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Normal CDF Calculator

Calculate cumulative probabilities, Z-scores, and percentiles for any normal distribution. Supports left-tail, right-tail, between, and outside modes — plus inverse CDF to find X from a probability.

P(X ≤ x) · P(X ≥ x) · P(a ≤ X ≤ b) · Inverse CDF · Live bell curve

Distribution Parameters

Define your normal distribution. Leave μ=0, σ=1 for standard normal.

μ =
σ =

Calculation Mode

Find P(X ≤ x) — the probability that a value falls below x

x =

Bell Curve Visualization

■ Shaded region = probability being calculated

Live Result
—
P(event)
DistributionN(0, 1)
ModeLeft Tail
Z-score—
Common Critical Values
90% CI (α=0.10)z = ±1.645
95% CI (α=0.05)z = ±1.960
99% CI (α=0.01)z = ±2.576
P(X≤x)=0.975z = 1.960
P(X≤x)=0.995z = 2.576
1σ (68.27%)z = ±1.000
2σ (95.45%)z = ±2.000
3σ (99.73%)z = ±3.000
68-95-99.7 Rule
μ ± 1σ 68.27%
μ ± 2σ 95.45%
μ ± 3σ 99.73%

For any normal distribution N(μ, σ²).

Key Formulas
Z-score
z = (x − μ) / σ
CDF (Φ)
Φ(z) = ½[1 + erf(z/√2)]
PDF
f(x) = e^(−z²/2) / (σ√2π)
Inverse (Φ⁻¹)
x = μ + σ · Φ⁻¹(p)