💡 Expected Value Calculator

Calculate the expected value and variance of a discrete probability distribution.

📊 Statistics📐 Formula included🧠 Practice questions✓ Free

Outcomes

Probabilities

Probabilities should sum to 1.

Please enter valid values.

Formula and Explanation

Calculate the expected value and variance of a discrete probability distribution.

Concept	Formula or rule
Expected Value Calculator	E(X)=Σxᵢpᵢ

How to Use This Calculator

Enter the sample, counts, probabilities, or summary statistics requested above.
Keep paired lists in the same order and separate values with commas.
Select the correct test direction or sample/population option when available.
Press Calculate and interpret the result together with the supporting statistics.

Worked Example

Example: A game pays 0 with probability 0.5 and 10 with probability 0.5.

✓ E(X)=0×0.5+10×0.5=5

Where This Is Used

School & coursework

Check homework, understand formulas, and verify step-by-step statistics exercises.

Research

Explore estimates, uncertainty, distributions, and relationships before using full statistical software.

Business & data analysis

Summarize variation, compare groups, and communicate data-driven findings clearly.

Common Mistakes to Avoid

⚠️

Probabilities not summing to 1
A valid distribution must total 1, apart from tiny rounding error.

⚠️

Interpreting expectation as a guaranteed result
Expected value is a long-run average.

⚠️

Ignoring losses
Include negative outcomes when appropriate.

Practice Questions

1. Outcomes 0 and 2 with equal probability. Find EV.

Answer: 1

2. Can EV exceed the maximum outcome?

Answer: No for nonnegative probabilities summing to 1.

3. What must probabilities sum to?

Answer: 1

Frequently Asked Questions

Can expected value be impossible as a single outcome?▾

Yes. It is an average, not necessarily an attainable value.

Can expected value be negative?▾

Yes.

How is variance calculated?▾

Σp(x−E[X])².