Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

2.3 Measurement

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 9  Distributions of Estimates

segmentChapter 10  Confidence Intervals and Their Uses

segmentChapter 11  Model Comparison with the F Ratio

segmentChapter 12  What You Have Learned

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list Introduction to Statistics: A Modeling Approach
2.3 Measurement
Measurement is the process of turning variation in the world into data. When we measure, we assign numbers or category labels to some sample of cases in order to represent some attribute or dimension along which the cases vary.
Let’s make this more concrete by looking at some more measurements, in a data set called Fingers. A sample of college students filled in an online survey in which they were asked a variety of basic demographic questions. They also were asked to measure the length of each finger on their right hand.
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# A way to look at a data frame is to type its name
# Look at the data frame called Fingers
Fingers
ex() %>% check_output_expr("Fingers")
You’ll notice that trying to look at the whole data frame can be very cumbersome, especially for larger data sets.
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# Remember the head() command?
# Use it to look at the first six rows of Fingers
head(Fingers)
ex() %>% check_output_expr("head(Fingers)", missing_msg = "Did you call `head()` with `Fingers`?")
Sex RaceEthnic FamilyMembers SSLast Year Job MathAnxious Interest GradePredict Thumb Index Middle Ring Pinkie Height Weight
1 2 3 2 NA 3 1 4 1 3.3 66.00 79.0 84.0 74.0 57.0 62 188
2 2 3 4 9 2 2 5 3 4.0 58.42 76.2 91.4 76.2 63.5 70 145
3 2 3 2 3 2 2 2 3 4.0 70.00 80.0 90.0 70.0 65.0 69 175
4 2 1 5 7 2 1 1 3 3.7 59.00 83.0 87.0 79.0 64.0 72 155
5 2 5 2 9 3 1 5 3 4.0 64.00 76.0 89.0 76.0 69.0 70 180
6 2 3 7 7037 3 1 5 2 3.3 67.00 83.0 95.0 86.0 75.0 71 145
The command head()
shows you the first six rows of a data frame, but if you wanted to look at a different number of rows, you can just add in a number at the end like this.
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# Try it and see what happens
head(Fingers, 3)
head(Fingers, 3)
ex() %>% check_function("head") %>% check_arg("n") %>% check_equal()
Sex RaceEthnic FamilyMembers SSLast Year Job MathAnxious Interest GradePredict Thumb Index Middle Ring Pinkie Height Weight
1 2 3 2 NA 3 1 4 1 3.3 66.00 79.0 84.0 74.0 57.0 62 188
2 2 3 4 9 2 2 5 3 4.0 58.42 76.2 91.4 76.2 63.5 70 145
3 2 3 2 3 2 2 2 3 4.0 70.00 80.0 90.0 70.0 65.0 69 175
Notice that to answer these questions, you need to know something about how these numbers were measured. You need to know: Was Height measured with inches? What number represents which Sex? Does FamilyMembers include the person answering the question? (Sex can be a controversial variable; see here for more on this.)
We will be talking a lot about what measurements mean throughout the class. But before we go on, let’s learn one more way to take a quick look at a data frame.
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# Try using tail() to look at the last 6 rows of the Fingers data frame.
tail(Fingers)
ex() %>% check_function("tail") %>% check_result() %>% check_equal()
Sex RaceEthnic FamilyMembers SSLast Year Job MathAnxious Interest GradePredict Thumb Index Middle Ring Pinkie Height Weight
152 1 4 7 6 3 1 5 2 3.0 59 69 79 72 56 67.5 193
153 1 4 7 3 3 1 5 2 3.0 50 71 78 75 57 65.5 145
154 1 4 8 2354 2 2 3 2 2.7 64 70 76 70 51 59.0 114
155 1 4 3 789 1 1 4 2 2.7 50 70 85 74 55 64.0 165
156 1 3 8 0 3 2 4 2 3.7 57 67 73 65 55 63.0 125
157 1 1 6 NA 2 1 5 3 3.3 56 69 76 72 60 72.0 133
Levels of Measurement: Quantitative and Categorical Variables
Measures can be divided into two types, often referred to as “levels of measurement”: quantitative and categorical.
FamilyMembers and Height (which in this case was measured in inches) are examples of quantitative variables. The values assigned to quantitative variables represent some quantity (e.g., inches for height). And we can know that someone with a higher number (say, 62) is taller than someone with a lower number (say, 60). Moreover, the difference between the numbers actually tells us exactly how much taller one person is than another.
Categorical variables are quite different. Sex in this data set is a categorical variable. Students categorized themselves as male, female, or other. For purposes of analysis we might code each person in the following way: 1 if they are female; 2 if male; or 3 if other. The specific numbers we assign are arbitrary; we could have said other is 1, female is 2, and male is 3. The numbers don’t tell us anything about quantity; the numbers simply tell us which category the object belongs to.
While we use the terms quantitative and categorical, other writers will use other terms. They all mean roughly the same thing so you may not want to get hung up on these particular terms. Here are a few synonyms for quantitative variable and categorical variable that you may run across:
Quantitative Variable  Categorical Variable  

Numeric (num) variable  Nominal variable  
Continuous variable  Qualitative variable  
Scale variable  Factor 
Quantitative and Categorical Variables in R
Quantitative variables are always represented as numeric (or num) variables in R. Categorical variables could be either numeric or character (chr) variables in R, depending on what values they hold. If we were to code the variable Sex, for example, as 1 or 2 (for male and female) we could put the values in a numeric variable in R. If, on the other hand, we wanted to enter the values “male” or “female” into the variable Sex, R would represent it as a character variable. No matter what kind of variable we use in R, from the researcher’s point of view, the variable itself is still categorical.
R won’t necessarily know whether a variable is quantitative or categorical. A number could be used by a researcher to code a categorical variable (e.g., 1 for males and 2 for females), or it could represent units of some real quantitative measurement (1 sibling or 2 siblings). R will usually try to guess what kind of variable it is, but it may guess wrong!
For that reason, R has a way to let you specify whether a variable is categorical, using the factor()
command. A factor variable, in R, is always categorical. In the Fingers data frame, Sex is coded as 1 or 2. In order for R to know that it is categorical, we can tell it by using the command factor(Fingers$Sex)
. Remember, we also have to save the result of the command back into the Fingers data frame if we want R to remember it. We use the following code to turn Sex into a factor, and then replace the old version of the variable, which was numeric, with the new version, a factor:
Fingers$Sex < factor(Fingers$Sex)
We can also turn a factor back into a numeric variable by using the as.numeric()
function.
If the 1s and 2s in the Sex column were numbers, we could add them up using the code sum(Fingers$Sex)
. But if we tell R that Sex is a factor, it will assume the 1s and 2s refer to categories, and so it won’t be willing to add them up.
Try running this code to find the sum of Sex when it is coded as a factor:
allow_solution_error()
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# this turns Sex into a factor:
Fingers$Sex < factor(Fingers$Sex)
# click submit: this code will produce an error because you can't sum a factor
sum(Fingers$Sex)
# this turns Sex into a factor:
Fingers$Sex < factor(Fingers$Sex)
# click submit: this code will produce an error because you can't sum a factor
sum(Fingers$Sex)
ex() %>%
check_function("sum") %>%
check_output("not meaningful for factors")
That didn’t work! To get the sum of the Sex variable, you can use as.numeric()
to convert it back to numbers before using sum()
.
Depending on your goals, you may decide to treat a variable with numbers as both a quantitative and a categorical variable. If this is the case, it’s a good idea to make two copies of the variable, one numeric and one factor.
For example, Likert scales (those questions that ask you to rate something on a 5 or 7point scale) could be treated as quantitative variables in some situations, and categorical in other situations. In the Fingers data frame we have a variable called Interest, a rating by students of how interested they are in statistics. It is coded on a 3point scale from 0 (no interest) to 2 (very interested).
If you want to ask what the average rating is, you would need the variable to be numeric in R. But if you want to compare the group of people who gave a 0 rating with those who gave a 2, you want R to know that you consider Interest to be a factor.
require(tidyverse)
require(mosaic)
require(supernova)
Fingers < supernova::Fingers %>%
mutate_if(is.factor, as.numeric) %>%
arrange(desc(Sex)) %>%
{.[1, "FamilyMembers"] < 2; . } %>%
{.[1, "Height"] < 62; . }
# Interest has been coded numerically in the Fingers data.frame
# Modify the following to convert it to factor and store it as Interest.factor in Fingers
Fingers$Interest.factor <
Fingers$Interest.factor < factor(Fingers$Interest)
ex() %>%
check_object("Fingers") %>%
check_column("Interest.factor") %>%
check_equal()
If you made this new variable correctly, you won’t see anything appear in the R console. That’s because simply creating a new variable doesn’t cause R to print out anything. Sometimes while you are coding, you’ll feel like you did something wrong because nothing gets printed. It might just be that you didn’t tell R to print anything.
The str()
command tells you the type of each variable in a data frame. In the code you just wrote, you told R to make a new factor variable, Fingers$Interest.factor, based on the numeric variable, Fingers$Interest. If you wanted to check whether you were successful, you could type str(Fingers)
in the code window you were just working in.
The output shows that the Fingers data frame now includes a new variable, Fingers$Interest.factor, and also confirms that this new variable is a factor variable.
str(Fingers)
'data.frame': 157 obs. of 17 variables:
$ Sex : num 2 2 2 2 2 2 2 2 2 2 ...
$ RaceEthnic : num 3 3 3 1 5 3 1 4 3 3 ...
$ FamilyMembers : num 2 4 2 5 2 7 4 3 7 5 ...
$ SSLast : num NA 9 3 7 9 ...
$ Year : num 3 2 2 2 3 3 3 3 1 3 ...
$ Job : num 1 2 2 1 1 1 2 2 1 2 ...
$ MathAnxious : num 4 5 2 1 5 5 2 1 4 2 ...
$ Interest : num 1 3 3 3 3 2 2 3 2 1 ...
$ GradePredict : num 3.3 4 4 3.7 4 3.3 4 4 3 3.7 ...
$ Thumb : num 66 58.4 70 59 64 ...
$ Index : num 79 76.2 80 83 76 83 70 75 74 63 ...
$ Middle : num 84 91.4 90 87 89 95 76 83 83 70 ...
$ Ring : num 74 76.2 70 79 76 86 72 78 79 65 ...
$ Pinkie : num 57 63.5 65 64 69 75 55 60 64 56 ...
$ Height : num 62 70 69 72 70 71 67.5 69 68.5 65 ...
$ Weight : num 188 145 175 155 180 145 130 180 193 138 ...
$ Interest.factor: Factor w/ 3 levels "1","2","3": 1 3 3 3 3 2 2 3 2 1 ...