Normal Distribution Calculator

How it works

1. 1The bell curve f(x) = (1 / (σ√(2π))) · exp(−½((x − μ)/σ)²) gives the density at each point x for mean μ and standard deviation σ.
2. 2P(X ≤ x) is the area under the curve to the left of x, computed via the error-function approximation (accurate to ≈ 1.5 × 10⁻⁷).
3. 3The empirical 68–95–99.7 rule says roughly 68%, 95%, and 99.7% of values fall within 1σ, 2σ, and 3σ of the mean; Z = ±1.96 captures the central 95%.

Use cases

• Statistics students checking tail probabilities and critical values for hypothesis tests (e.g. Z = 1.96 → P = 0.975).
• Quality-control engineers finding the fraction of products within spec limits from the process mean and standard deviation.
• Data analysts standardising scores to a common scale by converting raw values to Z-scores.

Frequently asked questions

What is the difference between PDF and CDF?

The PDF f(x) is the height of the bell curve at a single point — a density, not a probability. The CDF P(X ≤ x) accumulates area from −∞ to x and is the actual probability that a draw falls at or below x.

Why does σ have a minimum of 0?

A standard deviation of 0 means all values are identical, making the distribution degenerate; negative σ has no meaning. Both cases return — (undefined) in the output.

How accurate is the CDF?

It uses the Abramowitz & Stegun 7.1.26 polynomial approximation to the error function, with maximum absolute error below 1.5 × 10⁻⁷ — ample for practical statistical or engineering use.