PSTAT 10 Data Science Principles

Lecture 2: Vectors

John Robin Inston

johninston@ucsb.edu

University of California, Santa Barbara

August 4, 2026

Introduction

🔁 Review: Lecture 1

👈 Last lecture we started by going through course logistics 🥱 before looking into:

The programming language R
The IDE RStudio
Quarto documents

We then looked at some basic programming concepts such as:

Data types
R calculations
The assignment operator (<-)

👀 Overview: Lecture 2

👇 This lecture we continue our introduction, looking at

Data structures
Filtering
Recycling
Sorting
Vectorization

Data Structures

🔹 Scalars

What is a scalar?

A scalar data form is an object holding one value.
- For example, from last lecture:

x <- 16 # numeric scalar
y <- "PSTAT 10" # character string scalar
z <- FALSE # Boolean scalar

Vectors, Matrices and Lists

Often we need to use different data structures that contain multiple values of multiple dimensions
- This including (possibly) a variety of data types.

Vectors

Matrices

Data frames

Lists

\[ \vec{x} = \begin{bmatrix} x_1 \\ \vdots \\ x_n\end{bmatrix};\qquad \mathbf{X} = \begin{bmatrix} x_{11} & \cdots & x_{1p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{np}\end{bmatrix}; \qquad L = \{ x,~\vec{x},~X \}. \]

🧮 Vectors

What is a vector?

A vector data form is an object holding multiple values all of the same datatype. — For example all values are numeric, or all strings.
- We therefore say that vectors are atomic.

There are a number of different ways of constructing vector objects in R.

Creating Vectors I

Combine Function `c()`

The simplest way to create a vector in R is using the combine function c().
- This function takes a comma separated list of values and combines them into a vector.
- Use the help function ?c to learn more about the c() function.

# numeric vector
num_vec <- c(1,2,3,4,5)
print(num_vec)

[1] 1 2 3 4 5

# character string vector
char_vec <- c("Dog", "Cat", "Parrot", "Monkey")
print(char_vec)

[1] "Dog"    "Cat"    "Parrot" "Monkey"

# Boolean vector
bool_vec <- c(TRUE, TRUE, FALSE, FALSE, TRUE)
print(bool_vec)

[1]  TRUE  TRUE FALSE FALSE  TRUE

❗Note that all elements of a vector must be of the same data type.
- If you try to combine different data types, R will coerce the values into a single data type (discussed later in this lecture).

Creating Vectors II

🔁 `rep()` Function

The repeat function rep() repeats a value a specified number of times.

# repeating 3 over 7 values
rep_1 <- rep(3, 7)
rep_1

[1] 3 3 3 3 3 3 3

# repeating "dog" over 4 values
rep_2 <- rep("dog", 4)
rep_2

[1] "dog" "dog" "dog" "dog"

🔢 `seq()` Function

The sequence function seq() creates a vector of consecutive values.
- There are a number of different combinations of arguments that can be used to create a vector using seq().

seq_1 <- seq(from = 1, to = 10, by = 1); seq_1

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

seq_2 <- seq(from = 1, to = 10, length.out = 5); seq_2

[1]  1.00  3.25  5.50  7.75 10.00

🧩 Combining Methods

Flexibity of Vector Creation

You can combine methods to efficiently produce any vector you need.
- Simply nest the functions inside one another.
For example, to create a vector of 12 values consisting of the numbers 1 through 4 repeated 3 times:

comb_1 <- rep(c(1,2,3,4), 3)
comb_1

 [1] 1 2 3 4 1 2 3 4 1 2 3 4

Or to create a vector of 2 TRUE values followed by 3 FALSE values:

comb_2 <- c(rep(TRUE, 2), rep(FALSE, 3))
comb_2

[1]  TRUE  TRUE FALSE FALSE FALSE

Or to create a vector of numbers from 1 to 3 in increments of 0.25, followed by two 1.7 values:

comb_3 <- c(seq(1,3,by=0.25), rep(1.7,2))
comb_3

 [1] 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 1.70 1.70

Try to always think about the most efficient way to create a vector, as there are often multiple ways to do so!

Vector Property Functions

📏 `length()` Function

If we want to know how many values are in a vector we can use the length() function.

length(comb_3)

[1] 11

length(rep_2)

[1] 4

🏷️ `typeof()` Function

To find the datatype of our vector we use the typeof() function:

typeof(bool_vec)

[1] "logical"

typeof(comb_1)

[1] "double"

`is.type` Functions

We can also use the is.numeric(), is.character, and is.logical functions to check the datatype of a vector:

is.numeric(comb_1)

[1] TRUE

Double Datatype

Did anybody notice something interesting about the typeof() function output for comb_1?

print(comb_1)

 [1] 1 2 3 4 1 2 3 4 1 2 3 4

typeof(comb_1)

[1] "double"

What is a double?

Numeric datatypes are split into two sub-types:
- Integer type — all integer values.
- Double type - a numeric datatype that can store decimal values.
  - Equivalently, a double is a floating point number.

typeof(seq(1,6))

[1] "integer"

typeof(seq(1,8, by = 2))

[1] "double"

Typically, numeric data is by default saved as a double unless we specify that it should be saved as an integer using the as.integer() function.

💪 Exercise — Coding Vectors

02:00

For the next 2 minutes consider the following example.

Your somewhat unhelpful team mate on a class project reports the result of their survey of 20 members of the public about their pets:

The first 10 people I asked said they had dogs, then it was back and forth between cat and dog for 8 people, and the last two were fish.

Create a vector called pets that contains the 20 values of the survey results.

✅ Solution — Coding Vectors

Combining Methods

The easiest way to create the vector is to combine the rep() and c() functions:

pets <- c(rep("dog", 10), rep(c("cat", "dog"), 4), rep("fish", 2))
print(pets)

 [1] "dog"  "dog"  "dog"  "dog"  "dog"  "dog"  "dog"  "dog"  "dog"  "dog" 
[11] "cat"  "dog"  "cat"  "dog"  "cat"  "dog"  "cat"  "dog"  "fish" "fish"

Side Note on Syntax

Here we have nested functions inside each other and our code is both a bit long and hard to read.
- To make it easier to read we can use line breaks and indentation to make the code more clear:

pets <- c(
  rep("dog", 10),
  rep(c("cat", "dog"), 4),
  rep("fish", 2)
)

Some standard conventions:
- Start new function arguments on a new line and to indent them (use Tab key in RStudio).
- If your code is longer than 80 characters, consider breaking it up into multiple lines.

Vectorization

⚡ Vectorization

What is vectorization?

One of the most powerful features of R is its ability to perform vectorized computations.
- This means that we can perform operations on entire vectors at once.
- Vectorized computations are fast, as they are much more efficient than alternatives such as looping (discussed in Lecture 4).

Vectorized Arithmetic

Recall that we can perform mathematical operations on scalars:

# mathematical operations on scalars
a <- 5
a^3 - 2

[1] 123

We can perform mathematical operations on vectors in the same way, and R will apply the operation to each element of the vector:

# Recall sequence 1
print(seq_1)

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

# mathematical operations
seq_1^3 - 2

 [1]  -1   6  25  62 123 214 341 510 727 998

💪 Exercise - Vectorized Computations

03:00

Suppose a class of 5 students have midterm scores of 73, 82, 86, 86, 94 and final exam scores 85, 71, 83, 92, 92.

Create two vectors named midterm_scores and final_scores storing the midterm and final score values.
Create a vector named final_grade consisting of the students’ final grades which are calculated as follows:

\[ \text{final} = \frac{(2\times \text{midterm}) + (3\times\text{final})}{5}. \]

✅ Solution — Coding Vectors

Defining the Vectors

Use combine function c() to define the vectors:

# define the vectors
midterm_scores <- c(73, 82, 86, 86, 94)
final_scores <- c(85, 71, 83, 92, 92)

Calculating Final Grades

Use vectorized arithmetic to calculate the final grades:

# calculate final grades
final_grade <- (2*midterm_scores + 3*final_scores)/5
print(final_grade)

[1] 80.2 75.4 84.2 89.6 92.8

Indexing

🎯 Accessing Vector Data

What is indexing?

If we need to access specific elements of a vector we do so by indexing.
- In R we use braces vec_1[...].

# recall the vectors we created earlier
print(comb_1)
print(comb_2)
print(comb_3)

 [1] 1 2 3 4 1 2 3 4 1 2 3 4

[1]  TRUE  TRUE FALSE FALSE FALSE

 [1] 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 1.70 1.70

We can index specific values as follows:

# index the 4th value of comb_3
comb_3[4]

[1] 1.75

We can also index multiple values by providing a vector of indices:

# index multiple values
comb_2[c(1,3,5)]

[1]  TRUE FALSE FALSE

comb_1[2:8] # NOTE: use of colon operator (look up)

[1] 2 3 4 1 2 3 4

🎯 Negating & Logical Indexing

Specifying what we don’t want! 👎

We can index everything but certain values by simply negating our index value / vector:

# Index everything but the 2nd value of comb_2
comb_2[-2]

[1]  TRUE FALSE FALSE FALSE

# Index everything but 2nd through 8th values of comb_3
comb_3[-(2:8)] # careful to use parenthesis

[1] 1.0 3.0 1.7 1.7

Boolean Indexing

We can also index using logical statements:

comb_2[c(FALSE, FALSE, TRUE, TRUE, TRUE)]

[1] FALSE FALSE FALSE

This is specifying that we want values 3, 4, and 5 of comb_2 because the logical vector is TRUE for those indices.

❗This will become very important when we look at filtering later in this lecture.

🏷️ Named Vector Indices

From numbers to names

For convenience we can assign names to each vector index.
The names() function can be used to both inspect the current names of vector indices, and assign new names to vector indices.

Grading Excercise Example

Suppose the students from our exercise on grades a few slides ago (in the correct order) are named: Alex, Becca, Ciar, Dobby, and Eliah.
We can inspect the current index names using the names() function (currently empty).

names(final_scores)

NULL

Assign new names to the vector indices using the names() function with the assignment operator:

names(final_scores) <- c("Alex", "Becca", "Ciar", "Dobby", "Eliah")
names(final_scores)

[1] "Alex"  "Becca" "Ciar"  "Dobby" "Eliah"

# Indexing using names
final_scores[c("Alex", "Dobby")]

 Alex Dobby 
   85    92

Filtering

🔍 Filtering

What is filtering?

Instead of indexing specific values we can return all values that meet some logical criteria
- This is called filtering.
- We input some logical expression into the index and any value for which the expression is true is returned.
For example, to return the final grades of students who scored 85% or higher on their final grade:

final_grade[final_grade >= 85]

[1] 89.6 92.8

🤔 Recall Boolean Indexing

Notice that the logical expression is creating the following logical vector:

final_grade >= 85

[1] FALSE FALSE FALSE  TRUE  TRUE

Our indexing is returning the last two elements.

💪 Exercise — Handling Vectors

03:00

For the next 3 minutes complete the following exercise.

The following 8 values are an individual’s monthly grocery bill from January through August:

326, 281, 287, 308, 405, 299, 421, 301

Create a vector called grocery with the 8 values. Check that all 8 are accounted for.
Name the vector indices as Jan, Feb, Mar, Apr, May, Jun, Jul, Aug.
Determine how many months had a grocery bill exceeding 300.
Determine how much was spent on groceries in Jan, Mar and Aug combined.

✅ Solution — Handling Vectors

Define the Vector

# define grocery vector
grocery <- c(326, 281, 287, 308, 405, 299, 421, 301)

Label the Indices

# label indices
names(grocery) <- c("Jan", "Feb", "Mar", "Apr", "May", "June", "Jul", "Aug")

Compute the Number of Months Exceeding 300

# determine how many months had a grocery bill > 300
length(grocery[grocery > 300])

[1] 5

Compute the Sum of Selected Months

# determine how much was spent on groceries in months
sum(grocery[c("Jan", "Mar", "Aug")])

[1] 914

Recycling

♻️ Recycling

What is Recycling?

We have seen the result of adding two numeric vectors of the same length:

c(1,2) + c(3,4)

[1] 4 6

When we add two numeric vectors of different length, R will recycle values on the shorter vector (looping from start to finish) until the computation is complete:

c(1,2) + rep(1,7)

[1] 2 3 2 3 2 3 2

🤔 Recall Vectorized Arithmetic

Recycling is actually what let us subtract a single scalar from an entire vector earlier in the lecture:

seq_1 - 2

 [1] -1  0  1  2  3  4  5  6  7  8

R silently recycles the length-1 vector 2 to match the length of seq_1, subtracting it from every element.

❗ When Recycling Goes Wrong

Multiples of Lengths

Recycling only works cleanly when the longer length is a whole-number multiple of the shorter length.
- If it isn’t, R still recycles — but warns you:

c(1,2,3) + c(1,2)

Warning in c(1, 2, 3) + c(1, 2): longer object length is not a multiple of
shorter object length

[1] 2 4 4

📌 The computation still runs, but this is a common source of silent bugs — always double-check your vector lengths before combining them.

Hint: If in doubt, the modulo operator %% tells you if one length divides evenly into the other — a non-zero result means expect a warning.

length(c(1,2,3)) %% length(c(1,2))

[1] 1

As a general rule it is easier to try to avoid recycling!

Coercion

🔀 What is Coercion?

What is Coercion?

Vector coercion happens when R changes a vector’s data type because you’re combining different data types.
For example, if we try to create a vector consisting of both character strings and numeric values:

combined_vec <- c("cat", "dog", 2, "monkey", 7, 9)
combined_vec

[1] "cat"    "dog"    "2"      "monkey" "7"      "9"

Notice the new vector consists of only character strings — even the numbers 2, 7 and 9 were converted:

typeof(combined_vec)

[1] "character"

🔀 Examples of Coercion

More Combinations

The same thing happens whenever we mix logical, numeric or character values in one vector:

Logical & Numeric Values

bool_num_vec <- c(TRUE, FALSE, TRUE, 12, 5, 6)
bool_num_vec

[1]  1  0  1 12  5  6

Now let’s try combining all three types together:

Logical, Numeric & Character Strings

triple_vec <- c("dog", TRUE, FALSE, 7, 6, "cat")
triple_vec

[1] "dog"   "TRUE"  "FALSE" "7"     "6"     "cat"

📌 We notice an order of priority: (1) Character strings; (2) Numeric; and (3) Boolean.

❗ Why Coercion Matters

A Common Gotcha

Coercion can silently break your calculations if you’re not paying attention:

nums <- c("1", "2", 3)
nums + 1

Error in nums + 1: non-numeric argument to binary operator

📌 Because nums was coerced to character, R can no longer do arithmetic on it — always check typeof() if a calculation errors unexpectedly.

Hint: use as.numeric(), as.character(), or as.logical() to explicitly convert a vector back to the type you need.

Sorting

🔃 Sorting

The `sort()` Function

When dealing with vectors we often wish to sort them into some kind of order — we can do so using the sort() function.

sort(c(5,23,7,87,45,65)) # numerical sorting

[1]  5  7 23 45 65 87

The same function works on character vectors — sorted alphabetically, with an optional decreasing = TRUE argument:

sort(c("a", "g", "b", "e", "z"), decreasing = TRUE) # alphabetical sorting

[1] "z" "g" "e" "b" "a"

And for logical vectors, FALSE is always treated as smaller than TRUE:

sort(c(TRUE, FALSE, TRUE, FALSE, TRUE, FALSE)) # logical sorting

[1] FALSE FALSE FALSE  TRUE  TRUE  TRUE

🔢 The `order()` Function

Sorting One Vector by Another

sort() only reorders the values of a single vector — but what if we want to reorder a second, related vector to match?
The order() function returns the indices that would sort a vector, rather than the sorted values themselves:

scores <- c(73, 95, 88, 60)
order(scores)

[1] 4 1 3 2

We can then use these indices to reorder a different vector — for example, a vector of the corresponding student names:

students <- c("Alex", "Becca", "Ciar", "Dobby")
students[order(scores)]

[1] "Dobby" "Alex"  "Ciar"  "Becca"

📌 This is the standard way to keep multiple related vectors in sync when sorting by one of them.

💪 Exercise — Final Problem

05:00

Spend the last 5 minutes completing the following exercises, combining all of the skills you have developed throughout the lecture.

Define the vector num_seq as a vector of consecutive integers starting at 1 and ending at 500.
Calculate the sum of the square roots of all numbers between 200 and 400.
Calculate the sum of the natural logarithm of every even number between 1 and 500.
Create a vector named rev_seq which consists of all numbers between 1 and 5 and between 495 and 500, displayed in descending order.

✅ Solution — Final Problem

Define the sequence

# define the sequence
num_seq <- seq(1, 500, by = 1)

Sum the square roots

# compute the sum of sqrts
sum(sqrt(num_seq[200:400]))

[1] 3464.785

Sum of natural logs

# compute the sum of natural logs
sum(log(num_seq[c(FALSE, TRUE)]))

[1] 1307.332

Create the reverse sequence

# create vector
rev_seq <- sort(c(num_seq[1:5], num_seq[495:500]), decreasing = TRUE); rev_seq

 [1] 500 499 498 497 496 495   5   4   3   2   1

Summary

✅ Topics Covered

Data structures
Indexing
Filtering
Recycling
Coercion
Sorting
Vectorization

📅 Next Class

Matrices
Arrays
Lists

PSTAT 10 Data Science Principles

Introduction

🔁 Review: Lecture 1

👀 Overview: Lecture 2

Data Structures

🔹 Scalars

What is a scalar?

Vectors, Matrices and Lists

🧮 Vectors

What is a vector?

Creating Vectors I

Combine Function c()

Creating Vectors II

🔁 rep() Function

🔢 seq() Function

🧩 Combining Methods

Flexibity of Vector Creation

Vector Property Functions

📏 length() Function

🏷️ typeof() Function

is.type Functions

Double Datatype

What is a double?

💪 Exercise — Coding Vectors

✅ Solution — Coding Vectors

Combining Methods

Side Note on Syntax

Vectorization

⚡ Vectorization

What is vectorization?

Vectorized Arithmetic

💪 Exercise - Vectorized Computations

✅ Solution — Coding Vectors

Defining the Vectors

Calculating Final Grades

Indexing

🎯 Accessing Vector Data

What is indexing?

🎯 Negating & Logical Indexing

Specifying what we don’t want! 👎

Boolean Indexing

🏷️ Named Vector Indices

From numbers to names

Grading Excercise Example

Filtering

🔍 Filtering

What is filtering?

🤔 Recall Boolean Indexing

💪 Exercise — Handling Vectors

✅ Solution — Handling Vectors

Define the Vector

Label the Indices

Compute the Number of Months Exceeding 300

Compute the Sum of Selected Months

Recycling

♻️ Recycling

What is Recycling?

🤔 Recall Vectorized Arithmetic

❗ When Recycling Goes Wrong

Multiples of Lengths

Coercion

🔀 What is Coercion?

What is Coercion?

🔀 Examples of Coercion

More Combinations

❗ Why Coercion Matters

A Common Gotcha

Sorting

🔃 Sorting

The sort() Function

🔢 The order() Function

Sorting One Vector by Another

💪 Exercise — Final Problem

✅ Solution — Final Problem

Define the sequence

Sum the square roots

Sum of natural logs

Create the reverse sequence

Summary

✅ Topics Covered

Combine Function `c()`

🔁 `rep()` Function

🔢 `seq()` Function

📏 `length()` Function

🏷️ `typeof()` Function

`is.type` Functions

The `sort()` Function

🔢 The `order()` Function