Variance measures how far the values of a random variable typically spread out from its expected value. It matters because two distributions can have the same mean but very different levels of risk, uncertainty, or consistency. A small variance means values cluster near the mean, while a large variance means values are more spread out.
Standard deviation is the square root of variance and is often easier to interpret because it uses the same units as the random variable.
To compute variance from a probability distribution, first find the expected value by weighting each outcome by its probability. Then measure each outcome's deviation from the mean, square that deviation, and weight it by the outcome's probability. Adding these weighted squared deviations gives the variance.
This process turns a probability table into a numerical description of spread, which is useful in science, finance, games of chance, and data analysis.
Key Facts
- Expected value: μ = E(X) = Σ xP(x)
- Variance: Var(X) = σ^2 = Σ (x - μ)^2P(x)
- Shortcut formula: Var(X) = E(X^2) - [E(X)]^2
- Second moment: E(X^2) = Σ x^2P(x)
- Standard deviation: σ = sqrt(Var(X))
- For a valid probability distribution, Σ P(x) = 1 and every P(x) must be between 0 and 1.
Vocabulary
- Random Variable
- A random variable is a variable whose possible values are numerical outcomes of a random process.
- Expected Value
- Expected value is the long-run average value of a random variable over many repetitions.
- Variance
- Variance is the probability-weighted average of the squared distances from the mean.
- Standard Deviation
- Standard deviation is the square root of variance and describes typical spread in the original units.
- Probability Distribution
- A probability distribution lists each possible value of a random variable and the probability of that value occurring.
Common Mistakes to Avoid
- Averaging the outcomes without using probabilities is wrong because more likely outcomes must contribute more to the mean.
- Using x - μ instead of (x - μ)^2 for variance is wrong because positive and negative deviations can cancel out.
- Forgetting to multiply squared deviations by P(x) is wrong because variance is a probability-weighted average, not a simple sum.
- Confusing variance with standard deviation is wrong because variance is in squared units, while standard deviation is in the original units.
Practice Questions
- 1 A random variable X has values 0, 1, and 2 with probabilities 0.2, 0.5, and 0.3. Find E(X), Var(X), and σ.
- 2 A game pays 1 with probability 0.75. Find the expected payout and the variance of the payout.
- 3 Two random variables have the same expected value of 10. Variable A has most probabilities near 10, while Variable B has high probabilities near 0 and 20. Which variable should have the larger variance, and why?