About this tool
The Normal Distribution Calculator works out probabilities for a normal (Gaussian) random variable X with mean μ and standard deviation σ. Enter μ, σ and a value x — and optionally a second value x2 — and it returns the z-score and the associated tail and interval probabilities as percentages, all computed in your browser with nothing uploaded.
The standardised score is z = (x − μ) / σ. The cumulative probability P(X ≤ x) is found from the standard normal CDF Φ(z), approximated with the Abramowitz–Stegun error-function formula; then P(X ≥ x) = 1 − Φ(z). If you supply x2 greater than x, the area between them is Φ((x2 − μ)/σ) − Φ(z). The probability density at x is (1 / (σ√(2π)))·e^(−z²/2).
The standard deviation σ must be greater than zero; a zero or negative σ is rejected as invalid. Because the CDF uses a rational approximation, probabilities are accurate to a few decimal places — plenty for coursework and everyday work, though not to the last digit of a printed z-table.