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Variance and Standard Deviation

Expectation gives you a fair price. It says nothing about risk, and two positions with identical expected value can be wildly different trades. Variance is what separates them.

Variance is the expected squared deviation from the mean:

Variance
Var(X)=E[(X−μ)2]\text{Var}(X) = E\left[(X - \mu)^2\right]

The average squared distance from the mean, which is why its units are squared and the standard deviation exists at all.

Standard deviation is its square root, σ=Var(X)\sigma = \sqrt{\text{Var}(X)}, and is usually the number quoted because it carries the same units as XX. If XX is a P&L in dollars, variance is in dollars squared, which means nothing intuitively, while σ\sigma is in dollars.

The formula you should actually use

Computing E[(X−μ)2]E[(X-\mu)^2] directly requires two passes. This identity does it in one and is the version to have memorised:

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