IN THIS CHAPTER

Summarizing categorical data effectively

Interpreting commonly used statistics

Realizing what statistics do and don’t say

Every data set has a story, and if statistics are used properly, they do a good job of uncovering and reporting that story. Statistics that are improperly used can tell a different story, or only part of it, so knowing how to make good decisions about the information you’re given is very important.

A descriptive statistic (or statistic for short) is a number that summarizes or describes some characteristic about a set of data. In this chapter and the next one, you see some of the most common descriptive statistics and how they are used, and you find out how to calculate them, interpret them, and put them together to get a good picture of a data set. You also find out what these statistics say and what they don’t say about the data.

Summing Up Data with Descriptive Statistics

Descriptive statistics take a data set and boil it down to a set of basic information. Summarized data are often used to provide people with information that is easy to understand and that helps answer their questions. Picture your boss coming to you and asking, “What’s our client base like these days, and who’s buying our products?” How would you like to answer that question? With a long, detailed, and complicated stream of numbers that are sure to glaze their eyes over? Probably not. You want clean, clear, and concise statistics that sum up the client base for them, so they can see how brilliant you are and then send you off to collect even more data to see how they can include more people in the client base. (That’s what you get for being efficient.)

Summarizing data has other purposes, as well. After all the data have been collected from a survey or some other kind of study, the next step is for the researcher to try to make sense out of the data. Typically, the first step researchers take is to run some basic statistics on the data to get a rough idea about what’s happening in it. Later in the process, researchers can do more analyses to formulate or test claims made about the population the data came from, estimate certain characteristics about the population (like the mean), look for links between variables they measured, and so on.

Another big part of research is reporting the results, not only to their peers, but also to the media and the general public. Although a researcher’s peers may be anxiously waiting to hear about all the complex analyses that were done on a data set, the general public is neither ready for nor interested in that. What does the public want? Basic information. Statistics that make a point clearly and concisely are typically used to relay information to the media and to the public.

If you really need to learn more from data, a quick statistical overview isn’t enough. In the statistical world, less is not more, and sometimes the real story behind the data can get lost in the shuffle. To be an informed consumer of statistics, you need to think about which statistics are being reported, what these statistics really mean, and what information is missing. This chapter focuses on these issues.

Crunching Categorical Data: Tables and Percents

Categorical data (also known as qualitative data) capture qualities or characteristics about an individual, such as a person’s eye color, gender, political party, or opinion on some issue (using categories such as Agree, Disagree, or No Opinion). Categorical data tend to fall into groups or categories pretty naturally. “Political Party,” for example, typically has four groups in the United States: Democrat, Republican, Independent, and Other. Categorical data often come from survey data, but they can also be collected in experiments. For example, in an experimental test of a new medical treatment, researchers may use three categories to assess the outcome of the experiment: Did the patient get better, worse, or stay the same while undergoing the treatment?

Practice Questions Answers and Explanations

1 Eleven shoppers prefer Brand A, and nine shoppers prefer Brand B. The survey results are shown in the following frequency table. Brand A got more votes, but the results are pretty close.

2 Frequencies are fine for summarizing data as long as you keep the total number in perspective.

1. The results are shown in the following frequency table. Because the total is 18,726, and the Yes count is 10,479, the No count is the difference between the two, which is .

   Vote	Frequency
   Y	10,479
   N	8,247
   Total	18,726

2. The total is important because it helps keep the frequencies in perspective when you compare them to each other.

3 This problem shows the importance of reporting not only the results of participants who responded, but also what percentage of the total actually responded.

1. The results are shown in the following frequency table:

   Gone to the Zoo in the Last Year?	Frequency
   Y	592
   N	198
   Nonrespondents	210
   Total	1,000

2. If you don’t show the nonrespondents, the total doesn’t add up to 1,000 (the number surveyed). An alternative way to show the data is to base it on only the respondents, but the results would be biased. You can’t definitively say that the nonrespondents would respond the same way as the respondents.

4 Relative frequencies do just what they say: They help you relate the results to each other (by finding percentages).

1. Eleven shoppers out of the twenty prefer Brand A, and nine shoppers out of the twenty prefer Brand B. The survey results are shown in the following relative frequency table. Brand A got more votes, but the results are pretty close, with 55 percent of the shoppers preferring Brand A and 45 percent preferring Brand B.

   Brand Preferred	Relative Frequency
   A	55%
   B	45%

2. You often have an easier time interpreting percents, because when you need to interpret counts, you have to put them in perspective in terms of “out of how many?”

5 The results are shown in the following relative frequency table. The Yes percentage is percent. Because the total is 100 percent, the No percentage is .

6 You can see the relative frequency table following this answer. Knowing the response rate is critical for interpreting the results of a survey. (The higher the response rate, the better.) The response rate is , that is, the total percentage of people who responded in any way (yes or no) to the survey. Note that 21 percent is the nonresponse rate.

7 Showing the percentages rather than counts means making a relative frequency table rather than a frequency table.

1. One advantage of a relative frequency table is that everything sums to 100 percent, making it easier to interpret the results, especially if you have a large number of categories.
2. One disadvantage of a relative frequency table is that if you see only the percents, you don’t know how many people participated in the study; therefore, you don’t know how precise the results are. Remember the commercial about ‘four out of five dentists surveyed’? Maybe the company only asked five dentists! You can get around this problem by putting the total sample size somewhere at the top or bottom of your relative frequency table.

When making a relative frequency table, include the total sample size somewhere on the table.

8 Be careful how you interpret tables where an individual can be in more than one category at the same time.

1. The frequencies don’t sum to 1,000, because people have the option to choose multiple locations or none at all, so each person doesn’t end up in exactly one group. If you take the grand total of all the frequencies (1,493) and divide each frequency by 1,493 to get a relative frequency, the relative frequencies sum to 1 (or 100 percent). But what does that mean? It makes it hard to interpret these percents because they don’t account for the total number of people.
2. One way you can summarize this data is by showing the percentage of people who have been at each location separately (compared to the percentage who haven’t been there before). These percents add up to 1 for each location. The following table shows the results summarized with this method. Note: The table isn’t a relative frequency table; however, it uses relative frequencies.

   Location	% Who Have Been There	% Who Haven’t Been There
   Disney World
   New Orleans
   Las Vegas
   New York City
   Washington, D.C.

Not all tables involving percents should sum to 1. Don’t force tables to sum to 1 when they shouldn’t; do make sure you understand whether each individual can fall under more than one category. In those cases, a typical relative frequency table isn’t appropriate.

9 You can always sum all the frequencies to get a total and then find each relative frequency by taking the frequency divided by the total. However, if you have only the percents, you can’t go back and find the original counts unless you know the total number of individuals.

Suppose you know that 80 percent of the people in a survey like ice cream. How many actual people in the survey like ice cream? If the total number of respondents is 100, then people like ice cream. If the total is 50, then you’re looking at positive answers. If the total is 5, then you’re dealing only with 4 respondents . This illustrates why relative frequency tables need to have the total sample size somewhere.

Watch for total sample sizes when you’re given a relative frequency table. Don’t be misled by percentages alone, thinking they’re always based on large sample sizes. Many are not.

If you’re ready to test your skills a bit more, take the following chapter quiz that incorporates all the chapter topics.

Whaddya Know? Chapter 4 Quiz

Quiz time! Complete each problem to test your knowledge on the various topics covered in this chapter. You can then find the solutions and explanations in the next section.

Answers to Chapter 4 Quiz

1 Yes; No. Relative frequency table:

1. Pizzas	On Time? Yes	On Time? No
   Percent (Relative Frequency)	50%	50%

2. Fifty percent of the 10 pizzas were delivered on time and 50% were not.

2

1. Clients	Thinking of Selling? Yes	Thinking of Selling? No
   Count (Frequency)	4	6

2. Four of the 10 clients are thinking of selling and 6 are not.

3

1. When you sum the frequencies in a frequency table, you get the total sample size, n.
2. When you sum the relative frequencies in a relative frequency table, you get 1.

4 The table is a relative frequency table because the results are shown in percentage form, and sum to .

5

1. Days	Dropped below 32? Yes	Dropped below 32? No
   Count (Frequency)

2. yes, yes, no, no, no, no, no, no, yes, no. Note that any combination of 3 yeses and 7 nos works here.

6

1. The total number of grades is 25; divide each frequency by 25 to get the relative frequency. The results should sum to 1.

   Grade	A	B	C	D	F
   Relative Frequency

2. The 25-member class had 40% As, 32% Bs, 20% Cs, 8% Ds, and 0% Fs.

7 A pie chart is similar to a relative frequency table because each slice of the pie is a group and each group has a percentage attached to it that indicates the percentage of individuals in that group. The percents sum to 1. A pie chart is a graphical representation of a relative frequency table.

8

1. .

2. 	Yes	No	No Response
   Percent (Relative Frequency)

3. The survey showed 50% of the students plan to go away from home for spring break; 30% do not plan to; and 20% did not respond.

9 A two-way table with rows and columns representing expired (yes, no), and proper weight (yes, no). One possible two-way table is the following:

10 The number of cells will be 4 (political parties) * 3 (age groups) = 12 category combinations = 12 cells.