25. Video: Summation

Aggregations

An aggregation is a way to turn multiple numbers into fewer numbers (commonly one number).

Summation is a common aggregation. The notation used to sum our values is a greek symbol called sigma \Sigma .

Example 1

Imagine we are looking at the amount of time individuals spend on our website. We collect data from nine individuals:

\bold{x_1} = 10, \bold{x_2} = 20 \bold{x_3} = 45 \bold{x_4} = 12 \bold{x_5} = 8 \bold{x_6} = 12, \bold{x_7} = 3 \bold{x_8} = 68 \bold{x_9} = 5

If we want to sum the first three values together in our previous notation, we write:

\bold{x_1} + \bold{x_2} + \bold{x_3}

In our new notation, we can write:

\sum\limits_{i = 1}^3 x_i .

Notice, our notation starts at the first observation ( i=1 ) and ends at 3 (the number at the top of our summation).

So all of the following are equal to one another:

\sum\limits_{i = 1}^3 x_i = \bold{x_1} + \bold{x_2} + \bold{x_3} = 10 + 20 + 45 = 75

Example 2

Now, imagine we want to sum the last three values together.

\bold{x_7} + \bold{x_8} + \bold{x_9}

In our new notation, we can write:

\sum\limits_{i = 7}^9 x_i .

Notice, our notation starts at the seventh observation ( i=7 ) and ends at 9 (the number at the top of our summation).

Other Aggregations

The \Sigma sign is used for aggregating using summation, but we might choose to aggregate in other ways. Summing is one of the most common ways to need to aggregate. However, we might need to aggregate in alternative ways. If we wanted to multiply all of our values together we would use a product sign ** \Pi **, capital Greek letter pi. The way we aggregate continuous values is with something known as integration (a common technique in calculus), which uses the following symbol \int which is just a long s. We will not be using integrals or products for quizzes in this class, but you may see them in the future!