Variance of a Random Variable
Measures the spread of a random variable around its mean.
Formula
Variables
Example
Larger variance = more spread
How it works
Variance tells you how far the values of a random variable tend to sit from its average, but it measures that spread in a mathematically convenient way. The identity Var(X) = E[X^2] - (E[X])^2 comes from expanding E[(X - E[X])^2]: when you square the deviation from the mean, the cross-term combines neatly into the square of the mean. This form is especially useful because it lets you compute variability using just the first and second moments of the distribution.
When to use it
Use this when you know, or can find, the expected value of X and X^2 and want the spread of a distribution without listing every deviation one by one. It is common in probability, statistics, finance, physics, and any problem comparing how variable different random outcomes are.
Worked example, step by step
Let X be a random variable with values 0, 1, and 2. P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4 Step 1: Find E[X] = 0(1/4) + 1(1/2) + 2(1/4) = 0 + 1/2 + 1/2 = 1 Step 2: Find E[X^2] = 0^2(1/4) + 1^2(1/2) + 2^2(1/4) = 0 + 1/2 + 1 = 3/2 Step 3: Use Var(X) = E[X^2] - (E[X])^2 Var(X) = 3/2 - (1)^2 = 3/2 - 1 = 1/2 Final answer: Var(X) = 0.5
Common mistakes to avoid
- using E[X]^2 as if it meant E[X^2]
- forgetting to square the mean when subtracting
- confusing variance with standard deviation, which is the square root of variance
Did You Know?
Standard deviation is the square root of variance.
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