What Is Variance of a Probability Distribution?
The variance of a probability distribution is the expected squared distance of the random variable from its mean. Each named distribution has a short formula in terms of its parameters.
Choose the distribution and enter its parameters. You get the mean, variance, standard deviation, the formula used with your numbers and a chart of the distribution.
Below are the formulas, a worked binomial example and answers on common variance results.
How to Use the Variance of a Probability Distribution Calculator
- Choose the distribution.
- Enter its parameters.
- Read the mean, variance and SD.
- Check the shape on the chart.
Variance of a Probability Distribution Formula
Variance definition
Binomial
Geometric
Variance of a Probability Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| E(X) | Mean of the distribution | |
| Var(X) | Variance | |
| n, p | Binomial parameters | |
| λ | Poisson or exponential rate |
Variance of a Probability Distribution Worked Example
For a binomial distribution with n = 20 and p = 0.3 the mean is 6 and the variance is 20 × 0.3 × 0.7 = 4.2, so the standard deviation is 2.049.
With the example values loaded in the calculator above, the steps are:
- Distribution: Binomial.
- Mean E(X) = 6.
- Var = n·p·(1 − p) = 20 × 0.3 × 0.7 = 4.2.
- SD = √Var = 2.0493902.
Result: Variance Var(X) = 4.2; Standard deviation = 2.0493902; Mean E(X) = 6.
How to Read the Variance of a Probability Distribution
Variance is the probability-weighted average squared distance from the mean, so it is always zero or more. A variance of 0 means the variable never changes. A larger value means outcomes land farther from the expected value, on average.
Variance is in squared units, which is awkward to read. Take the square root to get the standard deviation, which is in the same units as the variable. For a binomial with n = 20 and p = 0.3, the variance is 4.2 and the standard deviation is about 2.05 successes.
Where Variance of a Probability Distribution Is Used
Binomial and Poisson problems
Pick the family, enter its parameters and read the mean and variance from the closed forms np(1 − p) and λ. No table of probabilities is needed.
Uniform, exponential and normal models
The same tool covers the continuous families, so you can compare the spread of a uniform with that of a normal that has the same mean.
Risk comparison
Two models can have the same expected value and very different variances. The larger variance means a wider swing in results.
Exam practice
Check a hand calculation of E[X²] − (E[X])² against the formula the calculator shows for the chosen distribution.
Common Variance of a Probability Distribution Mistakes to Avoid
- Check that the parameters belong to the family: p must be between 0 and 1, n a whole number, and λ above 0.
- The variance formula depends on how a family is parametrised. An exponential with rate λ has variance 1 ÷ λ², but with mean β it is β².
- Subtract the mean squared when you work it out by hand: Var(X) = E[X²] − (E[X])². Forgetting this returns E[X²], which is too large.
- Do not divide by n − 1. That correction belongs to sample variance from data, not to a probability distribution.
Variance of a Probability Distribution Terms and Definitions
- Variance
- The expected squared distance of X from its mean.
Variance of a Probability Distribution FAQs
How do you find the variance of a probability distribution?
Find the mean μ = Σ x·P(x), then compute Σ (x − μ)² · P(x). The shortcut is E[X²] − μ², where E[X²] = Σ x²·P(x). Both give the same value, and the calculator shows the steps.
What is the variance of a binomial distribution?
It is n·p·(1 − p). For n = 20 and p = 0.3 that is 20 × 0.3 × 0.7 = 4.2. The variance is largest when p is 0.5 and shrinks as p moves toward 0 or 1.
What is the variance of a Poisson distribution?
It equals the mean, λ. A Poisson with λ = 4 has mean 4 and variance 4. If your count data have a variance well above the mean, a Poisson model is probably too tight.
What is the variance of a uniform distribution?
For a continuous uniform on [a, b] it is (b − a)² ÷ 12. For a discrete uniform on the whole numbers 1 to n it is (n² − 1) ÷ 12. A fair die has variance 35 ÷ 12 ≈ 2.92.
Can a variance be negative?
No. Every term is a probability times a square, so the total cannot go below zero. A negative result means an arithmetic slip, usually a wrong probability or a sign error in E[X²] − μ².