Normal Distribution Calculator — Overview

This normal distribution calculator computes the Z-score, cumulative probability (CDF), percentile rank, and left/right tail probabilities for any value from a normal distribution. It uses the error function (erf) for precise CDF calculation and renders the full bell curve for visual reference.

Key Formulas

Z-Score: Z = (X − μ) / σ

PDF (bell curve height): f(x) = (1/σ√2π) × e^(−(x−μ)²/2σ²)

CDF (left tail probability): Φ(Z) = (1/2)[1 + erf(Z/√2)]

The erf approximation used achieves maximum error of 1.5 × 10⁻⁷ — sufficient for all practical statistical work.

Practical Applications

IQ Testing: Mean = 100, SD = 15. A score of 130 → Z = 2.0 → 97.7th percentile — higher than 97.7% of the population. Manufacturing Quality: If a part diameter has mean = 50mm, SD = 0.1mm, a part at 50.2mm → Z = 2.0 → only 2.3% of parts are larger. Finance: If stock returns have mean = 10%, SD = 20%, probability of losing money (return < 0%) = Φ(−0.5) ≈ 30.9%.

The 68-95-99.7 Empirical Rule

For any normal distribution: 68.27% of values fall within ±1σ. 95.45% within ±2σ. 99.73% within ±3σ. 99.9937% within ±4σ (Six Sigma quality: fewer than 3.4 defects per million at 4.5σ process shift). This rule is fundamental to quality control, statistical process control, and hypothesis testing.