Probability Distribution Mean & Variance Calculator
Find the mean (expected value), variance and standard deviation of a discrete probability distribution from its values and probabilities.
How it works
μ = Σ x·P(x) · σ² = Σ (x − μ)²·P(x) · σ = √σ²
For a discrete random variable, the mean (expected value) is the probability-weighted average of every possible outcome, and the variance is the probability-weighted average of the squared distances from that mean. The standard deviation is the square root of the variance, back in the variable’s own units. Together they summarise where the distribution is centered and how widely its outcomes scatter.
Worked example
Values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2 give μ = 0×0.1 + 1×0.3 + 2×0.4 + 3×0.2 = 1.7. The variance is 0.1×(−1.7)² + 0.3×(−0.7)² + 0.4×(0.3)² + 0.2×(1.3)² = 0.81, so σ = 0.9.
Sources: Expected value — Wikipedia
Frequently asked questions
Do the probabilities need to sum to exactly 1?
Ideally yes, since a valid probability distribution sums to 1. This calculator rescales them if they are slightly off due to rounding, but a large discrepancy means the distribution is misspecified and worth rechecking.
Why weight the squared deviations by probability?
Unlike a plain data set where each point counts once, outcomes here occur with different chances. Weighting each squared deviation by its probability gives the true long-run average squared distance from the mean.
How does this relate to expected value?
The mean of a probability distribution is exactly its expected value, μ = Σ x·P(x). This calculator adds the variance and standard deviation, which expected value alone does not provide.
Related calculators
For education — check the assumptions (sample size, independence, distribution) before using these statistics in real decisions.