# Expected Value Calculator

Free expected value calculator. Calculate the expected value, variance, and standard deviation of a discrete random variable from outcome-probability.

## What this calculates

Calculate the expected value (weighted average) of a discrete random variable. Enter up to 6 outcomes with their probabilities to find E(X), variance, and standard deviation.

## Inputs

- **Outcome 1 Value**
- **Outcome 1 Probability** — min 0, max 1
- **Outcome 2 Value**
- **Outcome 2 Probability** — min 0, max 1
- **Outcome 3 Value**
- **Outcome 3 Probability** — min 0, max 1
- **Outcome 4 Value**
- **Outcome 4 Probability** — min 0, max 1
- **Outcome 5 Value**
- **Outcome 5 Probability** — min 0, max 1
- **Outcome 6 Value**
- **Outcome 6 Probability** — min 0, max 1

## Outputs

- **Expected Value E(X)** — The weighted average of all outcomes.
- **Variance Var(X)** — The variance of the distribution.
- **Standard Deviation** — The standard deviation (square root of variance).
- **Total Probability** — Sum of all probabilities (should equal 1).
- **Status** — formatted as text — Whether the probabilities sum to 1.

## Details

The expected value is the long-run average outcome of a random variable.

Formula:
E(X) = Σ xᵢ × P(xᵢ)

Variance:
Var(X) = Σ xᵢ² × P(xᵢ) - [E(X)]² = E(X²) - [E(X)]²

Requirements

- All probabilities must be between 0 and 1

- Probabilities should sum to 1

Applications: Decision analysis, gambling, insurance, investments, any scenario with uncertain outcomes.

## Frequently Asked Questions

**Q: What does expected value mean?**

A: Expected value is the average result you would get if you repeated an experiment infinitely many times. It is NOT the most likely outcome. For example, the expected value of a fair six-sided die is 3.5, even though you can never actually roll 3.5.

**Q: What if my probabilities do not sum to 1?**

A: If probabilities do not sum to 1, either you have missing outcomes or incorrect probabilities. The calculator will warn you. The mathematical definition requires the total probability to equal exactly 1 for a valid probability distribution.

**Q: How is expected value used in decision-making?**

A: Expected value helps compare options under uncertainty. Calculate the expected value of each option (sum of each outcome's value times its probability) and choose the option with the highest expected value. This is the foundation of expected utility theory.

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Source: https://vastcalc.com/calculators/statistics/expected-value
Category: Statistics
Last updated: 2026-04-21