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When you complete this section, you will be able to:

• Give examples of cases in a data set.

• Identify the variables in a data set.

• Demonstrate how a label can be used as a variable in a data set.

• Identify the values of a variable.

• Classify variables as categorical or quantitative.

• Describe the key characteristics of a set of data.

• Explain how a rate is the result of adjusting one variable to create another.

A statistical analysis starts with a set of data. We construct a set of data by first deciding what *cases,* or units, we want to study. For each case, we record information about characteristics that we call *variables*.

CASES, LABELS, VARIABLES, AND VALUES

**Cases** are the objects described by a set of data. Cases may be customers, companies, subjects in a study, units in an experiment, or other objects.

A **label** is a special variable used in some data sets to distinguish the different cases.

A **variable** is a characteristic of a case.

Different cases can have different **values** of the variables.

EXAMPLE 1.1

Restaurant discount coupons. A website offers coupons that can be used to get discounts for various items at local restaurants. Coupons for food are very popular. Figure 1.1 gives information for seven restaurant coupons that were available for a recent weekend. These are the cases. Data for each coupon are listed on a different line, and the first column has the coupons numbered from 1 to 7. The remaining columns gives the type of restaurant, the name of the restaurant, the item being discounted, the regular price, and the discount price.

COUPONS

Figure 1.1: Figure 1.1 Spreadsheet of food discount coupons, Example 1.1.

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Some variables, like the type of restaurant, the name of the restaurant, and the item simply place coupons into categories. The regular price and discount price columns have numerical values for which we can do arithmetic. It makes sense to give an average of the regular prices, but it does not make sense to give an “average” type of restaurant. We can, however, do arithmetic to compare the regular prices classified by type of restaurant.

CATEGORICAL AND QUANTITATIVE VARIABLES

A **categorical variable** places a case into one of several groups or categories.

A **quantitative variable** takes numerical values for which arithmetic operations such as adding and averaging make sense.

EXAMPLE 1.2

Categorical and quantitative variables for coupons. The restaurant discount coupon file has six variables: coupon number, type of restaurant, name of restaurant, item, regular price, and discount price. The two price variables are quantitative variables. Coupon number, type of restaurant, name of restaurant, and item are categorical variables.

COUPONS

An appropriate label for your cases should be chosen carefully. In our food coupon example, a natural choice of a label would be the name of the restaurant. However, if there are two or more coupons available for a particular restaurant, or if a restaurant is a chain with different discounts offered at different locations, then the name of the restaurant would not uniquely label each of the coupons. In the restaurant discount coupon file, the first variable, ID, is a unique label for each coupon.

The display in Figure 1.1 is from an Excel **spreadsheet**spreadsheet. Spreadsheets are very useful for doing the kind of simple computations that you will do in Exercise 1.2. You can type in a formula and have the same computation performed for each row.

Note that the names we have chosen for the variables in our spreadsheet do not have spaces. For example, instead of “Restaurant Name” for the name of the restaurant, we simply use Name. In some statistical software packages, however, spaces are not allowed in variable names. For this reason, when creating spreadsheets for eventual use with statistical software, it is best to avoid spaces in variable names. Another convention is to use an underscore (_) where you would normally use a space. For our data set, we could have used Regular_Price and Discount_Price for the two price variables.

USE YOUR KNOWLEDGE

1.1 Read the spreadsheet. Refer to Figure 1.1. Give the regular price and the discount price for the Smokey Grill ribs coupon.

1.2 How much is the discount worth? Refer to Example 1.1. Consider adding another column to the spreadsheet that gives the coupon savings. Explain how you would compute the entries in this column. Does the new column contain values for a categorical variable or for a quantitative variable? Explain your answer.

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Another important part of the description of any quantitative variable is its unit of measurement. For both RegPrice and DiscPrice, the **unit of measurement**unit of measurement is clearly dollars. In other settings, it may not be as obvious. For example, if we were measuring heights of children, we might choose to use either inches or centimeters. The units of measurement are an important part of the description of a quantitative variable.

Key characteristics of a data set

In practice, any set of data is accompanied by background information that helps us understand the data. When you plan a statistical study or explore data from someone else’s work, ask yourself the following questions:

1.

**Who?**What**cases**do the data describe?**How many**cases does the data set contain?2.

**What?**How many**variables**do the data contain? What are the**exact definitions**of these variables? What are the units of measurement for each quantitative variable?3.

**Why? What purpose**do the data have? Do we hope to answer some specific questions? Do we want to draw conclusions about cases other than the ones we actually have data for? Are the variables that are recorded suitable for the intended purpose?

EXAMPLE 1.3

Statistics class data. Suppose that you are a teaching assistant for a statistics class and one of your jobs is to keep track of the grades for students in two sections of the course. The cases are the students in the class. There are weekly homework assignments, two exams during the semester, and a final exam. Each of these components is given a numerical score, and the components are added to get a total score that can range from 0 to 1000. Cutoffs of 900, 800, 700, etc., are used to assign letter grades of A, B, C, etc.

Caiaimage/Chris Ryan/Getty Images

The spreadsheet for this course will have seven variables:

• An identifier for each student.

• The number of points earned for homework.

• The number of points earned for the first exam.

• The number of points earned for the second exam.

• The number of points earned for the final exam.

• The total number of points earned.

• The letter grade earned.

The student identifier is a label and the letter grade earned is a categorical variable. All the other variables are measured in “points.” Because we can do arithmetic with their values, these variables are quantitative variables.

In our example of statistics class data, the possible values for the grade variable are A, B, C, D, and F. When computing grade point averages, many colleges and universities translate these letter grades into numbers using A = 4, B = 3, C = 2, D = 1, and F = 0. The transformed variable with numeric values is considered to be quantitative because we can average the numerical values across different courses to obtain a grade point average.

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Sometimes, experts argue about numerical scales such as this. They ask whether or not the difference between an A and a B is the same as the difference between a D and an F. Similarly, many questionnaires ask people to respond on a 1 to 5 scale, with 1 representing strongly agree, 2 representing agree, etc. Again we could ask whether or not the five possible values for this scale are equally spaced in some sense. From a practical point of view, the averages that can be computed when we convert categorical scales such as these to numerical values frequently provide a very useful way to summarize data.

EXAMPLE 1.4

Who, what, and why for the statistics class data. The data set in Example 1.3 was constructed to keep track of the grades for students in an introductory statistics course. The cases are the students in the class. There are seven variables in this data set. These include a label for each student and scores for the various course requirements. There are no units for the label and grade. The other variables all have “points” as the unit.

USE YOUR KNOWLEDGE

1.3 Who, what, and why? For the restaurant discount coupon data of Example 1.1 (page 2), what cases do the data describe? How many cases are there? How many variables are there? What are their definitions and units of measurement? What purpose do the data have?

EXAMPLE 1.5

Statistics class data for a different purpose. Suppose that the data for the students in the introductory statistics class were also to be used to study relationships between student characteristics and success in the course. Here, we have decided to focus on the TotalPoints and Grade as the outcomes of interest. Other variables of interest would have been included—for example, Sex, PrevStat (whether or not the student has taken a statistics course previously), and Year (student classification as first, second, third, or fourth year). ID is a categorical variable, TotalPoints is a quantitative variable, and the remaining variables are all categorical.

USE YOUR KNOWLEDGE

1.4 Apartment rentals. A data set lists apartments available for students to rent. Information provided includes the monthly rent, whether or not cable is included free of charge, whether or not pets are allowed, the number of bedrooms, and the distance to the campus. Describe the cases in the data set, give the number of variables, and specify whether each variable is categorical or quantitative.

Often, the variables in a statistical study are easy to understand: height in centimeters, study time in minutes, and so on. But each area of work also has its own special variables. A psychologist uses the Minnesota Multiphasic Personality Inventory (MMPI), and a physical fitness expert measures “VO2 max” (the volume of oxygen consumed per minute while exercising at your maximum capacity). Both of these variables are measured with special **instruments**instrument. VO2 max is measured by exercising while breathing into a mouthpiece connected to an apparatus that measures oxygen consumed. Scores on the MMPI are based on a long questionnaire, which is also called an instrument.

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Part of mastering your field of work is learning what variables are important and how they are best measured. Because details of particular measurements usually require knowledge of the particular field of study, we will say little about them.

*Be sure that each variable really does measure what you want it to. A poor choice of variables can lead to misleading conclusions.* Often, for example, the **rate**rate at which something occurs is a more meaningful measure than a simple count of occurrences.

EXAMPLE 1.6

Comparing colleges based on graduates. Think about comparing colleges based on the numbers of graduates. This view tells you something about the relative sizes of different colleges. However, if you are interested in how well colleges succeed at graduating students they admit, it would be better to use a rate. For example, you can find data on the Internet on the six-year graduation rates of different colleges. These rates are computed by examining the progress of first-year students who enroll in a given year. Suppose that at College A there were 1000 first-year students in a particular year, and 800 graduated within six years. The graduation rate is

$\frac{800}{1000}=0.80$

or 80%. College B has 2000 students who entered in the same year, and 1200 graduated within six years. The graduation rate is

$\frac{1200}{2000}=0.60$

or 60%. How do we compare these two colleges? College B has more graduates but College A has a better graduation rate.

In Example 1.6, when we computed the graduation rate, we used the total number of students to adjust the number of graduates. We constructed a new variable by dividing the number of graduates by the total number of students. Computing a rate is just one of several ways of **adjusting one variable to create another**adjusting one variable to create another. We often divide one variable by another to compute a more meaningful variable to study. Example 1.20 (page 20) is another type of adjustment.

USE YOUR KNOWLEDGE

1.5 How should you express the change? Between the first exam and the second exam in your statistics course, you increased the amount of time that you spent working exercises. Which of the following three ways would you choose to express the results of your increased work: (a) give the grades on the two exams, (b) give the ratio of the grade on the second exam divided by the grade on the first exam, (c) take the difference between the grade on the second exam and the grade on the first exam, and express this as a percent of the grade on the first exam. Give reasons for your answer.

1.6 Which variable would you choose? Refer to Example 1.6 on colleges and their graduates.

(a) Give a setting in which you would prefer to evaluate the colleges based on the numbers of graduates. Give a reason for your choice.

(b) Give a setting in which you would prefer to evaluate the colleges based on the graduation rates. Give a reason for your choice.

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Exercises 1.5 and 1.6 illustrate an important point about presenting the results of your statistical calculations. *Always consider how to best communicate your results to a general audience.* For example, the numbers produced by your calculator or by statistical software frequently contain more digits that are needed. Be sure that you do not include extra information generated by software that will distract from a clear explanation of what you have found.