Z-score and normal distribution calculator

How to use it

Enter the mean and standard deviation of the normal distribution (0 and 1 for the standard normal), then a value. Switch the mode to find the probability between two values, or the value that has a given cumulative probability (a percentile).

Formulas

• z = (x − μ) / σ.
• P(X < x) = Φ(z), the standard normal cumulative distribution; P(X > x) = 1 − Φ(z) = Φ(−z); two tails P(|Z| > |z|) = 2Φ(−|z|).
• Between a and b: Φ(z_b) − Φ(z_a), computed from the smaller tails to avoid rounding.
• Percentile: x = μ + σ Φ⁻¹(p).

How it is computed

Φ(−|z|) = ½ Q(½, z² / 2), where Q is the regularized upper incomplete gamma function, so tail areas stay accurate to about 1e-13 relative even at z = −37 (p ≈ 6e-300). The inverse starts from Acklam’s rational approximation and is refined by Halley steps to full double precision.

Assumptions and limits

• The probabilities are only as good as the normal model for your variable; check a histogram or normal Q–Q plot first.
• z-scores of a sample (using its own mean and SD) describe position within the sample; probabilities then assume normality.

Worked example

IQ scores are scaled to mean 100 and SD 15. A score of 130 has z = (130 − 100) / 15 = 2; P(X < 130) = Φ(2) = 0.97725 (the 97.7th percentile) and P(X > 130) = 0.02275. The share between 130 and 145 is Φ(3) − Φ(2) = 0.02140. The 95th percentile is 100 + 1.644854 × 15 = 124.67.

References

• NIST/SEMATECH e-Handbook of Statistical Methods: normal distribution
• NIST/SEMATECH e-Handbook of Statistical Methods: cumulative distribution function of the standard normal
• R documentation: the normal distribution (pnorm, qnorm)