Summation

Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.

Summation is the operation of adding a sequence of numbers, the result being their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of such elements, summation always produces a well-defined sum.

The summation of the sequence $[1, 2, 4, 2]$ is an expression whose value is the sum of each of the members of the sequence. In the example, $1+2+4+2= 9$. Since addition is associative, the value does not depend on how the additions are grouped. For instance $(1+2)+(4+2)$ and $1+((2+4)+2)$ both have the value $9$; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so changing the order of the terms of a finite sequence does not change its sum.

Notation

There is no special notation for the summation of such explicit sequences as the example above, as the corresponding repeated addition expression will do. If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then a summation operator may be useful or even essential.

For the summation of the sequence of consecutive integers from 1 to 100 one could use an addition expression involving an ellipsis to indicate the missing terms: $1+2+3+4+\dots + 99+100$. In this case the reader easily guesses the pattern; however, for more complicated patterns, one needs to be precise about the rule used to find successive terms. This can be achieved by using the summation notation "$\Sigma$ " Using this sigma notation, the above summation is written as:

$\displaystyle \sum_{i=1}^{100}i$

In general, mathematicians use the following sigma notation: $\displaystyle \sum_{i=m}^{n}a_{i}$

In this notation, $i$ represents the index of summation, $a_i$ is an indexed variable representing each successive term in the series, $m$ is the lower bound of summation, and $n$ is the upper bound of summation. The "$i=m$" under the summation symbol means that the index $i$ starts out equal to $m$. The index, $i$, is incremented by 1 for each successive term, stopping when $i=n$.

Here is an example showing the summation of exponential terms (terms to the power of 2):

$\displaystyle \sum_{i=3}^{6}1^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86$

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:

 $\displaystyle \sum a_{i}^{2}=\sum_{i=1}^{n}a_{i}^{2}$

One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example, the sum of $f(k)$ over all integers $k$ in the specified range can be written as: $\displaystyle \sum_{0\leq k }$

The sum of $f(x)$ over all elements $x$ in the set $S$ can be written as: $\displaystyle \sum_{x\epsilon S}f(x)$