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Statistics & Data Science · pattern F, Dataset

Expected value and variance of a discrete distribution.

Enter each outcome with its probability, one per line, for the expected value, variance and standard deviation — with a check that the probabilities sum to one.

Inputs

Sample or population

Divide by n − 1 for a sample drawn from a wider population; by n only when the data is the entire population.

The formula used

E(X) = Σ x·P(x), Var(X) = Σ P(x)(x − E(X))²

One outcome per line as value, probability. Probabilities should sum to 1; the panel reports the sum so a missing outcome is visible.

Expected value

21.0000

5 outcomes, probabilities summing to 1.0000.

Expected value E(X)
21.00000
Variance
169.00000
Standard deviation
13.00000
Probabilities sum
1.00000
Highest-probability outcome
20.000
E(X²)
610.0000

Worked steps

  1. 10.000 × 0.1000 = 0.0000

    each outcome weighted by its probability

  2. 210.000 × 0.2000 = 2.0000

    each outcome weighted by its probability

  3. 320.000 × 0.4000 = 8.0000

    each outcome weighted by its probability

  4. 430.000 × 0.2000 = 6.0000

    each outcome weighted by its probability

  5. 550.000 × 0.1000 = 5.0000

    each outcome weighted by its probability

  6. 6E(X) = 21.00000

    the weighted sum — not the average of the outcomes

Expected value is a long-run average, not a prediction.

An expected value of 22 does not mean any single outcome will be 22, and it may not even be a possible outcome. It is what the average of many repetitions converges to, which is why it governs decisions taken repeatedly and misleads when applied to a one-off with a catastrophic downside. Variance is what tells you how far individual results scatter around it.

Questions about expected value

Can the expected value be an impossible outcome?
Yes, routinely. The expected number of heads in three fair tosses is 1.5, which no single trial can produce.
Why must probabilities sum to one?
Because the outcomes have to be exhaustive. If they sum to less, an outcome is missing and the expectation is understated in proportion.
How does this relate to a fair game?
A game is fair when the expected gain is zero. Every commercial betting market runs a negative expected value for the customer, which is the house margin.

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