Mathematics

Population variance measures spread around the mean, and its formula $\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}$ can be algebraically rewritten as $\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2$. Since you can compute the mean $\bar{x} = \frac{\sum x}{n}$ from the given sums, you have everything needed to plug into this computational form—no original data list required. This is why summary statistics are so powerful: they compress the information needed for variance into just three numbers.

Population variance measures spread around the mean, and its formula $\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}$ can be algebraically rewritten as $\sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2$. Since you can compute the mean $\bar{x} = \frac{\sum x}{n}$ from the given sums, you have everything needed to plug into this computational form—no original data list required. This is why summary statistics are so powerful: they compress the information needed for variance into just three numbers.