Sampling Distribution Probability Calculator

What This Sampling Distribution Probability Calculator Solves

This tool finds the probability that a sample mean falls below a value, above a value, or between two values. It uses the normal sampling distribution of the mean, so you can work from population mean, population standard deviation, and sample size without doing z-table lookups by hand.

Sampling Distribution of the Mean

If the population is normal, or if the sample size is large enough for the central limit theorem to apply, the sample mean $\bar{X}$ follows an approximately normal distribution with:

$$\mu_{\bar{X}} = \mu \qquad \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$$

The quantity $ \sigma_{\bar{X}} $ is called the standard error. It becomes smaller as the sample size grows, which means sample means cluster more tightly around the population mean.

Z-Score Formula for Sample Means

After computing the standard error, convert your boundary to a z-score:

$$z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$$

Once you have $z$, the corresponding standard normal probability gives the area to the left of that value. For intervals, subtract the lower cumulative area from the upper cumulative area.

Typical Uses

Sampling distribution probabilities appear in quality control, polling, lab measurements, and average-score analysis. For related topics, you may also want the Central Limit Theorem Calculator, Standard Error Calculator, and Confidence Interval Calculator.

How to Use This Tool

1. Enter the population mean $ \mu $ and population standard deviation $ \sigma $.
2. Enter the sample size $n$.
3. Choose whether you want a left-tail, right-tail, or between-values probability.
4. Provide the target value or both bounds.
5. Read the standard error, z-score, and final probability.