Calculate alpha and beta using several input methods. Inspect mean, variance, mode, and skewness quickly. Download neat tables, plots, and summaries for documentation needs.
Enter values and submit to estimate beta distribution parameters and derived measures.
The graph shows the beta probability density using the current alpha and beta values.
| Case | Input Method | Input Values | Alpha | Beta | Mean | Variance |
|---|---|---|---|---|---|---|
| 1 | Mean and variance | μ = 0.60, v = 0.02 | 6.60 | 4.40 | 0.60 | 0.02 |
| 2 | Mean and standard deviation | μ = 0.35, s = 0.12 | 5.18 | 9.61 | 0.35 | 0.0144 |
| 3 | Mean and concentration | μ = 0.75, k = 16 | 12.00 | 4.00 | 0.75 | 0.01103 |
| 4 | Alpha and beta | α = 3.50, β = 2.50 | 3.50 | 2.50 | 0.5833 | 0.03472 |
Mean: μ = α / (α + β)
Variance: v = αβ / [(α + β)²(α + β + 1)]
Standard deviation: s = √v
Mode: (α − 1) / (α + β − 2), when α > 1 and β > 1
From mean and variance: t = μ(1 − μ)/v − 1, then α = μt and β = (1 − μ)t
From mean and standard deviation: set v = s², then use the same mean-variance transformation
From mean and concentration: α = μk and β = (1 − μ)k, where k = α + β
This calculator estimates beta distribution parameters from several common input combinations. It supports mean with variance, mean with standard deviation, mean with concentration, and direct alpha-beta entry. That makes it useful for probability modeling, Bayesian work, reliability studies, forecasting, and bounded percentage analysis.
The beta distribution is defined on the interval from 0 to 1. Because of that range, it is often used for rates, probabilities, conversion fractions, risk proportions, and other bounded outcomes. The shape changes quickly with alpha and beta, so deriving those parameters correctly matters when fitting practical models.
After calculation, the page returns alpha, beta, mean, variance, standard deviation, mode, skewness, excess kurtosis, and the total concentration. The result section appears directly below the header and above the input form, making comparison easy when you test several scenarios one after another.
The included Plotly graph draws the probability density from the current parameter values. This helps you see whether the distribution is symmetric, left-skewed, right-skewed, sharply concentrated, or broadly spread. Visual inspection is especially helpful when you are choosing priors or checking whether assumptions look realistic.
Export tools are also built in. You can download current results as CSV, save the example table as CSV, or create a PDF summary for reporting. The example table shows sample cases so you can verify the workflow quickly before entering your own values.
In applied mathematics, the calculator is useful for parameter recovery and interpretation. When you know a target mean and uncertainty, the solver converts those summary values into alpha and beta. When parameters are already known, it turns them into interpretable moments and shape statistics for clearer communication.
It estimates beta distribution parameters and related summary measures. Depending on your input method, it can solve for alpha and beta or derive mean, variance, mode, skewness, and concentration from known parameters.
The beta distribution only models values inside the unit interval. Since its support is bounded by 0 and 1, any valid mean must also remain strictly inside that same range.
For a beta distribution with mean μ, the variance must stay below μ(1 − μ). If it does not, the implied alpha and beta values become nonpositive, which makes the model invalid.
Concentration is α + β. Larger concentration values make the distribution more tightly centered around the mean, while smaller values create broader and often flatter or more dispersed shapes.
The interior mode formula only works when alpha and beta are both greater than 1. If either parameter is 1 or less, the density may peak at a boundary or remain monotonic.
Use that option when your source gives average behavior and spread directly. The calculator squares the standard deviation to get variance, then solves for alpha and beta from those values.
The graph shows the density shape implied by alpha and beta. It helps you identify symmetry, skewness, concentration, and whether mass is pulled toward 0, 1, or the interior.
Yes. Beta distributions are common priors for probabilities and proportions. This calculator helps translate intuitive targets, such as a mean and confidence level, into workable alpha and beta parameters.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.