Range, Variance & Standard Deviation

At a glance 10 formulas

1. 1Range$\text{range} = \text{largest} - \text{smallest}$
2. 2Variance and SD$\sigma^2 = \dfrac{\sum (x_i - \bar{x})^2}{n},\quad \sigma = \sqrt{\sigma^2}$$\bar{x} = \text{mean};\ \text{divide by } n,\ \text{not } n - 1$
3. 3Shortcut form$\sigma^2 = \dfrac{\sum x_i^2}{n} - \bar{x}^2$$\text{so } \textstyle\sum x_i^2 = n(\sigma^2 + \bar{x}^2)$
4. 4SD of two values$\sigma = \dfrac{|a - b|}{2}$$a, b = \text{the two values};\ \text{range} = |a - b|$
5. 5Evenly spaced values$\sigma^2 = \dfrac{d^2(n^2 - 1)}{12}$$d = \text{gap},\ n = \text{count};\ 1, 2, \ldots, n:\ \dfrac{n^2 - 1}{12}$
6. 6Shift and scale$x_i \to ax_i + b:\ \sigma \to |a|\,\sigma,\ \sigma^2 \to a^2\sigma^2$$\text{range} \to |a| \times \text{range};\ \text{adding } b \text{ changes no spread}$
7. 7Adding the mean$\text{one more value} = \bar{x}:\ \sigma^2 \to \dfrac{n}{n + 1}\,\sigma^2$
8. 8Two groups merged$\sigma^2 = \dfrac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}$$n_i, \sigma_i = \text{group size, SD};\ d_i = \text{group mean} - \text{combined mean}$
9. 9Coefficient of variation$CV = \dfrac{\sigma}{\bar{x}} \times 100$$\text{lower CV} = \text{more consistent}$
10. 10Largest possible spreadWith every value between $L$ and $H$ and the mean fixed at $m$, the variance is largest when all values sit at $L$ or $H$: then $\sigma^2 = (H - m)(m - L)$.