🎲Expected Value Calculator

Calculate expected value, variance, and standard deviation from outcome values and probabilities.

Enter probabilities as decimals between 0 and 1 (e.g., 50% = 0.5). They must sum to 1.

Outcome (xᵢ)Probability (pᵢ)

How to Use the Expected Value Calculator

The Expected Value Calculator computes E[X], variance, and standard deviation for a discrete probability distribution. Enter each outcome value and its corresponding probability (as a decimal between 0 and 1). The probabilities must sum to 1. Add or remove rows to handle 2 to 10 outcomes.

It is used in statistics, game theory, insurance, finance, and anywhere you need to quantify uncertainty. The expected value is the long-run average, variance measures spread, and standard deviation expresses spread in the original units.

Formulas

E[X] = Σ(xᵢ × pᵢ). Var(X) = Σ(pᵢ × (xᵢ − E[X])²). σ = √Var(X). Example: coin flip with outcomes +1 (p=0.5) and −1 (p=0.5) → E[X] = 0, Var = 1, σ = 1.

Frequently Asked Questions

Can I enter probabilities as percentages?

No — enter probabilities as decimals: 50% = 0.5, 30% = 0.3, and so on.

Can outcome values be negative?

Yes. Negative outcomes (e.g., losses) are valid. Example: outcomes −100 and +200 for a risk-return analysis.

Is expected value the same as average?

When all probabilities are equal, the expected value equals the arithmetic mean. Otherwise it is a probability-weighted (i.e., unequal-weight) average.