1.1    Objectives

In this chapter, you will learn the following items:

• The difference between parametric and nonparametric statistics.
• How to rank data.
• How to determine counts of observations.

1.2    Introduction

If you are using this book, it is possible that you have taken some type of introductory statistics class in the past. Most likely, your class began with a discussion about probability and later focused on particular methods of dealing with populations and samples. Correlations, z-scores, and t-tests were just some of the tools you might have used to describe populations and/or make inferences about a population using a simple random sample.

Many of the tests in a traditional, introductory statistics text are based on samples that follow certain assumptions called parameters. Such tests are called parametric tests. Specifically, parametric assumptions include samples that

• are randomly drawn from a normally distributed population,
• consist of independent observations, except for paired values,
• consist of values on an interval or ratio measurement scale,
• have respective populations of approximately equal variances,
• are adequately large,* and
• approximately resemble a normal distribution.

If any of your samples breaks one of these rules, you violate the assumptions of a parametric test. You do have some options, however.

You might change the nature of your study so that your data meet the needed parameters. For instance, if you are using an ordinal or nominal measurement scale, you might redesign your study to use an interval or ratio scale. (See Box 1.1 for a description of measurement scales.) Also, you might seek additional participants to enlarge your sample sizes. Unfortunately, there are times when one or neither of these changes is appropriate or even possible.

If your samples do not resemble a normal distribution, you might have learned a strategy that modifies your data for use with a parametric test. First, if you can justify your reasons, you might remove extreme values from your samples called outliers. For example, imagine that you test a group of children and you wish to generalize the findings to typical children in a normal state of mind. After you collect the test results, most children earn scores around 80% with some scoring above and below the average. Suppose, however, that one child scored a 5%. If you find that this child speaks no English because he arrived in your country just yesterday, it would be reasonable to exclude his score from your analysis. Unfortunately, outlier removal is rarely this straightforward and deserves a much more lengthy discussion than we offer here.* Second, you might utilize a parametric test by applying a mathematical transformation to the sample values. For example, you might square every value in a sample. However, some researchers argue that transformations are a form of data tampering or can distort the results. In addition, transformations do not always work, such as circumstances when data sets have particularly long tails. Third, there are more complicated methods for analyzing data that are beyond the scope of most introductory statistics texts. In such a case, you would be referred to a statistician.

Fortunately, there is a family of statistical tests that do not demand all the parameters, or rules, that we listed earlier. They are called nonparametric tests, and this book will focus on several such tests.

1.3    The Nonparametric Statistical Procedures Presented in this Book

This book describes several popular nonparametric statistical procedures used in research today. Table 1.1 identifies an overview of the types of tests presented in this book and their parametric counterparts.

When demonstrating each nonparametric procedure, we will use a particular step-by-step method.

1.4    Ranking Data

Many of the nonparametric procedures involve ranking data values. Ranking values is really quite simple. Suppose that you are a math teacher and wanted to find out if students score higher after eating a healthy breakfast. You give a test and compare the scores of four students who ate a healthy breakfast with four students who did not. Table 1.3 shows the results.

To rank all of the values from Table 1.3 together, place them all in order in a new table from smallest to largest (see Table 1.4). The first value receives a rank of 1, the second value receives a rank of 2, and so on.

Notice that the values for the students who ate breakfast are in bold type. On the surface, it would appear that they scored higher. However, if you are seeking statistical significance, you need some type of procedure. The following chapters will offer those procedures.

1.5    Ranking Data with Tied Values

The aforementioned ranking method should seem straightforward. In many cases, however, two or more of the data values may be repeated. We call repeated values ties, or tied values. Say, for instance, that you repeat the preceding ranking with a different group of students. This time, you collected new values shown in Table 1.5.

Rank the values as in the previous example. Notice that the value of 90 is repeated. This means that the value of 90 is a tie. If these two student scores were different, they would be ranked 6 and 7. In the case of a tie, give all of the tied values the average of their rank values. In this example, the average of 6 and 7 is 6.5 (see Table 1.6).

Most nonparametric statistical tests require a different formula when a sample of data contains ties. It is important to note that the formulas for ties are more algebraically complex. What is more, formulas for ties typically produce a test statistic that is only slightly different from the test statistic formulas for data without ties. It is probably for this reason that most statistics texts omit the formulas for tied values. As you will see, however, we include the formulas for ties along with examples where applicable.

When the statistical tests in this book are explained using the computer program SPSS^® (Statistical Package for Social Scientists), there is no mention of any special treatment for ties. That is because SPSS automatically detects the presence of ties in any data sets and applies the appropriate procedure for calculating the test statistic.

1.6    Counts of Observations

Some nonparametric tests require counts (or frequencies) of observations. Determining the count is fairly straightforward and simply involves counting the total number of times a particular observations is made. For example, suppose you ask several children to pick their favorite ice cream flavor given three choices: vanilla, chocolate, and strawberry. Their preferences are shown in Table 1.7.

To find the counts for each ice cream flavor, list the choices and tally the total number of children who picked each flavor. In other words, count the number of children who picked chocolate. Then, repeat for the other choices, vanilla and strawberry. Table 1.8 reveals the counts from Table 1.7.

To check your accuracy, you can add all the counts and compare them with the number of participants. The two numbers should be the same.

1.7    Summary

In this chapter, we described differences between parametric and nonparametric tests. We also addressed assumptions by which nonparametric tests would be favorable over parametric tests. Then, we presented an overview of the nonparametric procedures included in this book. We also described the step-by-step approach we use to explain each test. Finally, we included explanations and examples of ranking and counting data, which are two tools for managing data when performing particular nonparametric tests.

The chapters that follow will present step-by-step directions for performing these statistical procedures both by manual, computational methods and by computer analysis using SPSS. In the next chapter, we address procedures for comparing data samples with a normal distribution.

1.8    Practice Questions

1.9    Solutions to Practice Questions

1.  The value ranks are listed in Table 1.11. Notice that there are no ties.

2.  The value ranks are listed in Table 1.12. Notice the tied values. The value of 80 occurred twice and required averaging the rank values of 5 and 6.

The value of 100 occurred three times and required averaging the rank values of 8, 9, and 10.

3.  Table 1.13 shows the passing scores and failing scores using 70 as a passing score. The counts (or frequencies) of passing scores is n_passing = 7. The counts of failing scores is n_failing = 3.

Participant	Value	Rank
1	0 : 36	7
2	0 : 28	6
3	1 : 41	14
4	0 : 37	8
5	1 : 01	13
6	2 : 30	15
7	0 : 44	9
8	0 : 47	10
9	0 : 13	4
10	0 : 24	5
11	0 : 51	11
12	0 : 09	2
13	−2 : 08	1
14	0 : 12	3
15	0 : 56	12

Participant	Score	Rank
1	100	9
2	60	3
3	70	4
4	90	7
5	80	5.5
6	100	9
7	80	5.5
8	20	1
9	100	9
10	50	2

Participant	Score	Pass/Fail
1	100	Pass
2	60	Fail
3	70	Pass
4	90	Pass
5	80	Pass
6	100	Pass
7	80	Pass
8	20	Fail
9	100	Pass
10	50	Fail