Normal Distribution Calculator
Enter μ, σ, and x to calculate cumulative probabilities and visualize the normal distribution curve.
Standardized z = (x − μ) / σ = (1 − 0) / 1 = 1.0000
P(X ≤ x)
84.1309%
Left-tail cumulative prob. (CDF)
P(X ≥ x)
15.8691%
Right-tail probability
P(μ−|x−μ| ≤ X ≤ μ+|x−μ|)
68.2618%
Two-tail symmetric interval
Normal Distribution Curve
Left-tail area P(X ≤ x)
Formula Reference
z = (x − μ) / σ ← standardized z-score
P(X ≤ x) = Φ(z) ← standard normal CDF
P(X ≥ x) = 1 − Φ(z)
P(|X − μ| ≤ |x − μ|) = Φ(|z|) − Φ(−|z|) = 2Φ(|z|) − 1
CDF approximated using Horner polynomial method (Abramowitz & Stegun 26.2.17), max error ≈ 7.5×10⁻⁸