Beta Distribution Calculator

The beta distribution is a continuous probability distribution defined on the interval from 0 to 1, which makes it the natural model for a proportion, a rate or a probability. Its shape is set by two positive parameters, alpha and beta, often written as the number of prior successes and failures. When alpha and beta are equal the distribution is symmetric around 0.5, when alpha is larger the weight shifts towards 1, and when beta is larger the weight shifts towards 0. This calculator takes your alpha and beta and returns the summary statistics of the distribution: the mean, the variance and standard deviation, the mode where the density peaks, and the skewness that measures how lopsided the curve is. It also evaluates the probability density function (PDF) and the cumulative distribution function (CDF) at any point x you choose between 0 and 1, so you can read off the height of the curve and the probability that a random draw falls at or below that point. The beta distribution turns up constantly in Bayesian statistics as the conjugate prior for the binomial, in project scheduling under the PERT method, and in modelling conversion rates, click-through rates and component reliability. Enter alpha, beta and a value of x, and the results update as you type. All inputs must be positive, and x must sit between 0 and 1.

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0.2857
distribution mean, α / (α + β)
Variance0.0255
Std deviation0.1597
Mode0.2000

At x = 0.30 the probability density f(x) is 2.1609 and the cumulative probability P(X ≤ x) is 0.5798. Distribution skewness is 0.5963.

Alpha and beta must be positive and x must be between 0 and 1. The mode is defined at an interior peak only when both alpha and beta are greater than 1.

How it works

A beta distribution with shape parameters alpha and beta has probability density f(x) = x to the power (alpha minus 1), times (1 minus x) to the power (beta minus 1), divided by the beta function B(alpha, beta). The beta function is the normalising constant that makes the total area under the curve equal to 1. From the two parameters the summary statistics follow directly: the mean is alpha / (alpha + beta), the variance is (alpha times beta) / ((alpha + beta) squared times (alpha + beta + 1)), and when both parameters exceed 1 the mode is (alpha minus 1) / (alpha + beta minus 2). The cumulative probability P(X is at or below x) is the regularised incomplete beta function, computed here with a continued fraction expansion so it stays accurate across the full range.

Worked example

Take alpha = 2 and beta = 5. The mean is 2 / (2 + 5) = 0.2857, and the variance is (2 × 5) / (7² × 8) = 10 / 392 = 0.0255, so the standard deviation is 0.1597. Because both parameters are above 1, the mode sits at (2 − 1) / (2 + 5 − 2) = 0.2000. Evaluating at x = 0.30, the density is 0.3¹ × 0.7⁴ / B(2, 5) = 0.07203 / 0.03333 = 2.1609, and the cumulative probability P(X ≤ 0.30) works out to 0.5798. The positive skewness of 0.5963 confirms the longer tail runs towards the higher values.

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Frequently asked questions

What is the mean of a beta distribution?

The mean of a beta distribution equals alpha divided by (alpha plus beta). For example, with alpha of 2 and beta of 5 the mean is 2 / 7, which is about 0.2857. Because the distribution is bounded between 0 and 1, the mean is always a value in that range.

How do you calculate the variance of a beta distribution?

The variance is (alpha times beta) divided by ((alpha plus beta) squared times (alpha plus beta plus one)). With alpha of 2 and beta of 5 that is 10 / (49 times 8), which is 10 / 392, or about 0.0255, giving a standard deviation of about 0.1597.

What is the beta distribution used for?

The beta distribution models a proportion or probability that must lie between 0 and 1, such as a conversion rate, a success rate, or the reliability of a component. It is widely used in Bayesian statistics as the conjugate prior for the binomial distribution, where alpha and beta act as counts of prior successes and failures.