StatReason

Expected Value Calculator

Given a set of possible outcomes and their probabilities, find the long-run average outcome and its variability.

What this answers

Expected value answers "what is the long-run average outcome if this exact scenario repeated many times?" A fair six-sided die has an expected value of 3.5, even though 3.5 is not a value the die can ever actually show; expected value describes a long-run average, not a single realistic outcome.

How it is calculated

Expected value is the sum of each outcome multiplied by its probability. Variance is the sum of each outcome's squared distance from that expected value, weighted by its probability, and standard deviation is the square root of variance. This calculator also checks that your probabilities sum to 1, the requirement for a valid probability distribution, and flags it directly rather than silently renormalizing your inputs if they do not.

Worked example

For a fair six-sided die (outcomes 1 through 6, each with probability 1/6): the expected value is exactly 3.5, the well-known long-run average roll. No single roll ever shows 3.5, illustrating that an expected value need not be an attainable outcome of any individual trial.

Assumption audit

Calculated from your data: whether your probabilities sum to exactly 1 (within a small numerical tolerance), and the resulting expected value, variance, and standard deviation.
Evidence to review: a warning appears below the result if your probabilities do not sum to 1, since this usually means an outcome was missed or a probability was mistyped.
You must verify: that your listed outcomes and probabilities genuinely and completely describe the scenario you are modeling; this calculator cannot detect a missing outcome you simply forgot to include.

Common mistakes

Treating a favorable expected value as a guarantee of a favorable single outcome is a common error, especially in gambling and insurance contexts: a game can have a positive expected value overall while most individual plays still lose, if a rare large win offsets many small losses. Expected value describes what happens on average across many repetitions, never what will happen on the next single trial.

Limitations

Expected value summarizes a long-run average, not what will happen on any single trial, and it can be a poor summary for a distribution with rare but extreme outcomes, where the typical result looks nothing like the expected value.