The beta distribution is a continuous probability distribution that models random variables with values falling inside a finite interval. Use it to model subject areas with both an upper and lower bound for possible values. Analysts commonly use it to model the time to complete a task, the distribution of order statistics, and the prior distribution for binomial proportions in Bayesian analysis.

The standard beta distribution uses the interval [0,1]. This range is ideal for modeling probabilities, particularly for experiments with only two outcomes. However, other intervals are possible.

**Related post**: Understanding Probability Distributions

## Beta Distribution Parameters and Notation

Unlike other distributions with shape and scale parameters, the beta distribution has two shape parameters, α and β. Both parameters must be positive values.

Additionally, statisticians denote the finite interval’s upper and lower bounds as *a* and *b*, respectively.

Let’s see how these parameters work by graphing the probability density function for this distribution!

### How to Calculate Beta Distribution Parameters

Statistical software can use maximum likelihood estimation to find the parameters for the beta distribution. This process estimates the parameters that produce the best fitting curve for your data. Alternatively, you can perform simple calculations using the outcome of a binomial experiment to find the appropriate parameters, which I show in the next section.

The beta distribution is particularly flexible at modeling different curves within the interval, including symmetrical, left and right-skewed, U and inverted U shapes, and straight lines. The standard form can illustrate all curves, so I’ll use it in the examples below. However, keep in mind that the upper and lower bounds do not need to be 0 and 1.

Note that my software refers to α and β as First and Second, respectively.

## Beta vs Binomial Distribution and Updating Prior Probabilities

The beta distribution has a close relationship with the binomial distribution. First, remember that the binomial distribution models the number of successes in a specific number of trials when you have binary data. Now, consider that the number of successes divided by the number of trials is a binomial proportion, which is a probability. The beta distribution models the likelihood of success in Bernoulli Trials and captures its uncertainty. Learn more about the Binomial Distribution.

Suppose you sell breakfast cereal and perform a simple experiment. You randomly select ten people to try your cereal and a competitor’s. When a subject says your cereal is better, it’s a success. Seven out of 10 (70%) said your cereal is better. Because there are only two possible outcomes (success/failure), it’s a binomial experiment. Let’s use the beta distribution to model the results.

For this type of experiment, calculate the beta parameters as follows:

- α = k + 1
- β = n – k + 1

Where:

- k = number of successes
- n = number of trials.

Additionally, use this method to update your prior probabilities in a Bayesian analysis after you obtain additional information from a new binomial experiment. Simply add the new successes and trials to parameters of the prior probability’s beta distribution. As α and β increase, the distribution narrows, reflecting the greater precision of the larger sample size.

For our experiment, we have 7 successes and 10 trials:

- α = 7 + 1 = 8
- β = 10 – 7 + 1 = 4

### Assessing the Results

Using these parameter values produces the following beta distribution.

In closing, I’ll emphasize the relationship between the beta and binomial distributions. The graph below displays the binomial distribution for our experiment. Notice the similarities? Both curves peak at 0.7 or 7/10 and are similarly left-skewed. In the binomial distribution, take the number of successes and divide by 10 (the number of trials) to obtain the probabilities in the beta distribution.

Pradeep Kandpal says

Thanks a ton, Jim. I was using excel’s beta.inv function to simulate a random variable from a dataset that follows a beta distributon and was wondering how the alpha and beta value are arrived at.

Norman Abraham says

Re: A flexible method for parameterizing ranked nodes in Bayesian networks using Beta distributions, Steven Mascaro and Owen Woodberry

Hello,

I would appreciate your comments regarding the above as to its comparison to the Ranked Node Method of Fenton and Neil.

Thank you,

Norm

Jim Frost says

Hi Norman, unfortunately, I’m not familiar with that method. Sorry.

Pierre V says

Hello Jim,

Thank you for you explanation. I am using Minitab too and I am trying to know which distribution fits the best my data. I already computed the identification of the law and none of the laws studied were ok (p-values smaller than 0,05).

I would like to know if my data fits a beta distribution but I do not know how to proceed. Please let me know if it is possible

Jim Frost says

Hi Pierre,

When you say “laws,” do you mean probability distributions? Laws is not really the correct word. It’s better to say probability distributions or function. Although, I realize that English might not be your first language. I just want to be sure that I understand the question. You mention determining whether your data fit the Beta distribution, but I’ll address your question in relation to all candidate distributions. After all, if your data don’t fit the Beta distribution, it might fit another.

If none of the candidate probability distributions fit your data, that’s a tough spot to be in. Here are two possibilities to consider.

Have you’ve also looked at the data transformations to see if they normalize your data?

Is your sample size fairly large? If so, the distribution tests might be flagging trivial departures from the probability distributions. Check the probability distribution plots. If you have a large sample size but the data points follow the straight line in a probability distribution plot, it’s evidence that your data follow that distribution even if the p-value is less than 0.05. For more information, read my post about probability distribution plots. That article focuses on the normal distribution, but the same ideas apply to the other distributions.

If those two points don’t help you out, I unfortunately don’t have any easy answers for you because this type of issue highly depends on the subject area. You might also research your subject area to determine what others have found. They might have helpful insights.

When your data don’t fit a known distribution, you might need to analyze them using either a nonparametric analysis or by bootstrapping. Those methods don’t assume that your data follow a particular distribution.

I wish I could provide you with more concrete help, but I hope that points you in the right direction!

Khaled Alrashidi says

Thank you, Jim, for a clear explanation to the beta distribution! I wanted to ensure I understood the example: Shouldn’t beta be 2 instead of 4 since 10 – 7 + 1 = 2? Please let me know if I got this right.

Jim Frost says

Hi Khaled,

Using the standard order of operations, it equals 4 because addition and subtraction have equal priority in the expression. Consequently, you just proceed from left to right. Type that into Excel to confirm that the result is 4. If I had put parentheses around (7 + 1), then the answer would’ve been 2.