1 Hello data
Scientists seek to answer questions using rigorous methods and careful observations. These observations—collected from the likes of field notes, surveys, and experiments—form the backbone of a statistical investigation and are called data. Statistics is the study of how best to collect, analyze, and draw conclusions from data, and in this first chapter, we focus on both the properties of data and on the collection of data.
Though we were calculating probabilities in the 16th century, and the first US Census was directed by Thomas Jefferson in 1790^{4}, the discipline of statistics as we know it came about in the 1800s. Up until the 21st century, the statistical investigation process looked something like this (adapted from Tintle et al. (2016)):
 Ask a research question.
 Design a study and collect data.
 Summarize and visualize the data.
 Use statistical analysis methods to draw inferences from the data.
 Communicate the results and answer the research question.
 Revisit and look forward.
With the rise of data science, however, we might not start with a research question, and instead start with a data set^{5}. In this case, the statistical investigation process looks more like the data exploration cycle found in Figure 1.1 taken from Wickham and Grolemund (2017).
In either case, the ideas, concepts, and methods presented in this book will provide you with the tools to work through the statistical investigation process, whether starting with a research question or starting with data.
1.1 Case study: using stents to prevent strokes
In this section, we introduce a classic challenge in statistics: evaluating the efficacy of a medical treatment. Terms in this section, and indeed much of this chapter, will all be revisited later in the text. The plan for now is simply to get a sense of the role statistics can play in practice.
Here, we will consider an experiment that studies effectiveness of stents in treating patients at risk of stroke (Chimowitz et al. 2011). Stents are small mesh tubes that are placed inside narrow or weak arteries to assist in patient recovery after cardiac events and reduce the risk of an additional heart attack or death.
Many doctors have hoped that there would be similar benefits for patients at risk of stroke. We start by writing the principal question the researchers hope to answer:
Does the use of stents reduce the risk of stroke?
The researchers who asked this question conducted an experiment with 451 atrisk patients. Each volunteer patient was randomly assigned to one of two groups:
 Treatment group. Patients in the treatment group received a stent and medical management. The medical management included medications, management of risk factors, and help in lifestyle modification.
 Control group. Patients in the control group received the same medical management as the treatment group, but they did not receive stents.
Researchers randomly assigned 224 patients to the treatment group and 227 to the control group. In this study, the control group provides a reference point against which we can measure the medical impact of stents in the treatment group.
Researchers studied the effect of stents at two time points: 30 days after enrollment and 365 days after enrollment.
The data collected on 5 of these patients are summarized in Table 1.1.
Patient outcomes are recorded as stroke
or no event
, representing whether or not the patient had a stroke during that time period.
The data from this study can be found in the openintro package: stent30
and stent365
.
Load these data into your RStudio session using the following commands:
library(openintro)
data(stent30)
data(stent365)
patient  group  30 days  365 days 

1  treatment  no event  no event 
2  treatment  stroke  stroke 
3  treatment  no event  no event 
4  treatment  no event  no event 
5  control  no event  no event 
Considering data from each of the 451 patients individually would be a long, cumbersome path towards answering the original research question. Instead, performing a statistical data analysis allows us to consider all of the data at once. Table 1.2 summarizes the raw data in a more helpful way. In this table, we can quickly see what happened over the entire study. For instance, to identify the number of patients in the treatment group who had a stroke within 30 days after the treatment, we look in the leftmost column (30 days), at the intersection of treatment and stroke: 33. To identify the number of control patients who did not have a stroke after 365 days after receiving treatment, we look at the rightmost column (365 days), at the intersection of control and no event: 199.
stroke  no event  stroke  no event  

treatment  33  191  45  179 
control  13  214  28  199 
Total  46  405  73  378 
The data summarized in this table can also be visualized with a barplot, seen in Figure 1.2:
Of the 224 patients in the treatment group, 45 had a stroke by the end of the first year. Using these two numbers, compute the proportion of patients in the treatment group who had a stroke by the end of their first year. (Note: answers to all Guided Practice exercises are provided in footnotes!)^{6}
We can compute summary statistics from the table to give us a better idea of how the impact of the stent treatment differed between the two groups. A summary statistic is a single number summarizing a large amount of data. For instance, the primary results of the study after 1 year could be described by two summary statistics: the proportion of people who had a stroke in the treatment and control groups.
 Proportion who had a stroke in the treatment (stent) group: \(45/224 = 0.20 = 20\)%.
 Proportion who had a stroke in the control group: \(28/227 = 0.12 = 12\)%.
These two summary statistics are useful in looking for differences in the groups, and we are in for a surprise: an additional 8% of patients in the treatment group had a stroke! This is important for two reasons. First, it is contrary to what doctors expected, which was that stents would reduce the rate of strokes. Second, it leads to a statistical question: do the data show a “real” difference between the groups?
This second question is subtle, and is the basis of what we call statistical inference. Suppose you flip a coin 100 times. While the chance a coin lands heads in any given coin flip is 50%, we probably won’t observe exactly 50 heads. This type of fluctuation is part of almost any type of data generating process. It is possible that the 8% difference in the stent study is due to this natural variation. However, the larger the difference we observe (for a particular sample size), the less believable it is that the difference is due to chance. So what we are really asking is the following: is the difference so large that we should reject the notion that it was due to chance?
While we don’t yet have our statistical tools to fully address this question on our own, we can comprehend the conclusions of the published analysis: there was compelling evidence of harm by stents in this study of stroke patients.
Be careful. Do not generalize the results of this study to all patients and all stents. This study looked at patients with very specific characteristics who volunteered to be a part of this study and who may not be representative of all stroke patients. In addition, there are many types of stents and this study only considered the selfexpanding Wingspan stent (Boston Scientific). However, this study does leave us with an important lesson: we should keep our eyes open for surprises.
1.2 Data basics
Effective presentation and description of data is a first step in most analyses. This section introduces one structure for organizing data as well as some terminology that will be used throughout this book.
1.2.1 Observations, variables, and data frames
Here, we will consider loans offered through the Lending Club, a peertopeer lending company. Such data could be used to explore characteristics of people receiving loans from the platform, such as job titles, annual income, or home ownership. Table 1.3 displays six rows of a data set for 50 randomly sampled loans. These observations will be referred to as the loan50
data set.
The data can be found in the openintro package: loan50
.
Load these data into your RStudio session using the following commands:
library(openintro)
data(loan50)
Each row in the table represents a single loan. The formal name for a row is a case or observational unit. Since there are 50 observational units in our data set, the sample size, denoted by \(n\), is 50 (\(n = 50\)). The columns represent characteristics of each loan, where each column is referred to as a variable. For example, the first row represents a loan of $7,500 with an interest rate of 7.34%, where the borrower is based in Maryland (MD) and has an income of $70,000.
A variable is something that can be measured on an individual observational unit. Be careful not to confuse summary statistics—calculated from a group of observational units—with variables.
What is the grade of the first loan in Table 1.3? What is the home ownership status of the borrower for that first loan? Reminder: for these Guided Practice questions, you can check your answer in the footnote.^{7}
In practice, it is especially important to ask clarifying questions to ensure important aspects of the data are understood.
For instance, it is always important to be sure we know what each variable means and its units of measurement.
Descriptions of the variables in the loan50
data set are given in Table 1.4.
loan_amount  interest_rate  term  grade  state  total_income  homeownership  

1  22000  10.90  60  B  NJ  59000  rent 
2  6000  9.92  36  B  CA  60000  rent 
3  25000  26.30  36  E  SC  75000  mortgage 
4  6000  9.92  36  B  CA  75000  rent 
5  25000  9.43  60  B  OH  254000  mortgage 
6  6400  9.92  36  B  IN  67000  mortgage 
variable  description 

loan_amount  Amount of the loan received, in US dollars. 
interest_rate  Interest rate on the loan, in an annual percentage. 
term  The length of the loan, which is always set as a whole number of months. 
grade  Loan grade, which takes on values A through G and represents the quality of the loan and its likelihood of being repaid. 
state  US state where the borrower resides. 
total_income  Borrower’s total income, including any second income, in US dollars. 
homeownership  Indicates whether the person owns, owns but has a mortgage, or rents. 
The data in Table 1.3 represent a data frame (or data matrix), which is a convenient and common way to organize data, especially if collecting data in a spreadsheet. Each row of a data frame corresponds to a unique case (observational unit), and each column corresponds to a variable.
When recording data, use a data frame unless you have a very good reason to use a different structure. This structure allows new cases to be added as rows or new variables as new columns.
The grades for assignments, quizzes, and exams in a course are often recorded in a gradebook that takes the form of a data frame. How might you organize a course’s grade data using a data frame?^{8}
We consider data for 3,142 counties in the United States, which include the name of each county, the state where it resides, its population in 2017, how its population changed from 2010 to 2017, poverty rate, and nine additional characteristics. How might these data be organized in a data frame?^{9}
The data described in the Guided Practice above represent the county data set, which is shown as a data frame in Table 1.5. The variables as well as the variables in the data set that did not fit in Table 1.5 are described in Table 1.6
name  state  pop2017  pop_change  unemployment_rate  median_edu 

Autauga County  Alabama  55504  1.48  3.86  some_college 
Baldwin County  Alabama  212628  9.19  3.99  some_college 
Barbour County  Alabama  25270  6.22  5.90  hs_diploma 
Bibb County  Alabama  22668  0.73  4.39  hs_diploma 
Blount County  Alabama  58013  0.68  4.02  hs_diploma 
Bullock County  Alabama  10309  2.28  4.93  hs_diploma 
variable  description 

name  Name of county. 
state  Name of state. 
pop2000  Population in 2000. 
pop2010  Population in 2010. 
pop2017  Population in 2017. 
pop_change  Population change from 2010 to 2017. 
poverty  Percent of population in poverty in 2017. 
homeownership  Homeownership rate, 20062010. 
multi_unit  Percent of housing units in multiunit structures, 20062010. 
unemployment_rate  Unemployment rate in 2017. 
metro 
Whether the county contains a metropolitan area, taking one of the values yes or no .

median_edu 
Median education level (20132017), taking one of the values below_hs , hs_diploma , some_college , or bachelors .

per_capita_income  Per capita (per person) income (20132017). 
median_hh_income  Median household income. 
smoking_ban 
Describes whether the type of countylevel smoking ban in place in 2010, taking one of the values none , partial , or comprehensive .

These data can be found in the usdata package: county
.
Load these data into your RStudio session using the following commands:
library(usdata)
data(county)
1.2.2 Types of variables
Examine the unemployment_rate
, pop2017
, state
, metro
, and median_edu
variables in the county
data set.
Each of these variables is inherently different from the others, yet some share certain characteristics.
First consider unemployment_rate
, which is said to be a quantitative or numerical variable since it can take a wide range of numerical values, and it is sensible to add, subtract, or take averages with those values.
On the other hand, we would not classify a variable reporting telephone area codes as quantitative since the average, sum, and difference of area codes doesn’t have any clear meaning.
The pop2017
variable is also quantitative, although it seems to be a little different than unemployment_rate
.
This variable of the population count can only take whole nonnegative numbers (0, 1, 2, …).
For this reason, the population variable is said to be discrete since it can only take numerical values with jumps.
On the other hand, the unemployment rate variable is said to be continuous.
The variable state
can take up to 51 values after accounting for Washington, DC: AL, AK, …, and WY.
Because the responses themselves are categories, state
is called a categorical variable, and the possible values are called the variable’s levels . The variable metro
is also categorical, but it only has two levels (yes or no). A categorical variable with only two levels is called a binary variable. When working with a generic binary variable, we often call the two possible levels “success” and “failure.”
Finally, consider the median_edu
variable, which describes the median education level of county residents and takes values below_hs
, hs_diploma
, some_college
, or bachelors
in each county.
This variable seems to be a hybrid: it is a categorical variable but the levels have a natural ordering.
A variable with these properties is called an ordinal variable, while a regular categorical variable without this type of special ordering is called a nominal variable.
To simplify analyses, any ordinal variable in this book will be treated as a nominal (unordered) categorical variable.
Data were collected about students in a statistics course. Three variables were recorded for each student: number of siblings, student height, and whether the student had previously taken a statistics course. Classify each of the variables as continuous quantitative, discrete quantitative, or categorical.
The number of siblings and student height represent quantitative variables. Because the number of siblings is a count, it is discrete. Height varies continuously, so it is a continuous quantitative variable. The last variable classifies students into two categories—those who have and those who have not taken a statistics course—which makes this variable categorical.
An experiment is evaluating the effectiveness of a new drug in treating migraines.
A group
variable is used to indicate the experiment group for each patient: treatment or control.
The num_migraines
variable represents the number of migraines the patient experienced during a 3month period. Classify each variable as either quantitative or categorical.^{10}
1.2.3 Relationships between variables
Many analyses are motivated by a researcher looking for a relationship between two or more variables. A social scientist may like to answer some of the following questions:
Does a higher than average increase in county population tend to correspond to counties with higher or lower median household incomes?
If homeownership is lower than the national average in one county, will the percent of multiunit structures in that county tend to be above or below the national average?
How useful a predictor is median education level for the median household income for US counties?
To answer these questions, data must be collected, such as the county
data set shown in Table 1.5.
Examining summary statistics could provide insights for each of the three questions about counties.
Additionally, graphs can be used to visually explore the data.
Scatterplots are one type of graph used to study the relationship between two quantitative variables.
Figure 1.4 displays the relationship between the variables homeownership
and multi_unit
, which is the percent of units in multiunit structures (e.g., apartments, condos).
Each point on the plot represents a single county (a single observational unit).
For instance, the highlighted dot corresponds to County 413 in the county
data set: Chattahoochee County, Georgia, which has 39.4% of units in multiunit structures and a homeownership rate of 31.3%.
The scatterplot suggests a relationship between the two variables: counties with a higher rate of multiunits tend to have lower homeownership rates.
We might brainstorm as to why this relationship exists and investigate each idea to determine which are the most reasonable explanations.
The multiunit and homeownership rates are said to be associated because the plot shows a discernible pattern. When two variables show some connection with one another, they are called associated variables. Associated variables can also be called dependent variables and viceversa.
Examine the variables in the loan50
data set, which are described in Table 1.4.
Create two questions about possible relationships between variables in loan50
that are of interest to you.^{11}
This example examines the relationship between the change in population from 2010 to 2017 and median household income for counties, which is visualized as a scatterplot in Figure 1.5. Are these variables associated?
The larger the median household income for a county, the higher the population growth observed for the county. While this trend isn’t true for every county, the trend in the plot is evident. Since there is some relationship between the variables, they are associated.
Because there is a downward trend in Figure 1.4—counties with more units in multiunit structures are associated with lower homeownership—these variables are said to be negatively associated.
A positive association is shown in the relationship between the median_hh_income
and pop_change
variables in Figure 1.5, where counties with higher median household income tend to have higher rates of population growth.
If two variables are not associated, then they are said to be independent. That is, two variables are independent if there is no evident relationship between the two.
Associated or independent, not both. A pair of variables are either related in some way (associated) or not (independent). No pair of variables is both associated and independent.
1.2.4 Explanatory and response variables
When we ask questions about the relationship between two variables, we sometimes also want to determine if the change in one variable causes a change in the other.
Consider the following rephrasing of an earlier question about the county
data set:
If there is an increase in the median household income in a county, does this drive an increase in its population?
In this question, we are asking whether one variable affects another. If this is our underlying belief, then median household income is the explanatory variable variable and the population change is the response variable variable in the hypothesized relationship.^{12}
Explanatory and response variables.
When we suspect one variable might causally affect another, we label the first variable the explanatory variable and the second the response variable. The main reason for this is that observational studies do not control for confounding variables. We will revisit this idea when we discuss experiments in the next chapter.
For many pairs of variables, there is no hypothesized relationship, and these labels would not be applied to either variable in such cases.
Bear in mind that the act of labeling the variables in this way does nothing to guarantee that a causal relationship exists. A formal evaluation to check whether one variable causes a change in another requires an experiment.
1.2.5 Introducing observational studies and experiments
There are two primary types of data collection: observational studies and experiments. We already encountered an experiment in the case study in Section 1.1, and an observational study with the Lending Club data in this section.
Researchers perform an observational study when they collect data in a way that does not directly interfere with how the data arise. For instance, researchers may collect information via surveys, review medical or company records, or follow a cohort of many similar individuals to form hypotheses about why certain diseases might develop. In each of these situations, researchers merely observe the data that arise. In general, observational studies can provide evidence of a naturally occurring association between variables, but they cannot by themselves show a causal connection.
When researchers want to investigate the possibility of a causal connection, they conduct an experiment. Usually there will be both an explanatory and a response variable. For instance, we may suspect administering a drug will reduce mortality in heart attack patients over the following year. To check if there really is a causal connection between the explanatory variable and the response, researchers will collect a sample of individuals and split them into groups. The individuals in each group are assigned a treatment. When individuals are randomly assigned to a group, the experiment is called a randomized experiment. For example, each heart attack patient in the drug trial could be randomly assigned, perhaps by flipping a coin, into one of two groups: the first group receives a placebo (fake treatment) and the second group receives the drug. Note that the case study in Section 1.1 did not use a placebo.
Association \(\neq\) Causation. In general, association does not imply causation, and causation can only be inferred from a randomized experiment.
1.3 Chapter review
Summary
This chapter introduced you to the world of data. Data can be organized in many ways but tidy data, where each row represents an observation and each column represents a variable, lends itself most easily to statistical analysis. Many of the ideas from this chapter will be seen as we move on to doing full data analyses. In the next chapter you’re going to learn about how we can design studies to collect the data we need to make conclusions with the desired scope of inference.
Terms
We introduced the following terms in the chapter. If you’re not sure what some of these terms mean, we recommend you go back in the text and review their definitions. We are purposefully presenting them in alphabetical order, instead of in order of appearance, so they will be a little more challenging to locate. However you should be able to easily spot them as bolded text.
associated  discrete  placebo 
barplot  experiment  positive association 
binary  explanatory variable  quantitative 
case  independent  randomized experiment 
categorical  level  response variable 
cohort  negative association  sample size 
continuous  nominal  statistical investigation process 
data  observational study  summary statistic 
data frame  observational unit  variable 
dependent  ordinal 
Key ideas
A data set is comprised of measurements of variables on observational units. The type of variable can be quantitative or categorical, and a variable’s role can be an explanatory variable or response variable.
In an observational study, we merely observe the behavior of the individuals in the study; we do not manipulate the variables on the individuals in any way. In a randomized experiment, we randomly assign values of an explanatory variable to the observational units, then observe the response variable. Random assignment allows us to investigate causal relationships because it balances out any potential confounding variables, on average. These ideas will be discussed in detail in the next chapter.