🔔 Normal Probability Sampling Distribution Calculator
Enter a population mean, population standard deviation, and sample size, plus a sample mean value, to find the standard error, z-score, and probability under the normal sampling distribution.
What this calculator does
The Central Limit Theorem tells us that the sampling distribution of a sample mean is approximately normal, centered at the population mean, with a spread that shrinks as sample size grows. This calculator finds the probability that a sample mean falls above or below a chosen value, using that normal sampling distribution.
How the calculation works
The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size (SE = sigma / sqrt(n)). The sample mean value is converted into a z-score using z = (x – mu) / SE, and the probability is found from the cumulative standard normal distribution, either as P(mean < value) or P(mean > value) depending on your selection.
This approach is most accurate when the sample size is reasonably large (commonly n of 30 or more) or the underlying population is itself approximately normal. This tool is for general statistics education and planning purposes.