Variance
Variance measures the average squared distance of data values from the mean, indicating how spread out the data is.
Formula
s^2 = \dfrac{\sum (x_i - \bar{x})^2}{n-1}
Definition
Variance is a number that measures how much the values in a dataset vary from the average: a bigger variance means the values are more spread out. The sample variance $s^2 = \sum (x_i-\bar{x})^2/(n-1)$ is the average squared deviation from the sample mean, always non-negative and equal to zero only when all values are identical; taking the square root gives the standard deviation, which is in the original units. Formally, the population variance $\sigma^2 = E[(X-\mu)^2] = E[X^2] - \mu^2$ is a fundamental property of a distribution, and the sample variance is unbiased: $E[s^2] = \sigma^2$. In ANOVA, total variance is decomposed into between-group and within-group components, enabling F-tests.
Example
Scores of $80$, $80$, $80$, $80$ have a variance of $0$ (no spread), while scores of $60$, $70$, $90$, $100$ have a large variance because they are far from the average of $80$. For data $2$, $4$, $4$, $4$, $5$, $5$, $7$, $9$ with mean $5$: squared deviations $9$, $1$, $1$, $1$, $0$, $0$, $4$, $16$ sum to $32$, giving variance $s^2 = 32/7 = 4.57$ and standard deviation $s = \sqrt{4.57} = 2.14$. The law of total variance, $\text{Var}(Y) = E[\text{Var}(Y|X)] + \text{Var}(E[Y|X])$, decomposes variance into an average within-group component and a between-group component, underlying random effects models and the analysis of variance framework.
Key Insight
Variance and standard deviation both measure spread, but standard deviation is in the original units (like inches or points), making it easier to interpret, while variance is in squared units; squaring the deviations eliminates negative signs and penalizes large deviations more heavily than small ones. The Cramer-Rao lower bound states that no unbiased estimator of $\theta$ can have variance less than $1/I(\theta)$, where $I(\theta)$ is the Fisher information, and for normal data $s^2$ achieves this bound, making it the minimum variance unbiased estimator (MVUE) of $\sigma^2$.