Library / Probability distributions / PRD-28

Probability distributions

Beta proportion: variance

Variance of a beta-distributed proportion.

Ref: PRD-28
Rev: v1
Field: Probability distributions
Result: Var[X]

V a r [X] = \frac{a b}{{(a + b)}^{2} (a + b + 1)}

The variance ab/[(a + b)²(a + b + 1)] shrinks as the total a + b grows, which is how more data narrow the uncertainty about a proportion. It is largest for a given total when a = b. Enter proportions on the 0 to 1 scale.

Variance of a beta-distributed proportion. It shrinks as the total a + b grows, which is how more data narrow a belief about a proportion.

Calculator

Solve for

Var[X]VarianceComputed

aShape a

0.510

bShape b

0.510

Standard deviation0.1597
Mean proportion0.2857
Total shape, a + b7

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Variance () current value 0.02551

bShape bKey driver

0.01070.0559

aShape a

0.01270.0262

The standard deviation of the proportion is 0.16, around a mean of 0.286.
Doubling both shapes roughly halves the variance, as twice the evidence is available.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
b Shape b	0.5 to 10	0.01068	0.05586	0.04517	+10%	−9.81 %
a Shape a	0.5 to 10	0.01271	0.02617	0.01346	+10%	+1.44 %

Sweep of Shape a

Shape a	Variance
0.5	0.01271
1.688	0.02454
2.875	0.02612
4.063	0.02458
5.25	0.02221
6.438	0.01978
7.625	0.01756
8.813	0.01559
10	0.01389

Sweep one variable

Vary

Variance ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Var[X]

Variance

Result ·

What the equation gives you. Everything else feeds it.

Shape a

· typical 0.5 to 10

The first shape parameter (a plain number). Think of it as a count of successes plus one in a Bayesian update. Pulls the proportion towards 1 and tightens the distribution. Raising it by 10 % (to 2.2) moves the result by +1.44 %, from 0.02551 to 0.02588.

Shape b

· typical 0.5 to 10

The second shape parameter (a plain number): a count of failures plus one in a Bayesian update. Pulls the proportion towards 0 and tightens the distribution. Raising it by 10 % (to 5.5) moves the result by −9.81 %, from 0.02551 to 0.02301.

Worked example with the values above

Numerator: ab = 2 × 5 = 10.
Denominator: (a + b)²(a + b + 1) = 49 × 8 = 392.
Var = 0.02551.

Assumptions and limits

The quantity is a proportion between 0 and 1 described by a beta distribution with a, b > 0.
The shapes are dimensionless and come from data or from a prior belief.

Common mistakes

Using percentages (0 to 100) for x: it must be on the 0 to 1 scale.
Applying the mode formula when a or b is 1 or less: the peak is then at an end of the range.
Reading a density above 1 as an error: densities can exceed 1 on a narrow range.

Statistics and uncertainty

A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.

Where it is used

Describing uncertainty in a yield, defect rate or failure fraction, and updating it as test results arrive.
Prior and posterior distributions for a probability in Bayesian reliability work.

Maths you need

Probability density: area under the curve gives probability
Beta function: a ratio of gamma functions

Related formulas

Practise this formula