Normal Distribution Calculator

Calculate probabilities for a normal distribution. Find P(X < x), P(X > x), or P(a < X < b) given mean and standard deviation.

Mean (μ)

Standard Deviation (σ)

X value

Enter mean, standard deviation, and X.

Normal Distribution

Z = (X − μ) / σ
P(X < x) = Φ(z)

The normal (Gaussian) distribution is symmetric and bell-shaped. About 68% of values fall within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.

Example: μ = 100, σ = 15, X = 130
Z = (130 − 100) / 15 = 2.0
P(X < 130) = 0.9772 (97.72%)

In a normal distribution: ~68% of data falls within 1σ of the mean, ~95% within 2σ, and ~99.7% within 3σ. This is also called the empirical rule.

Last updated: 2026-08-08