1.7 Summation

The sum of \(n\) numbers \(x_1, x_2, ..., x_n\) is represented by \(\sum_{i=1}^n {x_i} = x_1 + x_2 + \dotsb + x_n\), and reads ‘sum of x \(i\) from one to n’.

Example 1.6 (Number of steps) Suppose the variable \(X\): ‘number of steps to the nearest trash can’ was observed in the city of Porto Alegre on \(n = 6\) occasions, as shown in the table below.

\(x_{1}\) \(x_{2}\) \(x_{3}\) \(x_{4}\) \(x_{5}\) \(x_{6}\)
186 402 191 20 7 124

This table indicates that on the first occasion, 186 steps were walked to locate a trash can (represented by \(x_1=186\)), on the second occasion, 402 steps were walked (represented by \(x_2=402\)), and so on. To calculate the total number of steps walked, you can do

\[\begin{equation} \sum_{i=1}^6 {x_i} = x_1 + x_2 + \dotsb + x_6 = 186+402+191+20+7+124 = 930 \tag{1.1} \end{equation}\]

186+402+191+20+7+124            # R and RStudio are calculators
## [1] 930
x <- c(186,402,191,20,7,124)    # We can create a vector and assign it to x
sum(x)                          # Using the 'sum' function, presented in Equation (1.1)
## [1] 930
sum(x^2)                        # Sum of squares, represented by Equation (1.2)
## [1] 248506

The Greek letter \(\sum\) is the capital sigma, as per Section 1.9.1. In many cases the summation symbology is simplified, using \(\sum\), \(\sum_{x}\) or \(\sum_{i}\). Below are some more advanced examples of more sophisticated use of summation, which can be omitted on a first read.

\[\begin{equation} \sum_{i=1}^n x_{i}^2 = x_{1}^2 + x_{2}^2 + \ldots + x_{n}^2 \tag{1.2} \end{equation}\]