Variance

Variance measures spread by adding up the squared distances from every value to the mean and then dividing. For a sample:

Read $s^2$ as s squared, where $x_i$ are the observations, $\bar{x}$ (x bar) is the sample mean and $n$ is the sample size. For a full population the symbol becomes $\sigma^2$ (sigma squared), the mean becomes $\mu$ (mu), and the divisor is the population size rather than $n - 1$.

Work through 6, 9, 10, 12, 13. The mean is 10, so the deviations are -4, -1, 0, 2 and 3. Squared, those are 16, 1, 0, 4 and 9, summing to 30. Treated as a sample, $s^2 = 30 / 4 = 7.5$. Treated as an entire population, $\sigma^2 = 30 / 5 = 6$. On small data sets the gap is far larger than a rounding difference, so the sample-or-population question has to be settled before the arithmetic starts.

Squaring is also what makes variance awkward to report. If those five numbers are minutes, the variance is 7.5 square minutes, a unit with no physical meaning. Taking the square root undoes that and returns the standard deviation, $s = \sqrt{7.5} = 2.74$ minutes, which is why AP Statistics quotes $s$ in almost every context and keeps $s^2$ for the algebra underneath.

Two properties fall straight out of the definition. Variance is never negative, and it equals 0 exactly when every value is identical to the mean, since that is the only way for all the squared terms to vanish. And because the distances are squared before they are added, one far-out observation contributes out of all proportion to the rest, so variance is not resistant. When outliers are present the interquartile range is the spread measure to reach for, and the median rather than the mean is the matching center.

Variance appears directly in one of finance's standard measures. An asset's beta is the covariance of its returns with the market's, divided by the variance of the market's returns, so the denominator is exactly the quantity defined here: beta.