# Summation Operator

The capital letter Sigma Σ represents the summation operation in mathematics, often called sum for short. A sum has three parts: an initial value, an end value, and the expression being summed. The summation starts at the initial value and iterates, adding one to the value for each iteration, stopping when the value reaches the end value.

For example, the summation from to of the expression shown below is equal to one plus two plus three. Since the expression being summed is the variable i, then i starts at one, iterates to two, iterates to three, and then stops there since it is the ending value:

The summation operator will typically be applied to a set or list of expressions. For example, the summation of the set of numbers can be expressed using the summation operator.

Note, the subscript number next to the variable represent an index of a specific element within the list or set. The ellipsis “” represent elements not specifically shown but that exist within the set.

## Usage

A series is the operation of adding an infinite number expressions together. A series is denoted using the summation operator and the index variable k.

### Formulas

The arithmetic mean, also called the sample mean, is the average of a sample space. To calculate the arithmetic mean sum all the data points in a sample space and then divide by the number of elements.

To calculate the expected value of a discrete distribution multiply all of the events of the distribution with the probability of the element occuring.

The sample standard deviation formula is denoted by the greek lower case sigma symbol in the case of the population and the latin letter s for the sample.

## Examples

### Summation of a Finite Set of Numbers

Let’s say the set of numbers is then the summation of the set would look like.

### Calculate the Average of a list of Numbers

Let’s say that we have a list of 5 numbers and we want to calculate the arithmetic mean or average of the numbers. We can use the formula for the arithmetic mean which can be expressed using the summation operator.

First we set up the formula and expand the summation and find that the average is equal to 2.

### Summation of the Counting Numbers

Given a set of counting numbers what is the summation of the set?