Normal Distribution Calculator

How to use

1. 1Pick a mode. Probability returns the area under the curve for a region; Inverse returns the X value(s) that enclose a target probability.
2. 2Enter the mean (μ) and the standard deviation (σ). Use μ = 0 and σ = 1 to use the standard normal Z-table; use any other values to work in the units of your problem (IQ, SAT, dollars, millimeters, anything bell-shaped).
3. 3Probability mode: pick a region (X < a, X > a, a < X < b, or outside (a, b)) and fill in the bound(s). The result and the shaded bell curve update instantly.
4. 4Inverse mode: pick a tail (Left, Right, or Central) and enter a probability between 0 and 1. The calculator returns the X bound(s) that enclose that area, plus the matching z-score(s).
5. 5Click Copy report to copy the inputs, z-scores, and result to your clipboard, or load one of the quick examples (IQ, SAT, critical z, 90th percentile) to see a worked problem.

About this tool

Normal Distribution Calculator computes probabilities and inverse values for any normal (Gaussian) distribution defined by a mean and a standard deviation. Switch between two modes. Probability mode answers the classic textbook questions: P(X < a), P(X > a), P(a < X < b), and P(X outside (a, b)). Inverse mode (the textbook invNorm) returns the X value that has a given area to its left, a given area to its right, or the two X values that enclose a given central area symmetric around the mean. Set the mean to 0 and the standard deviation to 1 to work directly with the standard normal Z-table; pass any other mean and standard deviation to work in the units of your problem (IQ, SAT scores, blood pressure, manufacturing tolerances, finance returns, anything modeled by a bell curve). Every result is rendered live alongside a clean SVG bell curve with the relevant region shaded and tick marks at integer standard deviations, so you can see exactly which part of the distribution you are asking about. The step-by-step explanation panel shows the z-score conversion (z = (a − μ) / σ), the standard normal CDF Φ(z) value used at each bound, and the formula that combined them into the final probability, so students can match the working to their textbook layout. Internally the calculator uses the Abramowitz and Stegun 7.1.26 rational approximation for the error function (max absolute error ~1.5 × 10⁻⁷, which exceeds the precision of every printed Z-table) and Peter Acklam's rational approximation for the inverse standard normal (max relative error ~5 × 10⁻⁹ over the full unit interval), so probabilities match published tables and TI calculator output to at least four decimal places. Useful for AP Statistics and college statistics homework, six-sigma quality control limits, psychometric percentile scoring, finance value-at-risk style questions, A/B test power calculations, manufacturing tolerance windows, sports analytics, and any time you need a normalcdf or invNorm result without firing up a graphing calculator. All math runs in your browser; nothing you type is uploaded.

Free to use. Works in your browser. No signup, no login.