1.7 Summation

The summation 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.

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

Exercise 1.5 Consider the database available in the coronavirus11 package as per the following code.

library(coronavirus)    # calling the library 'coronavirus'
data(coronavirus)       # leaving the database available
dim(coronavirus)        # database dimensions (rows x columns)
## [1] 973836     15
head(coronavirus)       # showing the first 6 rows of the database
##         date province country     lat      long      type cases   uid iso2
## 1 2020-01-22  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
## 2 2020-01-23  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
## 3 2020-01-24  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
## 4 2020-01-25  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
## 5 2020-01-26  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
## 6 2020-01-27  Alberta  Canada 53.9333 -116.5765 confirmed     0 12401   CA
##   iso3 code3    combined_key population continent_name continent_code
## 1  CAN   124 Alberta, Canada    4413146  North America             NA
## 2  CAN   124 Alberta, Canada    4413146  North America             NA
## 3  CAN   124 Alberta, Canada    4413146  North America             NA
## 4  CAN   124 Alberta, Canada    4413146  North America             NA
## 5  CAN   124 Alberta, Canada    4413146  North America             NA
## 6  CAN   124 Alberta, Canada    4413146  North America             NA
  1. Get the sum of cases registered over the entire period.
  2. Obtain the squared sum of cases recorded over the entire period.
  3. Get the sum of cases registered over the entire period divided by type.
  4. Considering the variable \(X\): ‘number of registered cases’ in nrow(coronavirus) database rows, represent the items a. and b. using summation notation.
  5. Get the number of days between the first record available in date and the current date using the lubridate package.

  1. Johns Hopkins University Center for Systems Science and Engineering (JHU CCSE). https://systems.jhu.edu/research/public-health/ncov↩︎