Summation Operator

Summation Operator Annotated

The summation operator is represented by the symbol (capital sigma) and represents the operation of summing a sequence of expressions together. The operator is used in math to represent the sum of sequences and series. For example, the statement below represents the summation of the sequence of three numbers.

In plain language, this means sum the sequence of numbers together represented by the expression starting from and iterating until . At each step, the element of the sequence is calculated by substituting the current value of into the expression. The sequence of expressions is shown below.

Expression Variable

The expression on top of the summation operator, in this case , controls the length of the sequence being added together. Expanding the expressions as a summation gives the following.

Adding together the sequence of expressions gives the result.


The summation operator is used to represent sequences and series in math. Calculus makes extensive use of the operator.


Calculate Sum of Counting Numbers

To calculate the summation of the counting numbers 1 + 2 + 3 + ... + n the sequence can be visualized geometrically and solved for by finding the area of the shape formed.


Arithmetic Mean

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.

Expected Value Discrete Distribution

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

Sample Standard Deviation

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.