Beta Distribution Calculator
Find beta probabilities below, above or between values of a proportion, or the x that matches a given probability. Enter α and β; the mean, variance, median, mode and skewness come with a shaded density chart on the interval from 0 to 1.
Positive; larger α pushes the mass toward 1
Positive; larger β pushes the mass toward 0
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What the beta distribution describes
The beta distribution lives on the interval from 0 to 1, so it is the natural model for an unknown probability or proportion: a conversion rate, the fraction of defective parts, a batting average. Its two shape parameters α and β control where the mass sits and how spread out it is.
f(x) = x^(α−1) · (1 − x)^(β−1) / B(α, β), for 0 < x < 1
Mean = α / (α + β) Variance = αβ / ((α + β)² (α + β + 1))
Mode = (α − 1) / (α + β − 2) for α > 1 and β > 1
Skewness = 2(β − α) √(α + β + 1) / ((α + β + 2) √(αβ))
- α = β = 1 is the uniform distribution on [0, 1].
- α = β > 1 is a symmetric bell shape; α = β < 1 is U-shaped with the mass at both ends.
- α < β skews the curve to the right (small proportions are likely); α > β skews it to the left.
Beta as a prior for a proportion
In Bayesian statistics the beta is the conjugate prior for a binomial success probability. If the prior is Beta(α, β) and you observe s successes and f failures, the posterior is Beta(α + s, β + f). Beta(1, 1) is a flat prior, and the posterior mean (α + s)/(α + β + s + f) is a compromise between the prior mean and the observed proportion. See Bayes' theorem for the discrete version of the same update.
Worked example
For α = 2 and β = 5 the mean is 2/7 = 0.2857, the mode is (2 − 1)/(2 + 5 − 2) = 0.2 and the curve is skewed to the right. The probability that the proportion lies between 0.1 and 0.4 is F(0.4) − F(0.1) = 0.652455. Load example shows this value, its complement 0.347545 (the area outside the interval), the variance 0.0255 and the median 0.2644.
Software equivalents
| Software | Cumulative probability | Inverse |
|---|---|---|
| Excel / Sheets | BETA.DIST(x, alpha, beta, TRUE) | BETA.INV(p, alpha, beta) |
| R | pbeta(x, shape1 = α, shape2 = β) | qbeta(p, α, β) |
| Python (SciPy) | scipy.stats.beta.cdf(x, a=α, b=β) | scipy.stats.beta.ppf(p, α, β) |
| MATLAB | betacdf(x, a, b) | betainv(p, a, b) |
Related guides and calculators
The beta distribution with both parameters equal to 1 is the uniform distribution, and it is the usual prior for a probability that is then updated with binomial data (see the binomial distribution). The gamma distribution is its counterpart for positive quantities. Read Bayesian statistics and Bayes' theorem explained for how priors are updated.
Frequently Asked Questions
What do the α and β parameters of a beta distribution mean?
They are shape parameters. α pulls the distribution toward 1 and β pulls it toward 0. For a Bayesian prior on a success rate you can read α − 1 as prior successes and β − 1 as prior failures.
How do I find the mean and mode of a beta distribution?
The mean is α/(α + β). The mode is (α − 1)/(α + β − 2) when both parameters are above 1. When one is below 1 the density is unbounded at that end and the mode is reported as 0 or 1.
How do I calculate the beta CDF in Excel?
Use BETA.DIST(x, alpha, beta, TRUE). The inverse is BETA.INV(probability, alpha, beta). Both assume the interval from 0 to 1 unless you supply optional lower and upper bounds.
Why can the density be higher than 1?
A density is not a probability; only areas are. Because the interval has width 1, a narrow, tall curve such as Beta(30, 30) has a peak far above 1 while the total area still equals 1.
How is the beta distribution related to the binomial distribution?
The binomial CDF equals a beta tail probability: P(X ≤ k) for n trials with success probability p equals the beta CDF at 1 − p with parameters n − k and k + 1. That identity is how binomial probabilities are computed accurately.
Can I use the beta distribution for values outside 0 to 1?
Not directly. Rescale the variable to the interval from 0 to 1 first, for example (y − min)/(max − min), or use a four-parameter beta with explicit bounds in specialist software.
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