Averages

At a glance 11 formulas

1. 1Average and total$\text{total} = \text{average} \times \text{count}$
2. 2Assumed mean$\bar{x} = A + \dfrac{\sum (x_i - A)}{n}$
3. 3Evenly spaced numbers$\text{average} = \dfrac{\text{first} + \text{last}}{2}$$1, 2, \ldots, n:\ \dfrac{n + 1}{2}$
4. 4Same change to all$x_i \to ax_i + b \Rightarrow \bar{x} \to a\bar{x} + b$$\text{a fixed group's average age rises 1 a year}$
5. 5Combined average$\bar{x} = \dfrac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$$\dfrac{n_1}{n_2} = \dfrac{\bar{x}_2 - \bar{x}}{\bar{x} - \bar{x}_1}$
6. 6Someone joins$\text{newcomer} = \text{new avg} + n_{\text{old}} \times \text{rise}$
7. 7Someone leaves$\text{leaver} = \text{old avg} + n_{\text{remaining}} \times \text{drop}$
8. 8Replacement$\text{new} = \text{replaced} + n \times \text{rise in avg}$
9. 9Wrong entry$\text{true avg} = \text{avg} + \dfrac{\text{true} - \text{entered}}{n}$
10. 10Overlapping groups$\text{shared term} = S_{\text{first}} + S_{\text{last}} - S_{\text{all}}$
11. 11Largest possible valueTo make one value as large as possible, set every other value to its smallest allowed value (consecutive if they must differ), then subtract from the total.