Normal Distribution Calculator

Every question about a normal distribution N(μ, σ²) reduces to one function: the standard-normal cumulative distribution Φ. Because the normal family is a location–scale family, a raw value x is first standardised into a z-score (NIST §1.3.6.6.1):

z = (x − μ) / σ

The z-score is how many standard deviations the value sits from the mean. Once you have z, the left-tail probability is P(X < x) = Φ(z), and everything else follows:

• Right tail: P(X > x) = 1 − Φ(z)
• Two-tailed p-value: 2 · [1 − Φ(|z|)]
• Percentile of x: 100 · Φ(z)
• Between a and b: Φ(z_b) − Φ(z_a)
• Value at percentile p: x = μ + σ · Φ⁻¹(p)

Φ(z) is evaluated from the error function using the Abramowitz & Stegun §7.1.26 rational approximation, where Φ(z) = ½ · [1 + erf(z / √2)]. Its maximum absolute error is 1.5 × 10⁻⁷, finer than any printed Z-table and accurate deep into the tails where tables stop. The inverse, Φ⁻¹(p), uses Peter Acklam's rational approximation refined by a single Halley step against that same Φ, so pushing a percentile out and back round-trips exactly. Values on this page were cross-checked against the NIST/SEMATECH standard-normal table (§1.3.6.7.1), which lists Φ(1.96) = 0.9750 and Φ(2.00) = 0.9772, both reproduced to the displayed digits.

As a built-in sanity anchor the tool also shows the empirical (68–95–99.7) rule: the exact central coverages Φ(1) − Φ(−1) = 0.6827, Φ(2) − Φ(−2) = 0.9545 and Φ(3) − Φ(−3) = 0.9973. The shaded bell curve is drawn deterministically from the density φ(z) = (1/√(2π)) e^(−z²/2), so the picture and the numbers always agree.

• NIST/SEMATECH e-Handbook §1.3.6.6.1: Normal Distribution (PDF/CDF, standardisation)
• NIST/SEMATECH e-Handbook §1.3.6.7.1: Cumulative Distribution Function of the Standard Normal
• Abramowitz & Stegun, Handbook of Mathematical Functions §7.1.26: erf rational approximation
• P. J. Acklam: Algorithm for the inverse normal CDF (Φ⁻¹)

The Φ(z) values produced here were last cross-checked against the NIST standard-normal table on 2026-06-12. The formulas are mathematical constants, not policy, so they do not change with the calendar.