Normal Distribution / Z-Score
PDF, CDF, inverse CDF and Z-score for a normal distribution.
Overview
The Normal Distribution and Z-Score calculator evaluates the PDF, CDF and inverse CDF for any normal distribution N(μ, σ²), plus converts raw scores to z-scores and back. Pick a mean and standard deviation, supply a value, and the calculator returns the density at that point, the cumulative probability to the left and the corresponding z-score.
It is essential for statistics students, quality engineers using six-sigma metrics, finance professionals computing tail probabilities and machine-learning engineers calibrating uncertainty estimates. The bell curve underpins so much practical statistics that quick lookups save real time.
How it works
The PDF of N(μ, σ²) is f(x) = 1 / (σ * sqrt(2π)) * exp(-0.5 * ((x - μ) / σ)^2). The z-score z = (x - μ) / σ converts to the standard normal N(0, 1), where all the heavy lifting happens.
The CDF Φ(z) = P(Z ≤ z) has no closed form, so the calculator uses a rational approximation (Hart's algorithm or the Abramowitz-Stegun series) accurate to many decimal places. The inverse CDF for a probability p returns the z-value with Φ(z) = p.
Examples
x = 1.5 in N(0, 1) → PDF ≈ 0.1295, CDF ≈ 0.9332, z = 1.5
x = 110 in N(100, 15^2) → z ≈ 0.667, CDF ≈ 0.7475
P(Z ≤ 1.96) → 0.975 (the basis of 95% confidence)
inverse CDF at 0.99 → z ≈ 2.326
FAQ
What's a z-score?
The number of standard deviations a value lies above or below the mean. A z of 2 means "two SDs above the mean."
Why is the area under the curve 1?
It's a probability density, so total probability across the whole real line equals 1 by definition.
How accurate is the CDF?
About 7-8 decimal digits across the typical input range — well beyond practical needs.
Why do z = 1, 2, 3 correspond to 68%, 95%, 99.7%?
These are the empirical rule percentages — the proportion of the distribution within 1, 2 and 3 standard deviations of the mean.
What does the inverse CDF give me?
The critical value for a one-sided probability. Use it to find cutoffs for tests, confidence intervals or risk thresholds.