How this instrument works
The Beta distribution is the standard way to describe a random quantity that must fall somewhere between 0 and 1, such as an unknown success probability, a proportion of a population, or a fraction of a budget spent. Two positive shape parameters, alpha and beta, control its shape: roughly speaking, alpha pulls the distribution's weight toward 1 and beta pulls it toward 0, so a larger alpha relative to beta produces a distribution skewed toward higher values, and vice versa. When alpha equals beta, the distribution is symmetric around 0.5.
This is why the Beta distribution shows up constantly in Bayesian statistics: it's the natural way to represent a belief about an unknown probability before or after seeing data, and it happens to combine cleanly with binomial data (observing successes and failures simply shifts alpha and beta upward by the counts observed). Outside Bayesian work, it's also used to model bounded quantities directly, like the fraction of a project's time spent on a particular task, or the proportion of a batch that fails a test.
This instrument reports the three standard descriptive moments — mean (the distribution's balance point), variance, and standard deviation (its spread) — computed directly from alpha and beta with closed-form formulas. It does not compute the probability density function or a cumulative probability at a given x, since the exact Beta density needs the Beta function B(alpha, beta), which in turn needs the Gamma function for non-integer shape parameters — a genuinely different piece of numerical machinery this instrument doesn't include. If you need the shape of the curve itself or the probability of landing below some threshold, that's out of scope here; this page covers the summary statistics only.
- Enter the first shape parameter into Shape parameter alpha — larger values pull the distribution's weight toward 1.
- Enter the second shape parameter into Shape parameter beta — larger values pull the distribution's weight toward 0.
- Both parameters must be greater than zero; equal values (alpha = beta) produce a distribution symmetric around 0.5.
- Read Mean, E[X] for the distribution's balance point, and Variance, Var(X) and Standard deviation for how spread out it is.
Worked example — alpha = 2, beta = 5
A product page has been shown to visitors a handful of times, resulting in a rough Bayesian estimate of its conversion rate summarized by a Beta(2, 5) distribution — 2 pseudo-successes against 5 pseudo-failures. Entering alphaP = 2 and betaP = 5, Mean, E[X] reads 0.285714, computed as 2/(2+5) = 2/7 — the distribution's center of mass sits noticeably below 0.5, reflecting the larger beta parameter pulling it toward 0.
Variance, Var(X) reads 0.02551020, computed as (2×5)/((7²)×8) = 10/392, and Standard deviation reads 0.159719, the square root of that variance. Together these say the plausible conversion rate is centered around 28.6%, but with real uncertainty — a standard deviation of about 16 percentage points either side — reflecting how little data (7 pseudo-observations total) has gone into the estimate so far.
Questions
What do alpha and beta actually represent?
In the common Bayesian-estimation use case, alpha can be read as roughly 'successes plus one' and beta as roughly 'failures plus one' (with a uniform Beta(1,1) prior as the starting point before any data). More generally, they're just shape parameters: increasing alpha shifts the distribution's mass toward 1, increasing beta shifts it toward 0, and increasing both together (while keeping their ratio fixed) tightens the distribution around the same mean without changing where its center sits.
Does this calculator show the shape of the distribution curve or a cumulative probability?
No — it reports the mean, variance, and standard deviation only. The probability density function itself requires the Beta function B(alpha, beta), and a cumulative probability (the chance X falls below some threshold) requires integrating that density, both of which need the Gamma function for non-integer parameters. Neither is computed here; this instrument is scoped to the closed-form summary statistics.
What does it mean when alpha equals beta?
The distribution becomes symmetric around 0.5 — Beta(1,1) is the special case that's exactly the uniform distribution on [0,1], flat everywhere, while Beta(5,5) is bell-shaped and concentrated tightly around 0.5. Any Beta(a, a) has mean exactly 0.5 by the symmetry of the mean formula, alpha/(alpha+beta), whenever the two parameters match.
Why is the Beta distribution always between 0 and 1?
It's defined that way by construction — its support is fixed to the interval [0, 1], which is exactly what makes it the natural choice for modeling proportions, probabilities, and rates rather than unbounded quantities. If you need to model a value bounded by some other range, it can be rescaled from a Beta(0,1) variable, but that rescaling isn't something this instrument performs.
How does more data change alpha and beta in a Bayesian update?
Observing k successes and m failures out of a batch simply adds those counts to the existing alpha and beta: the updated distribution is Beta(alpha + k, beta + m). As alpha and beta both grow with more observed data, the variance formula's denominator grows faster than its numerator, so the distribution narrows — more data means a tighter, more confident estimate, even if the mean itself barely shifts.