Sigma notation and the arithmetic mean

\sum just means "add every one of these in." It's the same thing you do when you add up a receipt at the store. Writing it this way works whether there are 5 items or 5,000. Two habits make sums easier to work with. First, a big sum can split into two smaller sums added separately. Add the tax column and the item column on their own. Then add those two totals together. That gives the same answer as adding everything item by item. Second, suppose every item in a sum gets multiplied by the same number. That number can be pulled out front instead of applied one item at a time. One fact to remember for later: subtract the average from every value, then add up how far each value is from the average. The answer is always exactly zero. The amounts above average and the amounts below it cancel out perfectly.

For observations x_1, x_2, \ldots, x_n, the arithmetic mean is \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \cdots + x_n}{n}. \sum_{i=1}^{n} x_i means "add up x_i as i runs from 1 to n" — the subscript and superscript on \sum are just the start and stop of that count, not new mathematics. Three properties of \sum get used constantly once formulas start combining several sums:

• Linearity: \sum_{i=1}^n(x_i+y_i)=\sum_{i=1}^n x_i+\sum_{i=1}^n y_i, and a constant factor pulls straight out: \sum_{i=1}^n c\,x_i=c\sum_{i=1}^n x_i.
• Sum of a constant: \sum_{i=1}^n c = nc — adding the same number to itself n times.
• Deviations from the mean sum to zero: \sum_{i=1}^n(x_i-\bar x)=0. In three lines: split the sum by linearity, \sum(x_i-\bar x)=\sum x_i-\sum\bar x; the second sum is n\bar x by "sum of a constant"; and \sum x_i=n\bar x by the definition of \bar x above — so the difference is n\bar x-n\bar x=0.