Normal Distribution Calculator

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Normal Distribution Calculator

This normal distribution calculator finds the probability that a normally distributed variable falls below, above or between chosen values. Enter the mean and standard deviation, pick the type of probability and the value(s), and the calculator standardises to z-scores, looks up the cumulative areas and applies the right subtraction, with a shaded curve showing the region.

Typical uses include finding the proportion of students scoring above 115 on a test with mean 100 and SD 15 (about 15.9%), the share of parts within a tolerance band, or the chance a sample mean falls in a given range (use σ/√n as the standard deviation). The density at x is shown too, for questions about the height of the curve.

Because the distribution is continuous, P(X = x) is zero and P(X < x) equals P(X ≤ x). For the reverse problem, finding the value that cuts off a given area, use the inverse normal calculator. The z-table lets you check the look-up by hand.

Frequently asked questions

How do I find the probability between two values?

Convert both to z-scores, find the cumulative area for each, and subtract: P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a).

What is the 68-95-99.7 rule?

About 68% of a normal distribution lies within one SD of the mean, 95% within two and 99.7% within three. The calculator reproduces these exactly (68.27%, 95.45%, 99.73%).

Can I use this for a sampling distribution of the mean?

Yes. Enter the population mean as μ and σ/√n as the standard deviation.

Why is P(X = x) zero?

For a continuous variable the probability of any single exact value is zero; only intervals have positive probability. Use the density if you need the curve height.

Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.