Expected Value Calculator

E[X] = Σ pᵢ xᵢ Var(X) = Σ pᵢ (xᵢ − E[X])² σ = √Var(X)

xᵢ

One outcome — the value of the i-th result on the list. It may be negative, and it may be a decimal; the list is a series of numbers separated by semicolons, commas or spaces, up to 200 of them

pᵢ

The probability of that outcome, entered as a percentage from 0 to 100. The two lists are read side by side, so the third probability belongs to the third outcome, and they must be the same length

E[X]

The expected value: each outcome multiplied by its own probability and the products added up. This is the weighted average the panel reports first, and the only number on the page the others are measured against

Var(X)

The variance: for each outcome, the squared distance from the expected value, weighted by that outcome's probability. Squaring is why a single far-off outcome can dominate it

σ

The standard deviation, the square root of the variance. It comes back to the same units as the outcomes themselves, which is what makes it the one to read when you want to know how wide the spread is

Σ

Add up over every outcome on the list. Nothing is weighted twice and nothing is dropped: the probabilities summing to 100% is what makes the whole list a complete description of the possibilities

Use it whenever the outcomes are few enough to list and each one has a probability you can name: a game with a payout table, a warranty that costs a fixed amount and pays out on a known fraction of units, a decision with three scenarios. Reach for the arithmetic average instead when the numbers are observations rather than possibilities — a list of measurements needs a mean, not an expected value, and dividing by the count assumes every value is equally likely; that assumption is exactly what this page replaces with explicit probabilities. Reach for a distribution instead once the outcomes follow a pattern you can describe with a formula: a count of successes out of a fixed number of trials has its own mean, and a Poisson count has a mean equal to its variance, so you do not need to type out every count and its probability. The long run is the frame in which all of these numbers make sense: an expected value is what the average converges to over many repetitions, not a forecast of the next one.

1. Five outcomes with uneven chances

   1. Multiply each outcome by its probability as a fraction: 5 × 0.10 = 0.5, 10 × 0.20 = 2, 20 × 0.30 = 6, 40 × 0.25 = 10, 100 × 0.15 = 15
   2. Add the five products: 0.5 + 2 + 6 + 10 + 15 = 33.5 — that is the expected value
   3. For the variance, subtract 33.5 from each outcome, square it and weight it: 0.10 × 812.25 + 0.20 × 552.25 + 0.30 × 182.25 + 0.25 × 42.25 + 0.15 × 4422.25 = 920.25
   4. The square root of 920.25 is 30.3356, the standard deviation

   The probabilities are entered as percentages, and that is the single easiest thing to get wrong here: reading the 10 as a fraction rather than as ten per cent gives 3.35 instead of 33.5, and nothing about the wrong answer looks wrong. The spread is worth a second look too — a standard deviation of 30.34 against an expected value of 33.5 means the 100 outcome, unlikely as it is at 15%, is doing most of the work, and the variance of 920.25 says the same thing in squared units.

2. A fair die, where the answer cannot be rolled

   1. Six outcomes, each with probability 16.67% — the last one carries 16.65% so that the six add up to exactly 100
   2. Each product is 1 × 0.1667 through 6 × 0.1667, and they add up to 3.4995
   3. The variance is the average squared distance from 3.4995, which comes to 2.916
   4. The square root of 2.916 is 1.7076

   This is the example to keep in mind when an expected value looks impossible: 3.5 is the correct answer for a six-sided die and no face shows it. An expected value is a property of the whole distribution, not a prediction of a single roll. It is also the cleanest case of the standard deviation earning its keep — 1.7076 is measured in pips, the same units as the outcomes, so "the roll lands about 1.7 away from 3.5 on average" is a sentence you can act on, while the variance of 2.916 is in squared pips.

3. An insurance payout, and a negative expected value

   1. Two outcomes: nothing happens, or it does and costs 5,000 — written as −5000
   2. Weight them: 0 × 0.97 + (−5000) × 0.03 = −150
   3. The variance is 0.97 × (0 + 150)² + 0.03 × (−5000 + 150)² = 727,500
   4. Its square root is 852.9361

   A negative expected value is not an error and it is not a reason to avoid the calculation — it is the answer to a real question, and the most common one people bring here: if this event has a 3% chance of costing 5,000, then on average each period costs 150, and a premium above 150 is the price of not carrying the risk yourself. Notice how little the standard deviation of 852.94 tells you on its own: it is dominated by the same rare 3% branch, so it mostly restates that the loss is large and unusual rather than that the outcomes are spread out in an interesting way.

4. Why the weighted average is not the plain average

   1. The four outcomes add up to 160, so their plain average is 40
   2. Weighted instead: 10 × 0.40 + 20 × 0.30 + 30 × 0.20 + 100 × 0.10 = 4 + 6 + 6 + 10 = 26
   3. The variance is 0.40 × 256 + 0.30 × 36 + 0.20 × 16 + 0.10 × 5476 = 664
   4. Its square root is 25.7682

   The two numbers differ by more than a third, and the reason is entirely in the probabilities: the large outcome of 100 carries only 10% of the weight, while the small outcomes of 10 and 20 carry 70% between them. Whenever someone quotes an average without saying whether it is weighted, this is the gap that hides behind the word. It is also why a probability of 0 is a meaningful entry rather than an empty one: that outcome still occupies its place in the list and contributes nothing, which is the difference between "impossible" and "not asked about".

Three things about this page are easy to misread, and all three are about the inputs rather than the arithmetic. The probabilities must add up to 100%, and the page refuses to fix that for you: silently rescaling a 40/50 split into 44.4/55.6 would produce a perfectly ordinary-looking number that answers a question you did not ask. The tolerance is one half of the last decimal place per entry, which is why three equal shares written as 33.33 each are accepted at 99.99 and a 40/50 pair is not. The two lists must be the same length, because the entries are read side by side — a missing probability is not a zero, it is a misalignment, and the page says so instead of quietly pairing the wrong numbers. Beyond the inputs, the interpretation has limits. An expected value describes a long run of repetitions, so it says nothing useful about a single instance, and it can be dominated by a rare extreme outcome in a way the single number does not reveal — read the standard deviation or the variance next to it for that. It also assumes the probabilities you enter are the real ones: everything here is arithmetic on the numbers you supply, and a precise expected value built on a guessed probability is a precise answer to a different question. Finally, there is no table of common expected values on this page, because such a table cannot see the outcomes and probabilities you entered, and a table that disagrees with the panel above it would be worse than no table.

What does an expected value actually tell me?

It tells you the average outcome you would see if you repeated the same situation many times and averaged the results, weighting each outcome by how often it comes up. For a game with a payout table it is the break-even price: an expected value of −150 per round means a fair price to play is 150, and a price below that is in your favour. What it does not tell you is what happens next. One round produces one outcome from the list, never the average itself, so a positive expected value is entirely compatible with losing every time you play a handful of rounds.

Can the expected value be a number that can never come up?

Yes, and it usually is. Each of the six faces of a fair die has probability 16.67% and the expected value is 3.5, a face the die does not have. The reason is that the expected value is a weighted average of the whole list, not a member of it: it is the balance point of the distribution, and a balance point can sit in empty space between the weights. The same thing happens with a lottery, where the expected value is a small negative number that no ticket can produce.

Why do the probabilities have to add up to 100%?

Because two lists that describe the same set of possibilities have to account for all of them. If your probabilities add up to 90%, either an outcome is missing from the list or one of the probabilities is wrong, and the page has no way to tell which — so it reports the sum instead of guessing. It deliberately does not rescale your numbers to 100%: dividing 40 and 50 by their sum would turn them into 44.4 and 55.6 and produce an answer to a question you did not ask, with nothing on the panel to show that it happened. The tolerance is half of the last decimal place per entry, so three equal shares typed as 33.33 add up to 99.99 and are accepted.

How is this different from the arithmetic average?

The arithmetic average divides by the number of values, which is the same as assuming every value is equally likely. The expected value replaces that assumption with probabilities you choose, so a value with a 10% chance pulls the result ten times less than a value with a 100% chance. The two agree only when the probabilities are all equal — four outcomes at 25% each give the same number both ways. When they disagree, the difference is entirely in the weights, which is why an average quoted without saying whether it is weighted is worth asking about.

What does a negative expected value mean?

It means the outcomes on the losing side are large enough or likely enough to outweigh the rest, so repeating the decision averages out to a loss. It is not the same as saying you will lose: a 3% chance of a 5,000 loss against a 97% chance of nothing has an expected value of −150, and most periods cost nothing at all. What the negative number does say is what the risk costs on average, which is the number to compare against a premium, a price or an insurance quote. It is the field where this calculation is most often useful, and a negative answer there is a result rather than a failure.

Why is there no table of expected values on this page?

Because the answer depends entirely on the two lists you type in, and a reference table cannot see them. The table people want is a short list of familiar cases — a die, a lottery ticket, a roulette bet — but every one of those has its own outcomes and probabilities, so a printed row would be answering a different question from the one on the panel, and the two would sometimes disagree in public. The panel is the table: change the outcomes and the probabilities and the same three rows recompute. Pages that genuinely benefit from a fixed table are the ones whose table does not depend on the inputs, such as a grading scale or a set of cut-off ranges.

• Measures of the Center of the Data — Introductory Statistics 2e, section 2.5 (the mean as the balance point of a list of values, and how a single extreme value moves it — the reason an expected value can sit far from every outcome on the list) — OpenStax, Rice University
• Measures of the Spread of the Data — OpenStax, Introductory Statistics 2e, section 2.7 (states that the variance is the average of the squares of the deviations, which is the quantity this page weights by the probabilities instead of counting equally) — OpenStax, Rice University
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (what it means for an outcome to have a distribution, and how probabilities are read off the outcomes it assigns them to) — National Institute of Standards and Technology (NIST)