How this calculator works
A discrete expected value is a probability-weighted average of possible outcomes. It describes the centre of the distribution, even when that value is not itself possible. Variance measures squared spread around that expectation. Probability mode checks that the entered probabilities sum to one within a tolerance of 1e-10, then removes any tiny summation drift. Relative-weight mode divides each weight by the positive total. Lists are paired by position.
Formula
E(X) = sum(x_i p_i). Var(X) = sum[p_i(x_i - E(X))²]. Standard deviation = sqrt(Var(X)). With weights w_i, p_i = w_i/sum(w_i).
Worked example
An outcome of 0 with probability 0.75 and 10 with probability 0.25 has expected value 2.5. Its variance is 0.75 × 2.5² + 0.25 × 7.5² = 18.75.
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Frequently asked questions
Is the expected value guaranteed?
No. It is a weighted average over the model. A single observation can be far from it.
Can outcomes be negative?
Yes. Losses, costs or signed measurements can be represented by negative values.
Should I enter 25 or 0.25 for a 25% chance?
Use 0.25 in the probability list. Alternatively, use relative weights such as 25 and 75 in weight mode.