Expected Value Calculator
Find the expected value, variance and standard deviation of a discrete random variable from its outcomes and probabilities.
Expected value
1.1
- Variance
- 0.49
- Standard deviation
- 0.7
- Sum of probabilities
- 1
Across 3 outcomes, the expected value is 1.1, the long-run average of the random variable, with a standard deviation of 0.7.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| Expected value | Sum of x × P(x) over 3 outcomes | 1.1 |
| Variance | Sum of (x − 1.1)² × P(x) | 0.49 |
| Standard deviation | √0.49 | 0.7 |
How it works
Enter each outcome and its probability, in the same order. The probabilities must add up to 1.
Worked example
Days per week a soccer team practices: x = 0, 1, 2 with P(x) = 0.2, 0.5, 0.3 (OpenStax):
- Expected value: 0 × 0.2 + 1 × 0.5 + 2 × 0.3 = 1.1
- Variance: 0.2 × (0 − 1.1)² + 0.5 × (1 − 1.1)² + 0.3 × (2 − 1.1)² = 0.49
- Standard deviation: √0.49 = 0.7
Rounding
Results are shown to six decimal places. Calculations use the unrounded values.
Sources
Frequently asked questions
What is expected value?
The expected value of a discrete random variable is its long-run average, also called its mean. OpenStax defines it as the sum of each value multiplied by its probability, E(X) = Σ xP(x). It is what you would expect on average over many repeats.
How do you calculate expected value?
Multiply each outcome by its probability and add the products. For a soccer team that practices 0, 1 or 2 days a week with probabilities 0.2, 0.5 and 0.3, the expected value is 0 × 0.2 + 1 × 0.5 + 2 × 0.3 = 1.1 days.
Why must the probabilities add up to 1?
A probability distribution lists every possible outcome, so together they cover all the possibilities and total 100%. The calculator checks that the probabilities sum to 1 and shows an error if they do not, since the result would be meaningless.
How do I find the variance and standard deviation?
The variance is the sum of each squared difference from the mean times its probability, and the standard deviation is its square root. In the soccer example the variance is 0.49 and the standard deviation is 0.7 days.
Can the expected value be a value that never happens?
Yes. A fair die has an expected value of 3.5 even though no roll gives 3.5. It is an average over many repeats, not a prediction for a single trial. Negative expected values in a bet mean a loss on average.
Results are for information only. Expected value describes a long-run average, not the outcome of a single trial.