Normal Distribution Calculator

About this calculator

This calculator finds the z-score, z = (x − μ) / σ, which measures how many standard deviations a value x sits from the mean μ of a normal distribution with standard deviation σ, then converts that z-score into a percentile using the cumulative distribution function (CDF) of the standard normal distribution, Φ(z) — the area under the bell curve to the left of z. A z-score of 0 sits exactly at the mean (the 50th percentile), while z-scores of ±1, ±2, and ±3 correspond to the well-known 68-95-99.7 rule.

Statisticians and researchers use z-scores to standardize measurements from different scales for comparison, standardized-test companies use them to convert raw scores into percentile rankings, quality-control engineers use them in six-sigma process control to flag outliers, and students learning inferential statistics use them as the entry point into hypothesis testing and confidence intervals.