# Summation notation

The upper case Greek Sigma is used to denote that the following expression is to be added to itself or summed over the specified range of values of the index. We shall use it to calculate the mean and variance (hence standard deviation). An example of summation notation follows:
```

4
____
\
\         2             2           2           2
/   ((1-j) + 2) = ((1-2) + 2)+((1-3) + 2)+((1-4) + 2) = 20
/____
j=2

```
If the range of values is not specified (2 to 4 in the above example), sum over all possible values (e.g., for the mean, add up all the values and divide by n). If the index is not specified (j in the above example) sum over the obvious index (usually i).

Leonhard Euler introduced the notation Sigma for summation in 1755 (he also introduced e for the base of the natural logarithm, pi for the ratio of the circumference to diameter of a circle, and i for the square root of -1). Upper case S had been in general use for summation; Gottfried Wilhelm Leibnitz used the elongated S for both summations and integrals.

N.B.: Because of a limited character set, we shall use "*sum*" to denote the summation symbol (Sigma).

Competencies: If x1=2, x2=5, and x3=8, evaluate

```

3
____
\
\            2
/   ( xj - 3)
/____
j=1

```
Reflection: Can you sum an infinite number of numbers?

Challenge: Show that Formula 3.1 on page 149 is equivalent to the definition 3.6 on page 147.