Expected Value Calculator — mean, variance and SD of a draw

The expected value of a discrete random variable is the long-run average outcome: each possible value multiplied by its probability, all summed. This calculator computes it from a table of outcomes and probabilities, and returns the variance and standard deviation alongside it, because an expected value on its own says nothing about how far individual results are likely to fall from it.

Arb Digital built this page as the distribution-level companion to the summary tools elsewhere in the library. Our weighted average calculator performs arithmetic that looks almost identical — values times weights, divided by total weight — but it describes data you already observed. This page describes a variable that has not been drawn yet, and that difference in meaning is what makes the variance and standard deviation on this page worth reading rather than ignoring.

What This Expected Value Calculator Does

It takes a finite list of outcomes and their probabilities and computes the first two moments of the distribution: the mean, written E(X), and the variance, written Var(X). It also reports the standard deviation, the sum of the probabilities as a validity check, and the single most likely outcome, which is the mode of the distribution and is frequently a very different number from the mean.

Probabilities can be entered as fractions, decimals or percentages, and if they do not sum to 1 the calculator can rescale them for you. That makes it usable directly with raw counts — enter how many times each outcome occurred and the tool converts them to an empirical distribution. The probability bar chart underneath shows the shape those numbers describe.

How to Use It

1. List the outcomes. These are the numeric values the variable can take, in any order, separated however you like.
2. List the probabilities in the same order. Fractions such as 1/6, decimals such as 0.1667 and percentages such as 16.67% all work.
3. Decide whether to rescale. Leave rescaling on if you are entering counts or weights; turn it off if you want the sum checked strictly.
4. Read the expected value. This is the average outcome over many repetitions, not a prediction of any single draw.
5. Read the standard deviation next to it. It tells you the typical distance between an actual outcome and that expected value.

The Formula and How It's Calculated

The expected value is E(X) = Σ xᵢ·pᵢ — every outcome multiplied by its probability, summed across the whole distribution. The variance is Var(X) = Σ pᵢ·(xᵢ − E(X))², the probability-weighted average of the squared distances from the mean, which is equivalently E(X²) − [E(X)]². The standard deviation is the square root of the variance and is expressed in the same units as the outcomes.

Take the default, a fair six-sided die. Each face has probability one sixth, so E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. For the variance, E(X²) = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6 = 15.1667, so Var(X) = 15.1667 − 3.5² = 15.1667 − 12.25 = 2.9167, and the standard deviation is 1.7078. A typical roll therefore lands a little under two away from 3.5, which matches the intuition that a die is genuinely spread out rather than clustered near its middle. Wolfram MathWorld's entry on expectation values states the same definition in its general form.

The Expected Value Is Often Impossible

A fair die has an expected value of 3.5 and no roll can ever produce 3.5. A household has an expected 2.4 people and no household contains 2.4 people. This is not a quirk of the arithmetic; it is what an average of a discrete variable is. The expected value is a balance point, the place where the probability-weighted distances on either side cancel out, and there is no requirement for that point to coincide with an achievable outcome.

The mistake this leads to is planning capacity against the expected value alone. If the expected number of daily support tickets is 12, staffing for exactly 12 leaves you short roughly half the time, because on any given day the actual number is drawn from a distribution with real spread. The expected value tells you the long-run average; the standard deviation tells you how badly a single day can deviate from it, and any decision about a single instance needs both. That is why this calculator refuses to display the mean alone.

Why Variance Is the Number That Changes Decisions

Two distributions can share an expected value and be nothing alike. A variable that always returns exactly 10 and a variable that returns 0 or 20 with equal probability both have an expected value of 10. The first has zero variance; the second has a variance of 100 and a standard deviation of 10. Anyone treating those two as equivalent because the means match is ignoring the entire difference between them.

The gap matters most when the consequences are not symmetric — when a shortfall costs more than an equivalent surplus gains, or when a single bad draw ends the sequence entirely and no long run exists to average over. That is why the standard deviation appears in the grid rather than being tucked away, and why the probability bars are drawn: shape carries information that a single number cannot. To measure the same spread on data you have already collected rather than on a distribution, use our variance calculator or the standard deviation calculator.

Linearity: The Property That Makes Expectation Easy

Expected value is linear, and the practical power of that is easy to underrate. E(aX + b) = a·E(X) + b, so scaling and shifting a variable scales and shifts its mean in exactly the way you would hope. Better still, E(X + Y) = E(X) + E(Y) holds for any two variables, whether or not they are independent. The expected total of ten dice is simply ten times 3.5, or 35, with no joint distribution needed.

Variance does not behave nearly as well. Var(aX) = a²·Var(X), so the squared factor means doubling a variable quadruples its variance. And Var(X + Y) = Var(X) + Var(Y) only when the two are independent; when they are correlated, a covariance term appears and the sum can be much larger or much smaller than the parts suggest. This asymmetry between the two rules is the source of a great many modelling errors — adding means is always safe, adding variances is not. Our covariance calculator measures that missing term when the variables move together.

Entering Counts Instead of Probabilities

The rescaling option turns this into an empirical distribution tool. If you observed an outcome of 0 tickets on 40 days, 1 ticket on 30 days, 2 on 20 and 3 on 10, type those counts straight into the probability field and leave rescaling on. The calculator divides each by the total of 100 and computes the mean and variance of the observed distribution.

Two cautions come with that convenience. First, an empirical distribution built from a small number of observations is a noisy estimate of the real one, and outcomes that simply never happened to occur get a probability of zero rather than a small positive number — which can badly understate the variance if the missing outcomes are the extreme ones. Second, if you turn rescaling off and the probabilities do not sum to 1, the calculator flags it rather than silently producing a meaningless number, because a distribution whose probabilities sum to 0.9 is not a distribution. Penn State's STAT 200 elementary statistics notes cover the axioms this requirement comes from.

Where Expected Value Stops Being the Right Tool

Expected value assumes the long run exists. If a process can be repeated many times and the results accumulate, the average is exactly the right summary. If a single draw can end the sequence — a bankruptcy, a total loss, an irreversible outcome — then averaging over repetitions that will never happen is a category error, and the expected value can point confidently in a direction no reasonable participant would go.

There is also a mathematical edge case worth knowing: some distributions have no finite expected value at all. The St. Petersburg game, where the payout doubles each round, has an infinite expectation because the shrinking probabilities are exactly offset by the growing payouts, and the sum never converges. This page handles finite lists of outcomes, so that case cannot arise here, but it is a reminder that "the expected value" is not always a number that exists. For repeated independent trials with two outcomes, our binomial distribution calculator handles the whole distribution at once, and the Poisson distribution calculator covers counts of rare events. The NIST/SEMATECH e-Handbook of Statistical Methods sets out the standard distributions and the moments attached to each.

Common Mistakes to Avoid

• Planning a single event around the expected value — the mean describes the long run, and a single draw lands somewhere in the spread around it.
• Reporting an expected value with no measure of spread — two wildly different distributions can share a mean, and the mean alone cannot distinguish them.
• Letting the probabilities sum to something other than 1 — either rescale them deliberately or fix the input, because the arithmetic is meaningless otherwise.
• Adding variances of correlated variables — that rule only holds under independence, and ignoring the covariance term understates or overstates the true spread.
• Applying expectation to non-repeatable events — when one bad outcome ends the sequence, there is no long run for the average to describe.

Related Free Tools From Arb Digital

Work out the chance of an event with the probability calculator, average observed data with the weighted average calculator, measure spread with the variance calculator, update a probability with new evidence using the Bayes' theorem calculator, or summarise a raw dataset with the descriptive statistics calculator. The free online tools hub lists every statistics calculator we publish.

Frequently Asked Questions

This page explains a statistical calculation for educational purposes only. It is not financial, gambling, medical, or legal advice, and no result from it should be used as the sole basis for a decision.