Normal distribution calculator

Probabilities at a point (P(X ≤ x), P(X > x), z-score, density), between two bounds, and critical values of a normal distribution with any µ and σ. Tails are computed in relative precision; P(Z > 6) keeps its significant digits.

The z-score maps any normal distribution to the standard one. Φ is evaluated through the error function, as a Maclaurin series near the centre and a Legendre continued fraction in the tails (Abramowitz & Stegun, Handbook of Mathematical Functions, §7.1); the two agree to machine precision in their overlap region. The quantile is obtained by Newton’s method on Φ, whose derivative is the density, bracketed by bisection.

z-score

the gap between x and the mean, in standard deviations.

Critical value

the z (or x) delimiting a given probability; 1.96 for the central 95% interval.

Tail

the probability beyond a point, computed directly (not as 1 − Φ) to keep its significant digits.

The variable is assumed normal; the normality of the data is not tested. The quantile requires a probability strictly between 0 and 1, bounds with no finite preimage. Probabilities below ~10⁻³⁰⁰ are shown as 0.

For a continuous distribution, the probability of an exact value is zero, and P(X ≤ x) = P(X < x). The density f(x) is a probability per unit of x; it exceeds 1 when σ is small. The one-sided 95% quantile is 1.645, against 1.96 for the central interval.