Expected Value Calculator
The expected value, variance and standard deviation of a set of outcomes and their probabilities, with a check that the probabilities add up to 100%.
Result
Expected value, E(X): 5.5
An average over many repeats. Any one outcome can be far from it, as the standard deviation shows.
- Variance
- 992.25
- Standard deviation
- 31.5
- Probabilities add up to
- 100%
https://calcgallery.com/statistics/expected-value-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Expected Value Calculator
- Choose whether you will type the probabilities as percentages or as decimals.
- Add one row per outcome, with its value, such as a payoff, a cost or a count, and its probability as the weight.
- Read the expected value, with the variance and standard deviation that show how far single results spread around it.
The formulas
E(X) = x1 × p1 + x2 × p2 + … + xn × pnVar(X) = p1 × (x1 − E(X))² + … + pn × (xn − E(X))²SD = √Var(X)Percentages are divided by 100 first. When the probabilities are within a tenth of a percentage point of 100%, each is divided by their total so they add up exactly, and the page says it did so.
Reading the result
An expected value below zero means that, repeated many times, the outcomes lose on average; above zero, they gain. It says nothing about any single try, and it assumes the probabilities are right. It is the balance point of the diagram: the bars at each value, as tall as their probability, would balance on the dashed line.
With your numbers
- E(X) = Σ x × p = 100 × 0.1 + -5 × 0.9 = 5.5
Quick reference
An average over many repeats. Any one outcome can be far from it, as the standard deviation shows.
| Example | Expected value, E(X) |
|---|---|
| 100 at 10%, −5 at 90% | 5.5 |
| A fair die, 16.6667% each | 3.5 |
| 0, 1 and 2 at 0.5, 0.25 and 0.25 | 0.75 |
| 495 at 0.2%, 45 at 2%, −5 at 97.8% | -3 |
| A coin toss, win 1 or lose 1 | 0 |
| Three outcomes at 20%, 30% and 50% | 23 |
Common questions about the Expected Value Calculator
- What is an expected value?
- The long-run average of a random result: each possible value times its probability, added up. A ticket that pays 100 with a 10% chance and loses 5 otherwise has an expected value of 100 × 0.1 − 5 × 0.9 = 5.5. No single ticket pays 5.5; it is what the tickets average over many plays.
- Why must the probabilities add up to 100%?
- Because the rows should list every possible outcome, and one of them always happens. If they add up to less, an outcome is missing; if more, one is counted twice or a probability is wrong. The page allows a tenth of a percentage point either way for rounding, such as six outcomes at 16.6667%, and refuses anything further off.
- What do the variance and standard deviation add?
- How spread out the outcomes are around the expected value. Two choices can share an expected value of zero while one swings by 1 either way and the other by 1,000. The standard deviation, the square root of the variance, is in the same units as the values.
- Why is the column called weight?
- The expected value is a weighted average of the outcomes with the probabilities as weights, and the row editor is shared with the weighted average pages. Enter each outcome's probability in that column, in the form chosen above.