Variance¶

Formula¶

\[ \operatorname{Var}(X) = \mathbb{E}\big[(X-\mu)^2\big] = \mathbb{E}[X^2] - \mu^2 \]

Parameters¶

\(X\): random variable
\(\mu = \mathbb{E}[X]\): mean

What it means¶

Measures the average squared deviation from the mean.

What it's used for¶

Measuring spread around the mean.
Risk and uncertainty quantification.

Key properties¶

\(\operatorname{Var}(X)\ge 0\)
\(\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)\)
\(\operatorname{Var}(X)=0\) iff \(X\) is almost surely constant

Common gotchas¶

Variance depends on squared units of \(X\).
\(\mathbb{E}[X]^2\neq\mathbb{E}[X^2]\); don't drop the square.

Example¶

For \(X\in\{0,1\}\) fair coin, \(\operatorname{Var}(X)=0.25\).

How to Compute (Pseudocode)¶

Input: sample data (and any reference values needed by the statistic)
Output: statistic value

compute the summary quantities required by the formula (for example, mean, deviations, counts)
apply the statistic formula from the card
return the result

Complexity¶

Time: Typically \(O(n)\) for \(n\) samples for common one-pass or two-pass summary-statistic computations (sorting-based medians are \(O(n\log n)\) unless selection is used)
Space: \(O(1)\) to \(O(n)\) depending on whether values must be stored/sorted
Assumptions: Sample-statistic workflow shown; parameter-estimation and streaming/online algorithms can change constants and memory usage