Normal Distribution Calculator

The normal (Gaussian) distribution is the bell curve defined by two parameters: mean (mu) sets where it is centered and standard deviation (sigma) sets how wide it spreads. PDF: f(x) = (1 / (sigma x sqrt(2 pi))) x e^(-(x - mu)^2 / (2 sigma^2)). It models heights, measurement errors, test scores, and many other natural quantities.

PDF gives the height of the bell curve at a single x and is not a probability on its own. CDF gives the cumulative probability P(X <= x), the area under the curve to the left of x, always between 0 and 1. P(X = x) is exactly 0 for any continuous distribution.

Use the between mode with lower bound a and upper bound b. The calculator returns P(a <= X <= b) = CDF(b) - CDF(a). With mu = 0 and sigma = 1, P(-1.96 <= X <= 1.96) is about 0.95, the basis for the 95% confidence interval.

Given a probability p between 0 and 1, find x such that P(X <= x) = p. It is how you turn confidence levels into critical values. With mu = 0 and sigma = 1, p = 0.975 gives x = 1.96 and p = 0.025 gives x = -1.96.

About 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. With mu = 0 and sigma = 1, the probabilities for [-1, 1], [-2, 2], and [-3, 3] come out to roughly 0.6827, 0.9545, and 0.9973.