Expected value calculator

Enter each possible outcome and its percentage probability to find the long-run average and spread.

Enter percentages from 0 to 100. All rows must total 100%.

Probability
%
Probability
%
Probability
%

Expected value E(X)

20

Variance15,600
Standard deviation124.89996

Read explanation below

Contents

What expected value tells you

Several possible outcomes with different probability weights converging on one expected value

Expected value is the probability-weighted average of every possible numeric outcome. It describes the long-run result of repeated trials, not a promise about the next trial.

How to enter a complete distribution

Put one outcome and its probability in each row. Probabilities are percentages from 0 to 100 and must total 100%. Negative outcomes can represent losses.

E(X)=โˆ‘i=1nxipiE(X)=\sum_{i=1}^{n}x_i p_i

The calculator multiplies each outcome by its probability and adds the products. Variance and standard deviation show how widely outcomes sit around the average.

Varโก(X)=โˆ‘i=1npi(xiโˆ’E(X))2\operatorname{Var}(X)=\sum_{i=1}^{n}p_i(x_i-E(X))^2

Three checks with familiar numbers

For -100 at 40%, 0 at 30% and 200 at 30%, E(X) is 20, variance is 15,600 and standard deviation is about 124.9.

A fair die has expected value 3.5 although no face shows 3.5. For 0 at 99% and 100 at 1%, E(X) is 1 and standard deviation is about 9.95.

Rows must be mutually exclusive and cover all possibilities. Keep one unit throughout, include costs in net outcomes, and compare standard deviation as well as the mean.

Reading the answer without hiding risk

Rows must be mutually exclusive and cover all possibilities. Keep one unit throughout, include costs in net outcomes, and compare standard deviation as well as the mean.

Questions about expected value

These answers cover percentages, negative outcomes and interpretation.

Why must probabilities total 100%?

The rows must describe every possible outcome of one trial.

Can an outcome be negative?

Yes. Negative values can represent losses, costs or penalties.

How do I enter 12.5%?

Enter 12.5 without the percent sign.

Can E(X) be absent from the outcomes?

Yes. A weighted average can lie between possible results.

Does a positive expected value guarantee a gain?

No. A finite run can differ substantially from the long-run average.

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