Expected Value & Variance Calculator
Expected value, variance and standard deviation of a discrete distribution — the foundation of decision theory.
Expected value is the long-run average outcome — the basis of rational decision-making under uncertainty. But EV alone ignores risk: two choices with the same EV can have wildly different variance. The standard deviation quantifies that spread, and risk-averse decisions weigh both. (Probabilities should sum to 1.)
Formula
About Expected Value & Variance Calculator
Expected value is the long-run average of a random outcome and the cornerstone of rational decision-making under uncertainty — weight each outcome by its probability and sum. This calculator computes the expected value of a discrete distribution along with its variance and standard deviation, the measures of risk that EV alone ignores. Two decisions can share the same expected value yet differ enormously in spread, and sound choices weigh both: the expected payoff and the uncertainty around it. It's the foundational tool for decision theory, insurance, investing and any choice between uncertain options.
How to use Expected Value & Variance Calculator
- 1Enter your values into Expected Value & Variance Calculator — sensible, domain-typical defaults are pre-filled so you see a real result immediately.
- 2The result recomputes live using the formula shown on the page; there is no button to press.
- 3Adjust any input to compare scenarios, then read the worked example to see the substituted numbers.
Why use Expected Value & Variance Calculator?
- ✓Computes Expected Value & Variance instantly in your browser — no sign-up, no upload, no server round-trip.
- ✓100% free and unlimited, with the exact formula shown: E[X] = Σ xᵢ.
- ✓Runs entirely client-side, so every value you enter stays private on your device.
- ✓Live recompute as you type, with a worked example and authoritative references for trust.
Frequently asked questions
What does expected value actually represent?+
The average outcome if you repeated the situation many times — not what happens any single time. A bet with positive expected value is favorable over the long run, but variance means you can still lose in the short run. EV is the right basis for repeated decisions; for one-shot, high-stakes choices, the risk (variance) often matters more than the average.
Why isn't expected value enough for decisions?+
Because it ignores risk. A guaranteed $50 and a 50/50 shot at $0 or $100 have the same EV ($50), but very different risk profiles. Most people (and prudent businesses) are risk-averse and prefer the sure thing — which is why decision theory weighs both expected value and variance, and why utility (not raw dollars) often drives real choices.
What's the difference between variance and standard deviation?+
Variance is the average squared deviation from the mean — it's in squared units (dollars², say), making it hard to interpret directly. Standard deviation is its square root, back in the original units (dollars), so it's the practical measure of spread. A larger SD means outcomes are more spread out and the result is less predictable.
Do the probabilities have to sum to 1?+
Yes — for a valid probability distribution, they must sum to 1 (every possible outcome is accounted for). This tool shows the sum so you can check; if it's not 1, you've either missed an outcome or made an error. Expected value computed from probabilities that don't sum to 1 is meaningless, so the sum is a built-in sanity check.
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