Expected Value Calculator
Calculate the expected value E[X], variance, and standard deviation of any discrete probability distribution online for free. Enter your outcome–probability pairs and our expected value calculator instantly computes E[X], Var(X), σ, skewness, kurtosis, and a full distribution table — all locally in your browser. No signup required.
Enter each outcome value and its probability. Probabilities must sum to 1. The calculator instantly computes E[X], variance, standard deviation, and higher moments — all locally in your browser.
Why Use Our Expected Value Calculator?
Instant Expected Value Calculation
Compute E[X], variance, and standard deviation for any discrete probability distribution instantly in your browser. Our expected value calculator handles any number of outcomes — from simple coin flips to complex multi-outcome distributions.
Secure Expected Value Calculator Online
All expected value calculations run 100% locally in your browser. Your probability data never leaves your device — use our expected value calculator online with complete privacy and zero data collection.
Expected Value Calculator — No Installation
Use our expected value calculator directly in any browser with no downloads, plugins, or app installs required. Calculate expected value, variance, and standard deviation from any device, anywhere.
Full Distribution Table & Higher Moments
Our expected value calculator shows a complete probability distribution table with x·P(x) and (x−μ)²·P(x) columns, plus E[X²], skewness, kurtosis, and an interpretation panel — all in one place.
Common Use Cases for Expected Value Calculator
Probability & Statistics Coursework
Students learning probability theory use our expected value calculator to verify homework answers and understand how E[X], variance, and standard deviation are derived from a probability distribution. The step-by-step distribution table makes the calculation transparent.
Gambling & Game Theory Analysis
Analyze the expected value of casino games, lotteries, and betting strategies. Enter each outcome and its probability to instantly see whether a game has a positive or negative expected value — the foundation of rational decision-making under uncertainty.
Financial Risk & Investment Analysis
Finance professionals use expected value to evaluate investment returns, option payoffs, and portfolio risk. Enter each scenario (bull, base, bear) with its probability to compute the expected return, variance, and standard deviation of any financial instrument.
Decision Theory & Operations Research
Decision analysts use expected value to compare strategies under uncertainty. Our expected value calculator supports any discrete distribution, making it ideal for decision trees, expected utility calculations, and Bayesian decision analysis.
Business & Product Pricing
Product managers and business analysts use expected value to model demand scenarios, price elasticity, and revenue projections. Enter each demand level and its probability to compute the expected revenue and variance for any pricing strategy.
Insurance & Actuarial Science
Actuaries compute expected claim values and variance to price insurance products. Use our expected value calculator to model claim distributions — enter each payout amount and its probability to get the expected loss and standard deviation instantly.
Understanding Expected Value
What is Expected Value?
The expected value (also called the mean or mathematical expectation) of a discrete random variable X is the probability-weighted average of all possible outcomes. It is written as E[X] = Σ x · P(x) — the sum of each outcome value multiplied by its probability. The expected value represents the long-run average result if the random experiment were repeated infinitely many times. For example, the expected value of a fair six-sided die is (1+2+3+4+5+6)/6 = 3.5 — you will never roll 3.5, but over many rolls the average converges to 3.5. Our expected value calculator computes E[X] along with variance, standard deviation, and higher moments for any discrete probability distribution.
How Our Expected Value Calculator Works
- 1. Enter your outcomes: Add each possible outcome value and its probability in the table. Probabilities must be between 0 and 1, and must sum to exactly 1. Use the quick-example buttons to load a pre-filled distribution.
- 2. Instant browser-based calculation: Click Calculate and the tool computes E[X], E[X²], Var(X), σ, skewness, and kurtosis instantly. All calculations run locally in your browser — your data never leaves your device.
- 3. Review the full distribution table: The results include a complete probability distribution table showing x·P(x) and (x−μ)²·P(x) for each outcome, plus an interpretation panel explaining what the results mean.
Key Statistics Explained
The probability-weighted average of all outcomes. Represents the long-run mean of the distribution.
The expected value of X squared. Used to compute variance via the shortcut formula Var(X) = E[X²] − (E[X])².
Measures how spread out the distribution is around the mean. Higher variance means outcomes are more dispersed.
The square root of variance. Expressed in the same units as X, making it easier to interpret than variance.
Measures asymmetry. Positive skewness means a longer right tail; negative means a longer left tail. Zero means symmetric.
Measures the heaviness of the tails. A normal distribution has kurtosis = 3. Higher values indicate heavier tails (leptokurtic).
Probability Distribution Rules
| Rule | Requirement |
|---|---|
| Non-negativity | Each P(x) must be ≥ 0 |
| Boundedness | Each P(x) must be ≤ 1 |
| Normalization | Σ P(x) = 1 (all probabilities sum to 1) |
| Completeness | All possible outcomes must be listed |
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Frequently Asked Questions About Expected Value Calculator
An expected value calculator computes E[X] — the probability-weighted average of all possible outcomes of a discrete random variable. Our expected value calculator also computes variance, standard deviation, skewness, kurtosis, and shows a full probability distribution table — all running instantly in your browser with no signup required.
Expected value is calculated as E[X] = Σ x · P(x) — multiply each outcome value by its probability, then sum all the products. For example, a fair coin with H=1 (P=0.5) and T=0 (P=0.5) has E[X] = 1×0.5 + 0×0.5 = 0.5. Our expected value calculator does this automatically for any number of outcomes.
A valid probability distribution must account for all possible outcomes, and the total probability of all outcomes must equal 1 (certainty). If your probabilities sum to less than 1, some outcomes are missing. If they sum to more than 1, the distribution is invalid. Our calculator validates this and shows the live probability sum as you type.
For a probability distribution, expected value and mean are the same thing — both equal Σ x · P(x). The term "expected value" is used in probability theory, while "mean" is used in statistics. For a sample dataset (without probabilities), the mean is simply the sum divided by the count.
Variance measures how spread out the outcomes are around the expected value. A high variance means outcomes vary widely from the mean; a low variance means outcomes cluster tightly around the mean. Variance is computed as Var(X) = E[X²] − (E[X])², which our calculator shows in the distribution table.
Our expected value calculator requires probabilities as decimals between 0 and 1 (e.g. 0.25 for 25%). If you have percentages, divide by 100 before entering them. The live probability sum indicator shows whether your entries are valid.
Skewness measures the asymmetry of a distribution. A skewness of 0 means the distribution is symmetric. Positive skewness means a longer right tail (rare large positive outcomes). Negative skewness means a longer left tail. Skewness matters in finance and risk analysis where tail behavior is important.
Yes, completely. All calculations run 100% locally in your browser using JavaScript. Your probability data and outcome values are never sent to any server. Your data never leaves your device, ensuring complete privacy.
Yes. Our expected value calculator is 100% free with no signup, no account, and no usage limits. Calculate expected value, variance, and standard deviation for any probability distribution — completely free, forever.