Expected Value Calculator
Find the expected value of a discrete distribution from paired value and probability lists.
Expected value E(X)
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E(X) = Σ (x × p)
Expected value is the long-run average outcome: multiply each value by its probability and add up the products.
How it works
Expected value is the long-run average of a random outcome. Weight each possible value by how likely it is, add up those weighted values, and you get the number you'd expect to average toward if you ran the experiment over and over.
The formula is E(X) = Σ(x × p): multiply each value by its probability and sum the products. Enter your outcomes in one list and their probabilities in the other, one probability per value. The tool also reports the variance, Σ p(x − E(X))², and its square root, the standard deviation, so you can see how spread out the outcomes are around the average.
For a valid probability distribution the probabilities must add to exactly 1. If yours don't, the calculator flags it, because a sum above or below 1 means the weights are off and the expected value won't mean what you think.
Frequently asked questions
How is expected value calculated?
Multiply each outcome by its probability and add the results: E(X) = Σ(x × p). For a fair six-sided die, that's (1 + 2 + 3 + 4 + 5 + 6) × (1/6) = 3.5.
Why must the probabilities add up to 1?
Because they cover every possible outcome, and something has to happen. If they sum to more or less than 1, the distribution is invalid — the tool warns you so you can fix a missing outcome or a typo before trusting the result.
What do the variance and standard deviation tell me here?
They measure how far outcomes tend to land from the expected value. A small variance means results cluster tightly around E(X); a large one means the outcomes swing widely, so the average is a less reliable guide to any single result.