The Beta distribution calculator above evaluates a Beta distribution with shape parameters alpha and beta. It returns the probability density at a chosen point, the cumulative probability below and above that point, the mean, the variance, the standard deviation, the mode where one exists, and a quantile found by inverting the cumulative function numerically. Because the Beta distribution is defined only on the interval from zero to one, every input and every output on this page is bounded by those two numbers.
Arb Digital publishes this because the Beta distribution is the one members of the standard distribution family that describes a probability rather than a count or a measurement. Our binomial distribution calculator answers how many successes you will see given a known success rate. This page answers the reverse question: given the successes you have already seen, what does the success rate itself look like? That reversal is the whole reason the distribution exists, and it is why a conversion rate, a click-through rate or a defect rate is modelled with a Beta rather than a normal curve.
What This Beta Distribution Calculator Does
It computes the density function f(x) = xα−1(1−x)β−1 divided by the Beta function B(α,β), and the cumulative function, which is the regularised incomplete Beta function Ix(α,β). Both are evaluated with a continued-fraction expansion rather than by numerical integration, so the answers hold at extreme parameter values where a naive quadrature routine drifts badly.
Alongside those it reports the four summary quantities that a textbook exercise usually asks for in the same breath: the mean, which is α divided by (α+β); the variance; the standard deviation; and the quantile at a probability you choose. The quantile is found by bisection on the cumulative function to a tolerance far below the displayed precision, which is more reliable than the closed-form approximations that some tables use near the boundaries.
The bar panel plots the density at evenly spaced points across the interval. That is deliberately not a smooth curve — it is a set of readable heights, so you can see at a glance whether the distribution is bell-shaped, U-shaped, flat, or piled against one edge. Those four shapes cover essentially everything the Beta family can do, and which one you get depends only on whether alpha and beta are above, at, or below one.
How to Use It
- Enter alpha and beta. Both must be strictly positive. They do not have to be whole numbers, and non-integer values are common in practice.
- Set the value x you want to evaluate. It must lie between 0 and 1 because that is the entire support of the distribution.
- Choose the headline figure. Density gives the height of the curve; the two cumulative options give genuine probabilities.
- Set the quantile probability to find the cut-off below which that share of the distribution sits. Enter 0.95 for a one-sided upper bound, or 0.025 and 0.975 in turn for a two-sided interval.
- Read the working panel for the arithmetic behind the headline, written out rather than merely asserted.
The Formula and How It Is Calculated
The density is f(x) = xα−1 (1−x)β−1 / B(α,β), where B(α,β) = Γ(α)Γ(β) / Γ(α+β) is the Beta function and acts purely as the constant that makes the total area equal one. The NIST/SEMATECH e-Handbook of Statistical Methods sets out the same definition in section 1.3.6.6.17 on the Beta distribution, including the four-parameter version with lower and upper bounds other than 0 and 1.
Take Beta(2, 5) evaluated at x = 0.3, which is the default on this page. B(2,5) = Γ(2)Γ(5)/Γ(7) = 1 × 24 / 720 = 1/30. So f(0.3) = 0.31 × 0.74 × 30 = 0.3 × 0.2401 × 30 = 2.1609. That number is greater than one, which is not an error: a density is a height, and only the area under it is constrained to be a probability. The mean is 2/7 = 0.2857, and the variance is (2 × 5) / (72 × 8) = 10/392 = 0.02551, giving a standard deviation of about 0.1597.
The cumulative function has no elementary closed form in general, so the tool uses Lentz's continued-fraction evaluation of the regularised incomplete Beta function, with the standard symmetry swap Ix(a,b) = 1 − I1−x(b,a) applied whenever x sits on the slow side of the convergence boundary. The NIST Digital Library of Mathematical Functions covers the underlying object in section 8.17 on incomplete Beta functions. When alpha and beta are both whole numbers there is a shortcut worth knowing: Ix(a,b) equals the probability of at least a successes in a+b−1 binomial trials with success probability x. For Beta(2,5) at x = 0.3 that is P(at least 2 heads in 6 trials at p = 0.3) = 1 − 0.76 − 6(0.3)(0.75) = 1 − 0.117649 − 0.302526 = 0.579825, which is exactly what the tool reports.
What Beta Is For That the Other Distributions Are Not
Every other distribution in our library describes an outcome. The normal distribution calculator describes a measurement, the Poisson distribution calculator a count of events in a window, the exponential distribution calculator a waiting time, the geometric distribution calculator the trial on which the first success lands, and the hypergeometric distribution calculator a count drawn without replacement.
The Beta distribution describes none of those. It describes a probability — a number that is itself the parameter of some other distribution. That is a genuinely different object, and it is why the Beta lives on [0, 1] while everything else in the list ranges over counts or over the whole real line. When you say "I think the conversion rate is somewhere around 4 percent but I am not sure", the thing you have just described a spread of belief over is a probability, and a Beta distribution is the natural way to write that spread down.
The mechanism that makes it useful is conjugacy. Start with a Beta(α, β) description of a rate, observe s successes and f failures, and the updated description is exactly Beta(α+s, β+f). No integration, no simulation, no approximation — the family is closed under binomial evidence, and the update is addition. That is why the parameters are so often described as pseudo-counts: alpha behaves like successes you have already seen and beta like failures. Our Bayes theorem calculator handles the discrete version of the same updating logic on individual hypotheses.
Reading the Shape From the Two Parameters
You can predict the shape without plotting anything. If both alpha and beta exceed one the density is a single hump with an interior mode at (α−1)/(α+β−2). If both are below one the density is U-shaped, rising to infinity at both ends, which describes a quantity that is usually near zero or near one and rarely in between. If both equal exactly one the density is flat: Beta(1,1) is the uniform distribution on [0,1], which is why the tool offers it as a preset.
Mixed cases give the J shapes. Alpha at or below one with beta above one piles the mass against zero; the reverse piles it against one. Beta(1, 5) is a decreasing curve from a height of 5 down to zero — the description of a rate you believe is small but have not pinned down. Where alpha is exactly one the density at zero is finite; where alpha is below one it diverges, and the mode is at the boundary rather than inside the interval. The tool reports "no interior mode" in that case rather than printing a meaningless number, because the formula for the mode is only valid when both parameters exceed one.
Symmetry is easy too: alpha equal to beta gives a distribution symmetric about 0.5, and the larger the common value the tighter it is. Beta(2,2) is a gentle arch; Beta(50,50) is nearly a narrow bell around a half. That tightening with no change to the mean is the visual signature of accumulating evidence, and it is the single most useful thing to watch on this page.
The Interval It Gives You Is Not a Confidence Interval
Take the 2.5 percent and 97.5 percent quantiles of a Beta and you get an interval that looks like a familiar 95 percent range. It is not the same object. A frequentist confidence interval is a statement about a procedure — repeat the experiment many times and 95 percent of the intervals so constructed will cover the true value. The Beta interval is a statement about the parameter given the data and the prior you chose, and it is usually called a credible interval to keep the two apart.
The practical difference shows up at the boundaries and at small samples. With zero successes in twenty trials, the normal approximation used by many quick calculators gives an interval of zero width, which is plainly wrong. A Beta(1, 21) posterior gives an upper bound of about 0.133, which is a sensible answer to "how high could this rate plausibly be?" Our confidence interval calculator builds the frequentist version and is the right tool when a report or a course requires that construction specifically; use this page when the question is about belief over a rate.
One more caution: the answer depends on the prior you started from, and Beta(1,1), Beta(0.5,0.5) and Beta(0,0) are all defended in the literature as the "uninformative" choice. They give different answers at small samples and converge as data accumulates. Stating which one you used is not optional if anyone else has to reproduce your number.
Where It Shows Up in Practice
Split testing is the obvious case. Two variants with different observed conversion rates produce two Beta posteriors, and the probability that one is better than the other is an integral over the pair rather than a p-value. Our A/B test calculator runs the significance-testing version of that comparison; the Beta view is what sits underneath most of the "probability to be best" numbers that testing platforms report.
Project scheduling uses a rescaled Beta as the basis for the three-point PERT estimate, where optimistic, most likely and pessimistic durations are mapped onto a Beta on that range. Reliability engineering uses it for the pass proportion of a component. Quality control uses it to describe a defect rate that is known to be small but has been observed only a handful of times. In finance and elsewhere it is used for any quantity naturally bounded at both ends — a recovery rate, a utilisation percentage, a share of a total.
Note the name collision. The stock beta calculator on this site computes a completely unrelated quantity: the sensitivity of an asset's returns to market returns. It shares only a Greek letter with this page. If you arrived looking for volatility relative to an index, that is the page you want.
Arb Digital builds measurement into campaigns so the numbers in the report carry an honest range rather than a single misleading point estimate.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Treating the density as a probability — f(x) is a height and routinely exceeds 1. Only areas under the curve are probabilities, which is what the cumulative options give you.
- Using the mode formula when a parameter is below one — (α−1)/(α+β−2) is only valid when both parameters exceed one; otherwise the mode sits at a boundary.
- Evaluating outside [0, 1] — the density is exactly zero outside the unit interval, not undefined and not small. Rescale your data first if it lives on a different range.
- Calling the quantile interval a confidence interval — it is a credible interval and it answers a different question, however similar the two numbers look.
- Not stating the prior — two analysts starting from Beta(1,1) and Beta(0.5,0.5) will publish different intervals from identical data, and neither is wrong until the prior is hidden.
Related Free Tools From Arb Digital
Model a fixed number of trials with the binomial distribution calculator, work backwards from a probability to a z value with the inverse normal distribution calculator, build a binned table from raw data with the frequency distribution calculator, size a study with the sample size calculator, or check a spread with the standard deviation calculator. The full free online tools hub lists every statistics calculator we publish.
Frequently Asked Questions
Because a density is a height rather than a probability. Only the area under the curve is constrained to sum to one, and on an interval only one unit wide the curve has to rise above one to enclose that area.
They act like pseudo-counts. Alpha behaves as successes already observed and beta as failures, which is why observing s more successes and f more failures updates the distribution to Beta(alpha plus s, beta plus f).
The uniform distribution on the interval from zero to one. Every value is equally dense, which is why it is often used as the starting point when nothing at all is known about a rate.
The binomial describes how many successes you get from a known rate. The Beta describes the rate itself given the successes you have seen. They are two halves of the same problem pointing in opposite directions.
No. It is a credible interval, which is a statement about the parameter given your data and prior. A confidence interval is a statement about the long-run behaviour of a procedure, and the two answer different questions.
Only when both parameters exceed one, in which case it sits at alpha minus one over alpha plus beta minus two. When either parameter is one or below, the density peaks at a boundary instead and the formula does not apply.
Not directly. The four-parameter version rescales the distribution onto any lower and upper bound, so shift and divide your data onto the unit interval first and then map the answers back.
No. Stock beta measures how much an asset moves relative to the market and has no connection to this distribution beyond sharing a Greek letter.
This page explains a statistical distribution for educational purposes. It is not investment, medical or business advice, and any decision based on an estimated rate should account for how the data were collected.
