Normal Distribution

Bell curve PDF and CDF with shaded probabilities

X ~ N(μ = 0, σ = 1) · P(X ≤ 1)

P(X ≤ x) — left tail

0.841345

z-score

Density f(x)

0.241971

Percentile

84.1345

P(X ≤ x)

0.841345

P(X ≥ x)

0.158655

About this calculator

The Normal Distribution Calculator draws the bell curve for any mean and standard deviation and shades the probability you choose — the left tail P(X ≤ x), the right tail P(X ≥ x), or the area between two bounds P(a ≤ X ≤ b). It also reports the density f(x), the z-score, and the percentile. Useful for statistics, z-scores, hypothesis testing, and exam grading.

Common examples

Standard normal N(0, 1): P(Z ≤ 1) = 0.8413 (84th percentile)
Right tail P(Z ≥ 1.5) = 0.0668
One-sigma rule: P(−1 ≤ Z ≤ 1) = 0.6827
Two-sigma rule: P(−2 ≤ Z ≤ 2) = 0.9545
IQ scores N(100, 15²): P(X ≤ 130) = 0.9772 (z = 2)