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A measure of spread for a distribution of a random variable that determines the degree to which the values of a random variable differ from the expected value.
The variance of random variable X is often written as Var(X) or σ2 or σ2x.
For a discrete random variable the variance is calculated by summing the product of the square of the difference between the value of the random variable and the expected value, and the associated probability of the value of the random variable, taken over all of the values of the random variable.
In symbols, Var(X) = (x - µ)2 P(X = x)
An equivalent formula is, Var(X) = E(X2) – [E(X)]2
The square root of the variance is equal to the standard deviation.
Example
Random variable X has the following probability function:
| x | 0 | 1 | 2 | 3 |
| P(X = x) | 0.1 | 0.2 | 0.4 | 0.3 |
Using Var(X) = (x - µ)2 P(X = x)
µ = 0 x 0.1 + 1 x 0.2 + 2 x 0.4 + 3 x 0.3
= 1.9
Var(X) = (0 – 1.9)2x 0.1 + (1 – 1.9)2x 0.2 + (2 – 1.9)2x 0.4 + (3 – 1.9)2x 0.3
= 0.89
Using Var(X) = E(X2) – [E(X)]2
E(X) = 0 x 0.1 + 1 x 0.2 + 2 x 0.4 + 3 x 0.3
= 1.9
E(X2) = 02 × 0.1 + 12 × 0.2 + 22 × 0.4 + 32 × 0.3
= 4.5
Var(X) = 4.5 – 1.92
= 0.89
See: population variance, variance
Curriculum achievement objectives reference
Probability: (Level 8)
