Standard Normal Distribution Calculator

The Bell Curve, Reduced to Its Essential Form

Nearly every normal distribution you'll encounter — test scores, measurement errors, heights, manufacturing tolerances — can be converted to a single reference distribution with a mean of 0 and a standard deviation of 1. Once converted, a single set of formulas answers every question about it: the probability below a value, above it, between two values, its density at a point, or the reverse problem of finding the z-score for a target probability.

The Formulas

Φ(z), the cumulative distribution function, gives the probability that a standard normal random variable falls below z. The inverse mode runs this in reverse, numerically searching for the z that produces a target cumulative probability.

Reference Table

The 1.645, 1.96, and 2.576 rows are the critical values behind 90%, 95%, and 99% two-tailed confidence intervals, respectively — they show up constantly throughout inferential statistics.

Where This Matters

• Hypothesis testing — converting a test statistic to a p-value relies directly on the standard normal CDF.
• Confidence intervals — the critical z-values used to build intervals come from the inverse CDF of this distribution.
• Process capability — six-sigma quality frameworks are built entirely around standard normal probabilities.
• Risk modeling — value-at-risk and other financial risk metrics often assume normally distributed returns.

How to Use This Calculator

1. Select a mode from the dropdown: P(Z < z), P(Z > z), P(z1 < Z < z2), the density φ(z), or Inverse.
2. Enter the z-score(s) required for your chosen mode, or a target probability for Inverse mode.
3. Select Calculate to get the probability or z-score.

Related Calculations

Convert your own raw data point into a z-score first with the Z-Score Calculator, then bring it here. To build a full confidence interval, use the Confidence Interval Calculator.