Variance of a random variable
The variance Var(X) measures spread around the expected value: Var(X) = E(X squared) - (E(X))^2. The standard deviation is its square root.
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Analogy
Two games can both pay 10 on average. The one with wild swings has the larger variance.
Example
x: 0, 1, 2; P: 0.5, 0.3, 0.2E(X) = 0.7E(X squared) = 0 + 0.3 + 0.8 = 1.1Var = 1.1 - 0.49 = 0.61SD = square root of 0.61 = 0.78
In practice
Variance measures risk. Investments with the same expected return can differ hugely in variance.
Practice
Flashcards
- Formula for variance of X?Var(X) = E(X squared) - (E(X))^2.
Questions people ask
Why is variance measured in squared units?
Variance averages squared gaps, so the units are squared too: metres become square metres. That is awkward to interpret, which is why we usually quote the standard deviation, the square root of the variance, in the original units. Then a spread of 2 metres is easy to picture.
What does a variance of zero mean?
It means the variable never changes: every outcome equals the expected value. There is no randomness at all. Any real chance process has positive variance. A larger variance means outcomes are more likely to land far from the average, so the same expected value can hide very different risks.