Objects

Suppose a relatively math unsavvy student asks us for help solving several quadratic equations of the form \(ax^2+bx+c = 0\). You—a savvy student—recall that the quadratic formula gives us the solutions:

\[ \frac{-b + \sqrt{b^2 - 4ac}}{2a}\,\, \mbox{ and } \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]

which of course depend on the values of \(a\), \(b\), and \(c\). That is, the quadratic equation represents a function with three arguments.

One advantage of programming languages is that we can define variables and write expressions with these variables, similar to how we do so in math, but obtain a numeric solution. We will write out general code for the quadratic equation below, but if we are asked to solve \(x^2 + x -1 = 0\), then we define:

a <- 1
b <- 1
c <- -1

which stores the values for later use. We use <- to assign values to the variables.

We can also assign values using = instead of <-, but some recommend against using = to avoid confusion.⁴

To see the value stored in a variable, we simply ask R to evaluate a and it shows the stored value:

a

## [1] 1

A more explicit way to ask R to show us the value stored in a is using print like this:

print(a)

## [1] 1

We use the term object to describe stuff that is stored in R. Variables are examples, but objects can also be more complicated entities such as functions, which are described later.

The workspace

As we define objects in the console, we are actually changing the workspace. You can see all the variables saved in your workspace by typing:

ls()

## [1] "a"      "b"      "c"      "filter" "select"

(Note that one of my variables listed above comes from generating the graphs above). In RStudio Posit, the Environment tab shows the values:

We should see a, b, and c. If you try to recover the value of a variable that is not in your workspace, you receive an error. For example, if you type x you will receive the following message: Error: object 'x' not found.

Now since these values are saved in variables, to obtain a solution to our equation, we use the quadratic formula:

(-b + sqrt(b^2 - 4*a*c) ) / ( 2*a )

## [1] 0.618034

(-b - sqrt(b^2 - 4*a*c) ) / ( 2*a )

## [1] -1.618034

Functions

Once you define variables, the data analysis process can usually be described as a series of functions applied to the data. R includes several zillion predefined functions and most of the analysis pipelines we construct make extensive use of the built-in functions. But R’s power comes from its scalability. We have access to (nearly) infinite functions via install.packages and library. As we go through the course, we will carefully note new functions we bring to each problem. For now, though, we will stick to the basics.

Note that you’ve used a function already: you used the function sqrt to solve the quadratic equation above. These functions do not appear in the workspace because you did not define them, but they are available for immediate use.

In general, we need to use parentheses to evaluate a function. If you type ls, the function is not evaluated and instead R shows you the code that defines the function. If you type ls() the function is evaluated and, as seen above, we see objects in the workspace.

Unlike ls, most functions require one or more arguments. Below is an example of how we assign an object to the argument of the function log. Remember that we earlier defined a to be 1:

log(8)

## [1] 2.079442

log(a)

## [1] 0

You can find out what the function expects and what it does by reviewing the very useful manuals included in R. You can get help by using the help function like this:

help("log")

For most functions, we can also use this shorthand:

?log

The help page will show you what arguments the function is expecting. For example, log needs x and base to run. However, some arguments are required and others are optional. You can determine which arguments are optional by noting in the help document that a default value is assigned with =. Defining these is optional.⁵ For example, the base of the function log defaults to base = exp(1)—that is, log evaluates the natural log by default.

If you want a quick look at the arguments without opening the help system, you can type:

args(log)

## function (x, base = exp(1)) 
## NULL

You can change the default values by simply assigning another object:

log(8, base = 2)

## [1] 3

Note that we have not been specifying the argument x as such:

log(x = 8, base = 2)

## [1] 3

The above code works, but we can save ourselves some typing: if no argument name is used, R assumes you are entering arguments in the order shown in the help file or by args. So by not using the names, it assumes the arguments are x followed by base:

log(8,2)

## [1] 3

If using the arguments’ names, then we can include them in whatever order we want:

log(base = 2, x = 8)

## [1] 3

To specify arguments, we must use =, and cannot use <-.

There are some exceptions to the rule that functions need the parentheses to be evaluated. Among these, the most commonly used are the arithmetic and relational operators. For example:

2 ^ 3

## [1] 8

You can see the arithmetic operators by typing:

help("+")

or

?"+"

and the relational operators by typing:

help(">")

or

?">"

Other prebuilt objects

There are several datasets that are included for users to practice and test out functions. You can see all the available datasets by typing:

data()

This shows you the object name for these datasets. These datasets are objects that can be used by simply typing the name. For example, if you type:

co2

R will show you Mauna Loa atmospheric \(CO^2\) concentration data.

Other prebuilt objects are mathematical quantities, such as the constant \(\pi\) and \(\infty\):

pi

## [1] 3.141593

Inf+1

## [1] Inf

Variable names

We have used the letters a, b, and c as variable names, but variable names can be almost anything. Some basic rules in R are that variable names have to start with a letter, can’t contain spaces, and should not be variables that are predefined in R. For example, don’t name one of your variables install.packages by typing something like install.packages <- 2. Usually, R is smart enough to prevent you from doing such nonsense, but it’s important to develop good habits.

A nice convention to follow is to use meaningful words that describe what is stored, use only lower case, and use underscores as a substitute for spaces. For the quadratic equations, we could use something like this:

solution_1 <- (-b + sqrt(b^2 - 4*a*c)) / (2*a)
solution_2 <- (-b - sqrt(b^2 - 4*a*c)) / (2*a)

For more advice, we highly recommend studying (Hadley Wickham’s style guide)[http://adv-r.had.co.nz/Style.html].

Saving your workspace

Values remain in the workspace until you end your session or erase them with the function rm. But workspaces also can be saved for later use. In fact, when you quit R, the program asks you if you want to save your workspace. If you do save it, the next time you start R, the program will restore the workspace.

We actually recommend against saving the workspace this way because, as you start working on different projects, it will become harder to keep track of what is saved. Instead, we recommend you assign the workspace a specific name. You can do this by using the function save or save.image. To load, use the function load. When saving a workspace, we recommend the suffix rda or RData. In RStudio, you can also do this by navigating to the Session tab and choosing Save Workspace as. You can later load it using the Load Workspace options in the same tab. You can read the help pages on save, save.image, and load to learn more.

Motivating scripts

To solve another equation such as \(3x^2 + 2x -1\), we can copy and paste the code above and then redefine the variables and recompute the solution:

a <- 3
b <- 2
c <- -1
(-b + sqrt(b^2 - 4*a*c)) / (2*a)
(-b - sqrt(b^2 - 4*a*c)) / (2*a)

By creating and saving a script with the code above, we would not need to retype everything each time and, instead, simply change the variable names. Try writing the script above into an editor and notice how easy it is to change the variables and receive an answer.

The answer you get from the 4th and 5th lines will depend on the values of a, b, and c. If you were to type new numbers directly into your console: c = 5.33 then re-run the last two lines, you will get a different answer. Your R “environment” is affected by what is run from a script and by what you type in the console. It is good (and necessary) practice to write all your code in a script (or in your Rmarkdown document), and run from the script. Always. Periodically running a script fresh from the start (clearing everything out of the environment first) is a good idea as well.

Commenting your code

If a line of R code starts with the symbol #, it is not evaluated. We can use this to write reminders of why we wrote particular code. For example, in the script above we could add:

## Code to compute solution to quadratic equation of the form ax^2 + bx + c
## define the variables
a <- 3
b <- 2
c <- -1

## now compute the solution
(-b + sqrt(b^2 - 4*a*c)) / (2*a)
(-b - sqrt(b^2 - 4*a*c)) / (2*a)

n <- 1000
x <- seq(1, n)
sum(x)

## Data types

Variables in R can be of different types. For example, we need to distinguish numbers from character strings and tables from simple lists of numbers. The function class helps us determine what type of object we have:

a <- 2
class(a)

## [1] "numeric"

To work efficiently in R, it is important to learn the different types of variables and what we can do with these.

Data frames

Up to now, the variables we have defined are just one number. This is not very useful for storing data. The most common way of storing a dataset in R is in a data frame. Conceptually, we can think of a data frame as a table with rows representing observations and the different variables reported for each observation defining the columns. Data frames are particularly useful for datasets because we can combine different data types into one object.

A large proportion of data analysis challenges start with data stored in a data frame. For example, we stored the data for our motivating example in a data frame. You can access this dataset by loading the dslabs library and loading the murders dataset using the data function:

library(dslabs)

## Warning: package 'dslabs' was built under R version 4.2.3

## 
## Attaching package: 'dslabs'

## The following object is masked from 'package:gapminder':
## 
##     gapminder

data(murders)

To see that this is in fact a data frame, we type:

class(murders)

## [1] "data.frame"

Examining an object

The function str is useful for finding out more about the structure of an object:

str(murders)

## 'data.frame':	51 obs. of  5 variables:
## $ state : chr "Alabama" "Alaska" "Arizona" "Arkansas" ...
## $ abb : chr "AL" "AK" "AZ" "AR" ...
## $ region : Factor w/ 4 levels "Northeast","South",..: 2 4 4 2 4 4 1 2 2 2 ...
## $ population: num 4779736 710231 6392017 2915918 37253956 ...
## $ total : num 135 19 232 93 1257 ...

This tells us much more about the object. We see that the table has 51 rows (50 states plus DC) and five variables. We can show the first six lines using the function head:

head(murders)

##        state abb region population total
## 1    Alabama  AL  South    4779736   135
## 2     Alaska  AK   West     710231    19
## 3    Arizona  AZ   West    6392017   232
## 4   Arkansas  AR  South    2915918    93
## 5 California  CA   West   37253956  1257
## 6   Colorado  CO   West    5029196    65

In this dataset, each state is considered an observation and five variables are reported for each state.

Before we go any further in answering our original question about different states, let’s learn more about the components of this object.

The accessor: $

For our analysis, we will need to access the different variables represented by columns included in this data frame. To do this, we use the accessor operator $ in the following way:

murders$population

##  [1]  4779736   710231  6392017  2915918 37253956  5029196  3574097   897934
##  [9]   601723 19687653  9920000  1360301  1567582 12830632  6483802  3046355
## [17]  2853118  4339367  4533372  1328361  5773552  6547629  9883640  5303925
## [25]  2967297  5988927   989415  1826341  2700551  1316470  8791894  2059179
## [33] 19378102  9535483   672591 11536504  3751351  3831074 12702379  1052567
## [41]  4625364   814180  6346105 25145561  2763885   625741  8001024  6724540
## [49]  1852994  5686986   563626

But how did we know to use population? Previously, by applying the function str to the object murders, we revealed the names for each of the five variables stored in this table. We can quickly access the variable names using:

names(murders)

## [1] "state"      "abb"        "region"     "population" "total"

It is important to know that the order of the entries in murders$population preserves the order of the rows in our data table. This will later permit us to manipulate one variable based on the results of another. For example, we will be able to order the state names by the number of murders.

Tip: R comes with a very nice auto-complete functionality that saves us the trouble of typing out all the names. Try typing murders$p then hitting the tab key on your keyboard. This functionality and many other useful auto-complete features are available when working in RStudio.

Vectors: numerics, characters, and logical

The object murders$population is not one number but several. We call these types of objects vectors. A single number is technically a vector of length 1, but in general we use the term vectors to refer to objects with several entries. The function length tells you how many entries are in the vector:

pop <- murders$population
length(pop)

## [1] 51

This particular vector is numeric since population sizes are numbers:

class(pop)

## [1] "numeric"

In a numeric vector, every entry must be a number.

To store character strings, vectors can also be of class character. For example, the state names are characters:

class(murders$state)

## [1] "character"

As with numeric vectors, all entries in a character vector need to be a character.

Another important type of vectors are logical vectors. These must be either TRUE or FALSE.

z <- 3 == 2
z

## [1] FALSE

class(z)

## [1] "logical"

Here the == is a relational operator asking if 3 is equal to 2. In R, if you just use one =, you actually assign a variable, but if you use two == you test for equality. Yet another reason to avoid assigning via =… it can get confusing and typos can really mess things up.

You can see the other relational operators by typing:

?Comparison

In future sections, you will see how useful relational operators can be.

We discuss more important features of vectors after the next set of exercises.

Advanced: Mathematically, the values in pop are integers and there is an integer class in R. However, by default, numbers are assigned class numeric even when they are round integers. For example, class(1) returns numeric. You can turn them into class integer with the as.integer() function or by adding an L like this: 1L. Note the class by typing: class(1L)

Factors

In the murders dataset, we might expect the region to also be a character vector. However, it is not:

class(murders$region)

## [1] "factor"

It is a factor. Factors are useful for storing categorical data. We can see that there are only 4 regions by using the levels function:

levels(murders$region)

## [1] "Northeast"     "South"         "North Central" "West"

In the background, R stores these levels as integers and keeps a map to keep track of the labels. This is more memory efficient than storing all the characters. It is also useful for computational reasons we’ll explore later.

Note that the levels have an order that is different from the order of appearance in the factor object. The default in R is for the levels to follow alphabetical order. However, often we want the levels to follow a different order. You can specify an order through the levels argument when creating the factor with the factor function. For example, in the murders dataset regions are ordered from east to west. The function reorder lets us change the order of the levels of a factor variable based on a summary computed on a numeric vector. We will demonstrate this with a simple example, and will see more advanced ones in the Data Visualization part of the book.

Suppose we want the levels of the region by the total number of murders rather than alphabetical order. If there are values associated with each level, we can use the reorder and specify a data summary to determine the order. The following code takes the sum of the total murders in each region, and reorders the factor following these sums.

region <- murders$region
value <- murders$total
region <- reorder(region, value, FUN = sum)
levels(region)

## [1] "Northeast"     "North Central" "West"          "South"

The new order is in agreement with the fact that the Northeast has the least murders and the South has the most.

Warning: Factors can be a source of confusion since sometimes they behave like characters and sometimes they do not. As a result, confusing factors and characters are a common source of bugs.

Lists

Data frames are a special case of lists. We will cover lists in more detail later, but know that they are useful because you can store any combination of different types. Below is an example of a list we created for you:

record

## $name
## [1] "John Doe"
## 
## $student_id
## [1] 1234
## 
## $grades
## [1] 95 82 91 97 93
## 
## $final_grade
## [1] "A"

class(record)

## [1] "list"

As with data frames, you can extract the components of a list with the accessor $. In fact, data frames are a type of list.

record$student_id

## [1] 1234

We can also use double square brackets ([[) like this:

record[["student_id"]]

## [1] 1234

You should get used to the fact that in R there are often several ways to do the same thing. such as accessing entries.⁶

You might also encounter lists without variable names.

record2

## [[1]]
## [1] "John Doe"
## 
## [[2]]
## [1] 1234

If a list does not have names, you cannot extract the elements with $, but you can still use the brackets method and instead of providing the variable name, you provide the list index, like this:

record2[[1]]

## [1] "John Doe"

We won’t be using lists until later, but you might encounter one in your own exploration of R. For this reason, we show you some basics here.

Matrices

Matrices are another type of object that are common in R. Matrices are similar to data frames in that they are two-dimensional: they have rows and columns. However, like numeric, character and logical vectors, entries in matrices have to be all the same type. For this reason data frames are much more useful for storing data, since we can have characters, factors, and numbers in them.

Yet matrices have a major advantage over data frames: we can perform matrix algebra operations, a powerful type of mathematical technique. We do not describe these operations in this class, but much of what happens in the background when you perform a data analysis involves matrices. We describe them briefly here since some of the functions we will learn return matrices.

We can define a matrix using the matrix function. We need to specify the number of rows and columns.

mat <- matrix(1:12, 4, 3)
mat

##      [,1] [,2] [,3]
## [1,]    1    5    9
## [2,]    2    6   10
## [3,]    3    7   11
## [4,]    4    8   12

You can access specific entries in a matrix using square brackets ([). If you want the second row, third column, you use:

mat[2, 3]

## [1] 10

If you want the entire second row, you leave the column spot empty:

mat[2, ]

## [1]  2  6 10

Notice that this returns a vector, not a matrix.

Similarly, if you want the entire third column, you leave the row spot empty:

mat[, 3]

## [1]  9 10 11 12

This is also a vector, not a matrix.

You can access more than one column or more than one row if you like. This will give you a new matrix.

mat[, 2:3]

##      [,1] [,2]
## [1,]    5    9
## [2,]    6   10
## [3,]    7   11
## [4,]    8   12

You can subset both rows and columns:

mat[1:2, 2:3]

##      [,1] [,2]
## [1,]    5    9
## [2,]    6   10

We can convert matrices into data frames using the function as.data.frame:

as.data.frame(mat)

##   V1 V2 V3
## 1  1  5  9
## 2  2  6 10
## 3  3  7 11
## 4  4  8 12

You can also use single square brackets ([) to access rows and columns of a data frame:

data("murders")
murders[25, 1]

## [1] "Mississippi"

murders[2:3, ]

##     state abb region population total
## 2  Alaska  AK   West     710231    19
## 3 Arizona  AZ   West    6392017   232

library(dslabs)
data(murders)

class(murders$region)

Creating vectors

We can create vectors using the function c, which stands for concatenate. We use c to concatenate entries in the following way:

codes <- c(380, 124, 818)
codes

## [1] 380 124 818

We can also create character vectors. We use the quotes to denote that the entries are characters rather than variable names.

country <- c("italy", "canada", "egypt")

In R you can also use single quotes:

country <- c('italy', 'canada', 'egypt')

But be careful not to confuse the single quote ’ with the back quote, which shares a keyboard key with ~.

By now you should know that if you type:

country <- c(italy, canada, egypt)

you receive an error because the variables italy, canada, and egypt are not defined. If we do not use the quotes, R looks for variables with those names and returns an error.

Names

Sometimes it is useful to name the entries of a vector. For example, when defining a vector of country codes, we can use the names to connect the two:

codes <- c(italy = 380, canada = 124, egypt = 818)
codes

##  italy canada  egypt 
##    380    124    818

The object codes continues to be a numeric vector:

class(codes)

## [1] "numeric"

but with names:

names(codes)

## [1] "italy"  "canada" "egypt"

If the use of strings without quotes looks confusing, know that you can use the quotes as well:

codes <- c("italy" = 380, "canada" = 124, "egypt" = 818)
codes

##  italy canada  egypt 
##    380    124    818

There is no difference between this function call and the previous one. This is one of the many ways in which R is quirky compared to other languages.

Sequences

Another useful function for creating vectors generates sequences:

seq(1, 10)

##  [1]  1  2  3  4  5  6  7  8  9 10

The first argument defines the start, and the second defines the end which is included. The default is to go up in increments of 1, but a third argument lets us tell it how much to jump by:

seq(1, 10, 2)

## [1] 1 3 5 7 9

If we want consecutive integers, we can use the following shorthand:

1:10

##  [1]  1  2  3  4  5  6  7  8  9 10

When we use these functions, R produces integers, not numerics, because they are typically used to index something:

class(1:10)

## [1] "integer"

However, if we create a sequence including non-integers, the class changes:

class(seq(1, 10, 0.5))

## [1] "numeric"

Subsetting

We use square brackets to access specific elements of a vector. For the vector codes we defined above, we can access the second element using:

codes[2]

## canada 
##    124

You can get more than one entry by using a multi-entry vector as an index:

codes[c(1,3)]

## italy egypt 
##   380   818

The sequences defined above are particularly useful if we want to access, say, the first two elements:

codes[1:2]

##  italy canada 
##    380    124

If the elements have names, we can also access the entries using these names. Below are two examples.

codes["canada"]

## canada 
##    124

codes[c("egypt","italy")]

## egypt italy 
##   818   380

Not availables (NA)

This “topic” seems to be wholly unappreciated and it has been our experience that students often panic when encountering an NA. This often happens when a function tries to coerce one type to another and encounters an impossible case. In such circumstances, R usually gives us a warning and turns the entry into a special value called an NA (for “not available”). For example:

x <- c("1", "b", "3")
as.numeric(x)

## Warning: NAs introduced by coercion

## [1]  1 NA  3

R does not have any guesses for what number you want when you type b, so it does not try.

While coercion is a common case leading to NAs, you’ll see them in nearly every real-world dataset. Most often, you will encounter the NAs as a stand-in for missing data. Again, this a common problem in real-world datasets and you need to be aware that it will come up.

sort

Say we want to rank the states from least to most gun murders. The function sort sorts a vector in increasing order. We can therefore see the largest number of gun murders by typing:

library(dslabs)
data(murders)
sort(murders$total)

##  [1]    2    4    5    5    7    8   11   12   12   16   19   21   22   27   32
## [16]   36   38   53   63   65   67   84   93   93   97   97   99  111  116  118
## [31]  120  135  142  207  219  232  246  250  286  293  310  321  351  364  376
## [46]  413  457  517  669  805 1257

However, this does not give us information about which states have which murder totals. For example, we don’t know which state had 1257.

order

The function order is closer to what we want. It takes a vector as input and returns the vector of indexes that sorts the input vector. This may sound confusing so let’s look at a simple example. We can create a vector and sort it:

x <- c(31, 4, 15, 92, 65)
sort(x)

## [1]  4 15 31 65 92

Rather than sort the input vector, the function order returns the index that sorts input vector:

index <- order(x)
x[index]

## [1]  4 15 31 65 92

This is the same output as that returned by sort(x). If we look at this index, we see why it works:

x

## [1] 31  4 15 92 65

order(x)

## [1] 2 3 1 5 4

The second entry of x is the smallest, so order(x) starts with 2. The next smallest is the third entry, so the second entry is 3 and so on.

How does this help us order the states by murders? First, remember that the entries of vectors you access with $ follow the same order as the rows in the table. For example, these two vectors containing state names and abbreviations, respectively, are matched by their order:

murders$state[1:6]

## [1] "Alabama"    "Alaska"     "Arizona"    "Arkansas"   "California"
## [6] "Colorado"

murders$abb[1:6]

## [1] "AL" "AK" "AZ" "AR" "CA" "CO"

This means we can order the state names by their total murders. We first obtain the index that orders the vectors according to murder totals and then index the state names vector:

ind <- order(murders$total)
murders$abb[ind]

##  [1] "VT" "ND" "NH" "WY" "HI" "SD" "ME" "ID" "MT" "RI" "AK" "IA" "UT" "WV" "NE"
## [16] "OR" "DE" "MN" "KS" "CO" "NM" "NV" "AR" "WA" "CT" "WI" "DC" "OK" "KY" "MA"
## [31] "MS" "AL" "IN" "SC" "TN" "AZ" "NJ" "VA" "NC" "MD" "OH" "MO" "LA" "IL" "GA"
## [46] "MI" "PA" "NY" "FL" "TX" "CA"

According to the above, California had the most murders.

max and which.max

If we are only interested in the entry with the largest value, we can use max for the value:

max(murders$total)

## [1] 1257

and which.max for the index of the largest value:

i_max <- which.max(murders$total)
murders$state[i_max]

## [1] "California"

For the minimum, we can use min and which.min in the same way.

Does this mean California is the most dangerous state? In an upcoming section, we argue that we should be considering rates instead of totals. Before doing that, we introduce one last order-related function: rank.

rank

Although not as frequently used as order and sort, the function rank is also related to order and can be useful. For any given vector it returns a vector with the rank of the first entry, second entry, etc., of the input vector. Here is a simple example:

x <- c(31, 4, 15, 92, 65)
rank(x)

## [1] 3 1 2 5 4

To summarize, let’s look at the results of the three functions we have introduced:

Beware of recycling

Another common source of unnoticed errors in R is the use of recycling. We saw that vectors are added elementwise. So if the vectors don’t match in length, it is natural to assume that we should get an error. But we don’t. Notice what happens:

x <- c(1,2,3)
y <- c(10, 20, 30, 40, 50, 60, 70)
x+y

## Warning in x + y: longer object length is not a multiple of shorter object
## length

## [1] 11 22 33 41 52 63 71

We do get a warning, but no error. For the output, R has recycled the numbers in x. Notice the last digit of numbers in the output.

library(dslabs)
data("murders")

temp <- c(35, 88, 42, 84, 81, 30)
city <- c("Beijing", "Lagos", "Paris", "Rio de Janeiro",
          "San Juan", "Toronto")
city_temps <- data.frame(name = city, temperature = temp)

data("na_example")
str(na_example)

##  int [1:1000] 2 1 3 2 1 3 1 4 3 2 ...

mean(na_example)

## [1] NA

Rescaling a vector

In R, arithmetic operations on vectors occur element-wise. For a quick example, suppose we have height in inches:

inches <- c(69, 62, 66, 70, 70, 73, 67, 73, 67, 70)

and want to convert to centimeters. Notice what happens when we multiply inches by 2.54:

inches * 2.54

##  [1] 175.26 157.48 167.64 177.80 177.80 185.42 170.18 185.42 170.18 177.80

In the line above, we multiplied each element by 2.54. Similarly, if for each entry we want to compute how many inches taller or shorter than 69 inches, the average height for males, we can subtract it from every entry like this:

inches - 69

##  [1]  0 -7 -3  1  1  4 -2  4 -2  1

Two vectors

If we have two vectors of the same length, and we sum them in R, they will be added entry by entry as follows:

\[ \begin{pmatrix} a\\ b\\ c\\ d \end{pmatrix} + \begin{pmatrix} e\\ f\\ g\\ h \end{pmatrix} = \begin{pmatrix} a +e\\ b + f\\ c + g\\ d + h \end{pmatrix} \]

The same holds for other mathematical operations, such as -, * and /.

This implies that to compute the murder rates we can simply type:

murder_rate <- murders$total / murders$population * 100000

Once we do this, we notice that California is no longer near the top of the list. In fact, we can use what we have learned to order the states by murder rate:

murders$abb[order(murder_rate)]

##  [1] "VT" "NH" "HI" "ND" "IA" "ID" "UT" "ME" "WY" "OR" "SD" "MN" "MT" "CO" "WA"
## [16] "WV" "RI" "WI" "NE" "MA" "IN" "KS" "NY" "KY" "AK" "OH" "CT" "NJ" "AL" "IL"
## [31] "OK" "NC" "NV" "VA" "AR" "TX" "NM" "CA" "FL" "TN" "PA" "AZ" "GA" "MS" "MI"
## [46] "DE" "SC" "MD" "MO" "LA" "DC"

temp <- c(35, 88, 42, 84, 81, 30)
city <- c("Beijing", "Lagos", "Paris", "Rio de Janeiro",
          "San Juan", "Toronto")
city_temps <- data.frame(name = city, temperature = temp)

Subsetting with logicals

We have now calculated the murder rate using:

murder_rate <- murders$total / murders$population * 100000

Imagine you are moving from Italy where, according to an ABC news report, the murder rate is only 0.71 per 100,000. You would prefer to move to a state with a similar murder rate. Another powerful feature of R is that we can use logicals to index vectors. If we compare a vector to a single number, it actually performs the test for each entry. The following is an example related to the question above:

ind <- murder_rate < 0.71

If we instead want to know if a value is less or equal, we can use:

ind <- murder_rate <= 0.71

Note that we get back a logical vector with TRUE for each entry smaller than or equal to 0.71. To see which states these are, we can leverage the fact that vectors can be indexed with logicals.

murders$state[ind]

## [1] "Hawaii"        "Iowa"          "New Hampshire" "North Dakota" 
## [5] "Vermont"

In order to count how many are TRUE, the function sum returns the sum of the entries of a vector and logical vectors get coerced to numeric with TRUE coded as 1 and FALSE as 0. Thus we can count the states using:

sum(ind)

## [1] 5

Logical operators

Suppose we like the mountains and we want to move to a safe state in the western region of the country. We want the murder rate to be at most 1. In this case, we want two different things to be true. Here we can use the logical operator and, which in R is represented with &. This operation results in TRUE only when both logicals are TRUE. To see this, consider this example:

TRUE & TRUE

## [1] TRUE

TRUE & FALSE

## [1] FALSE

FALSE & FALSE

## [1] FALSE

For our example, we can form two logicals:

west <- murders$region == "West"
safe <- murder_rate <= 1

and we can use the & to get a vector of logicals that tells us which states satisfy both conditions:

ind <- safe & west
murders$state[ind]

## [1] "Hawaii"  "Idaho"   "Oregon"  "Utah"    "Wyoming"

which

Suppose we want to look up California’s murder rate. For this type of operation, it is convenient to convert vectors of logicals into indexes instead of keeping long vectors of logicals. The function which tells us which entries of a logical vector are TRUE. So we can type:

ind <- which(murders$state == "California")
murder_rate[ind]

## [1] 3.374138

%in%

If rather than an index we want a logical that tells us whether or not each element of a first vector is in a second, we can use the function %in%. Let’s imagine you are not sure if Boston, Dakota, and Washington are states. You can find out like this:

c("Boston", "Dakota", "Washington") %in% murders$state

## [1] FALSE FALSE  TRUE

Note that we will be using %in% often throughout the course

library(dslabs)
data(murders)