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.
Understanding Statistics: Variance of a Random Variable
The squaring step has two important effects. First, values below the expected value have negative deviations, while values above it have positive deviations. If these were simply added, they would cancel out and hide the spread.
Squaring makes every contribution nonnegative. Second, it gives unusually distant outcomes much more influence. An outcome twice as far from the mean contributes four times as much to the variance.
This is useful when extreme outcomes are important, such as a large insurance claim, a severe storm, or a big error in a measurement. It also means variance can be strongly affected by rare outliers.
Probabilities control how much each possible outcome matters. A very distant outcome with a tiny probability may contribute less than a moderately distant outcome that occurs often. For example, a machine may usually make parts close to the target size but occasionally make a badly wrong part.
That occasional error can raise the variance noticeably. In a probability distribution, variance describes the long-run pattern across many repetitions.
It does not predict the exact result of one trial. A high variance process can still produce a typical result on any particular day.
Variance has squared units, which students should notice. If a random variable measures length in centimetres, its variance is measured in square centimetres. If it measures time in seconds, its variance is measured in square seconds.
Squared units can feel less natural when describing a real situation. This is why standard deviation is commonly reported alongside variance. Standard deviation returns to the original unit, making comparisons easier.
A scientist can compare the consistency of repeated measurements, while a factory can compare variation in product sizes. In both cases, a lower spread often means more predictable results, though the acceptable amount depends on the task.
The shortcut method can save time, especially when a table has many outcomes. Find the expected value of the squared outcomes, then subtract the square of the expected value. Keep the order of operations clear, since squaring the mean is not the same as averaging squared values.
A useful check is that variance can never be negative. A negative answer usually signals an arithmetic mistake, a missing probability, or rounding too early. Another important rule concerns changes of scale.
Adding a fixed amount to every outcome shifts the center but leaves variance unchanged. Multiplying every outcome by a number multiplies the variance by the square of that number. Converting metres to centimetres therefore makes the numerical variance much larger, even though the physical spread has not changed.
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?