Expected Value Calculator

Compute the expected value E[X] from a discrete probability distribution table.

About Expected Value Calculator

The expected value E[X] is the long-run average of a random variable. For a discrete distribution, it is the probability-weighted sum of all possible values: E[X] = Σ xᵢ × P(xᵢ).

Formulas

Expected Value

E[X] = Σ xᵢ × P(xᵢ)

xᵢ: Possible value
P(xᵢ): Probability of that value

Variance

Var(X) = Σ P(xᵢ) × (xᵢ − E[X])²

xᵢ: Value
E[X]: Expected value

Worked example

With the default values below, Expected Value Calculator returns Expected Value E[X] of 3.500000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Values (comma separated): 1, 2, 3, 4, 5, 6
Probabilities (comma separated): 0.1, 0.15, 0.25, 0.25, 0.15, 0.1

Results

Expected Value E[X]: 3.500000
Variance Var(X): 2.050000
Standard Deviation: 1.431782

Step by step

1Values used
2Expected Value
3Variance
4Expected Value E[X]= 3.500000
5Variance Var(X)= 2.050000
6Standard Deviation= 1.431782

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Does the expected value have to be a possible outcome?

No. For example, the expected value of a fair die roll is 3.5, which is not a possible outcome. It represents the long-run average.

What is the relationship between expected value and mean?

For a probability distribution, the expected value IS the mean (μ). For sample data, the sample mean estimates the population expected value.