Math & statistics
Expected Value Calculator
Type the outcomes of a discrete random variable and their probabilities — get the mean, variance and standard deviation with every step of the table filled in.
Expected value E(X)
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variance σ²
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standard deviation σ
| x | P(x) | x·P(x) | (x−μ)²·P(x) |
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Frequently asked questions
How do I calculate expected value?
Multiply each outcome by its probability and add: E(X) = Σ x·P(x). A raffle paying $100 with probability 0.01 and $0 otherwise has E(X) = 100×0.01 = $1 — so a $2 ticket loses $1 on average.
How do I find the standard deviation of a probability distribution?
First variance: σ² = Σ (x − μ)²·P(x), where μ is the expected value. Then σ = √σ². This calculator shows both, plus each (x − μ)²·P(x) term.
What if my probabilities do not add to 1?
A valid probability distribution must sum to exactly 1. The calculator warns you when yours does not — check for a missing outcome or a typo before trusting the result.
What does expected value mean in real life?
It is the long-run average per play, not what happens once. A casino game with E(X) = −$0.50 does not lose you 50 cents every game — it loses you about $50 per 100 games on average.
Method: E(X) = Σ x·P(x), σ² = Σ (x−μ)²·P(x), σ = √σ² — the discrete random variable formulas from every introductory statistics course. Probabilities must sum to 1.