Compute expected value, variance, and standard deviation from outcomes and probabilities
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When you’re making decisions under uncertainty whether it’s evaluating an investment, analyzing a game of chance, or assessing a business opportunity, the expected value tells you what outcome you’d average over many repetitions. It’s the weighted average of all possible outcomes, where each outcome is weighted by its probability. But expected value alone doesn’t tell the whole story; variance and standard deviation measure how much the outcomes can vary from the average.
This expected value calculator handles all three essential statistics in one tool. Enter your outcomes and probabilities, the calculator computes the expected value, variance, and standard deviation. Add as many outcomes as you need, and the tool updates results instantly. Whether you’re a statistics student learning about probability distributions, a gambler analyzing odds, or a business analyst evaluating risk, this calculator delivers accurate results. All calculations run locally in your browser, keeping your data private.
Enter the first outcome in the “Outcome (x)” field, this is the value associated with a specific event.
Enter the probability of that outcome in the “Probability P(x)” field, this must be between 0 and 1.
Add more outcomes using the “Add Outcome” button include every possible outcome of your scenario.
Review the Expected Value (EV), Variance, and Standard Deviation in the results section.
Use the Copy button to save results or Share to send them to others.
Click one of the quick example buttons to load common scenarios like a coin flip, die roll, or game of chance.
The calculator applies the fundamental formulas for expected value, variance, and standard deviation. These are the building blocks of probability and statistics.
Formula: Expected Value E(X) = Σ [x · P(x)]
The expected value is the weighted average of all possible outcomes, where each outcome is weighted by its probability. It’s what you’d expect to get “on average” over many repetitions.
Formula: Variance Var(X) = E(X²) − (E(X))²
Variance measures the spread of the distribution — how far the outcomes tend to deviate from the expected value. It’s the average of the squared deviations from the mean.
Formula: Standard Deviation σ = √Var(X)
Standard deviation is the square root of the variance. It’s in the same units as the outcomes, making it more interpretable than variance.
The calculator validates that probabilities sum to approximately 1 and handles any number of outcomes you add.
Consider a simple game where you flip a fair coin. If it lands heads, you win $1. If it lands tails, you lose $1. What’s the expected value, variance, and standard deviation?
Step 1: Identify the outcomes and probabilities
Heads: x = $1, P = 0.5
Tails: x = −$1, P = 0.5
Step 2: Calculate the expected value
E(X) = (1 × 0.5) + (−1 × 0.5) = 0.5 − 0.5 = 0
Step 3: Calculate the variance
E(X²) = (1² × 0.5) + (−1² × 0.5) = 0.5 + 0.5 = 1
Var(X) = 1 − 0² = 1
Step 4: Calculate the standard deviation
σ = √1 = 1
Interpretation: The expected value is $0 — over many games, you’d break even on average. The variance is 1, and the standard deviation is $1, meaning your actual outcomes typically differ from the expected value by about $1 in either direction.
Expected value is the weighted average of all possible outcomes of a random variable, where each outcome is weighted by its probability. It’s what you’d expect to get “on average” over many repetitions.
The formula is E(X) = Σ [x · P(x)], where x is each outcome and P(x) is its probability. The calculator applies this formula automatically.
Expected value is a weighted average where each outcome is weighted by its probability. A simple average treats all outcomes equally. In a fair coin flip, the expected value is 0, but a simple average of +1 and −1 is also 0, they coincide for symmetric distributions.
Variance is a measure of how spread out the outcomes are from the expected value. It’s the average of the squared deviations from the mean. A higher variance means more uncertainty.
Standard deviation is the square root of the variance. It’s in the same units as the outcomes, making it more interpretable than variance. A standard deviation of $1 means outcomes typically deviate from the expected value by about $1.
A negative expected value means that, on average, you’d lose money over many repetitions. This is common in gambling games where the house has an edge. In decision-making, a negative EV suggests you should avoid the option.
Variance measures the spread of the distribution. A low variance means outcomes are close to the expected value. A high variance means outcomes are spread out, more uncertainty. The calculator provides both variance and standard deviation.
Yes, if the outcomes are negative on average (weighted by probabilities), the expected value will be negative. For example, a game where you win $1 with 40% probability and lose $2 with 60% probability has an expected value of −$0.80.
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