tests Mathulate calculators against published sources.

Reviewed byMarc LefebvreMathematics teacher. Checks each formula and worked example by hand.

Last updated November 1, 2026.