This?page?intentionally?left?blank NONPARAMETRIC STATISTICS FOR NON-STATISTICIANS This?page?intentionally?left?blank NONPARAMETRIC STATISTICS FOR NON-STATISTICIANS A Step-by-Step Approach GREGORY W. CORDER DALE I . FOREMAN WILEY A JOHN WILEY Townsend Gaito, 1980; Velleman however, this measure of normality does not suffice for strict levels of defensible analyses. In this chapter, we present three quantitative measures of sample normality. First, we discuss the properties of the normal distribution. Then, we describe how to examine a samples kurtosis and skewness. Next, we describe how to perform and interpret a Kolmogorov-Smirnov one-sample test. In addition, we describe how to perform each of these procedures using SPSS. 2.3 DESCRIBING DATA AND THE NORMAL DISTRIBUTION An entire chapter could easily be devoted to the description of data and the normal distribution and many books do so. However, we will attempt to summarize the concept and begin with a practical approach as it applies to data collection. In research, we often identify some population we wish to study. Then, we strive to collect several independent, random measurements of a particular variable associated with our population. We call this set of measurements a sample. If we use good experimental technique and our sample adequately represents our population, we can study the sample to make inferences about our population. For example, during a routine checkup, your physician draws a sample of your blood instead of all of your blood. This blood sample allows your physician to evaluate all of your blood even though he or she tested only the sample. Therefore, all of your bodys blood cells represent the population about which your physician makes an inference using only the sample. While a blood sample leads to the collection of a very large number of blood cells, other fields of study are limited to small sample sizes. It is not uncommon to collect less than 30 measurements for some studies in the behavioral and social sciences. Moreover, the measurements lie on some scale over which the measurements vary about the mean value. This notion is called variance. For example, a researcher uses some instrument to measure the intelligence of 25 children in a math class. It is highly unlikely that every child will have the same intelligence level. In fact, a good instrument for measuring intelligence should be sensitive enough to measure differ- ences in the levels of the children. The variance, s2, can be expressed quantitatively. It can be calculated using Formula 2.1. =S(*/-*) 2 (2.1) n 1 14 TESTING DATA FOR NORMALITY where x is an individual value in the distribution, x is the distributions mean, and n is the number of values in the distribution. As mentioned in Chapter 1, parametric tests assume that the variances of samples being compared are approximately the same. This idea is called homogeneity of variance. To compare sample variances, Field (2005) suggested that we obtain a variance ratio by taking the largest sample variance and dividing it by the smallest sample variance. The variance ratio should be less than 2. Similarly, Pett ( 1997) indicated that no samples variance be twice as large as any other samples variance. If the homogeneity of variance assumption cannot be met, one would use a nonparametric test. A more common way of expressing a samples variability is with its standard deviation, s. Standard deviation is the square root of variance where s = v ? . In other words, standard deviation is calculated using Formula 2.2. s-f 0.05) indicated that the two testing conditions were not significantly different. Therefore, based on this study, the use of bilingual dictionaries on a math test did not significantly improve scores among limited English proficient students. 2. The results from the analysis are displayed in the SPSS Outputs. Ranks companion - alone Negative Ranks Positive Ranks Ties Total N 8“ 2* 0 0.05) indicated that the two conditions were not significantly different. Therefore, based on this study, the presence of a companion in the woods at night did not significantly influence the males pulse rates. 3. The results from the analysis are displayed in the SPSS Outputs below. Ranks Treatment - Treatment! Negative Ranks Positive Ranks Ties Total N 2 6 00.05) indicated that the two treatments were not significantly different. Therefore, based on this study, neither treatment program resulted in a significantly higher weight loss among obese female teenagers. 4. The results from the analysis are as follows: T = 50 xr =105 and sr = 26.79 z* = -2.05 ES = 0.46 This is a reasonably high effect size, which indicates a strong measure of association. 5. For our example, =10 and p = 0.05/2. Thus, T = 8 and K = 9. The ninth value from the bottom is 1.0 and the ninth value from the top is 7.0. Based on these findings, it is estimated with 95% confidence that the difference in students number of activities before and afte