Variance

The variance of a variable X, denoted $\text{Var}(X)$ or $\sigma^2$, is defined as the arithmetic mean of the squares of the deviations of all the observations from their own arithmetic mean: $\text{Var}(X) = \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2$ Working directly with this definition means computing every deviation $(x_i-\bar{x})$ and squaring it, which can be tedious. A more convenient computational formula is obtained by expanding the square: $\text{Var}(X) = \frac{1}{n}\sum_{i=1}^{n}(x_i^2 - 2x_i\bar{x} + \bar{x}^2) = \frac{1}{n}\sum x_i^2 - 2\bar{x}\cdot\frac{1}{n}\sum x_i + \bar{x}^2\cdot\frac{1}{n}\sum 1$ Since $\frac{1}{n}\sum x_i = \bar{x}$ and $\frac{1}{n}\sum 1 = 1$ (there are n terms, each equal to 1, divided by n), this becomes $\text{Var}(X) = \frac{1}{n}\sum x_i^2 - 2\bar{x}^2 + \bar{x}^2 = \frac{1}{n}\sum x_i^2 - \bar{x}^2$ So the two formulas $\sigma^2 = \frac{1}{n}\sum(x_i-\bar{x})^2$ and $\sigma^2 = \frac{1}{n}\sum x_i^2 - \bar{x}^2$ are math …