1. Introduction to Nonparametric Statistics
CD-ROM Chapter 15
Introduction to Nonparametric Statistics

2. Chapter 15 - Chapter Outcomes
After studying the material in this chapter, you should be able to:
Recognize when and how to use the runs test and testing for randomness.
Know when and how to perform a Mann-Whitney U test.
Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it in a decision-making context.
Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA.

3. Nonparametric Statistics
Nonparametric statistical procedures are those statistical methods that do not concern themselves with population distributions and/or parameters.

4. The Runs Test
The runs test is a statistical procedure used to determine whether the pattern of occurrences of two types of observations is determined by a random process.

5. The Runs Test
A run is a succession of occurrences of a certain type preceded and followed by occurrences of the alternate type or by no occurrences at all.

6. The Runs Test (Table 15-1)

7. The Runs Test (Small Sample Example)
H0: Computer-generated numbers are random between 0.0 and 1.0.
HA: Computer-generated numbers are not random .
Runs:
There are r = 10 runs
From runs table (Appendix K) with n1 = 9 and n2 = 11, the critical value of r is 6

8. The Runs Test (Small Sample Example)
Test Statistic:
r = 10 runs
Critical Values from Runs Table:
Possible
Runs:
Reject H0
Reject H0
Do not reject H0
Decision:
Since r = 10, we do not reject the null hypothesis.

9. MEAN AND STANDARD DEVIATION FOR r
Large Sample Runs Test
MEAN AND STANDARD DEVIATION FOR r
where:
n1 = Number of occurrences of first type
n2 = Number of occurrences of second type

10. TEST STATISTIC FOR LARGE SAMPLE RUNS TEST

11. Large Sample Runs Test (Example 15-2)
Table 15-2
OOOUOOUOUUOOUUOOOOUUOUUOOO
UUUOOOOUUOOUUUOUUOOUUUUU
OOOUOUUOOOUOOOOUUUOUUOOOU
OOUUOUOOUUUOUUOOOOUUUOOO
n1 = 53 “O’s”
n2 = 47 “U’s”
r = 45 runs

12. Large Sample Runs Test (Example 15-2)
H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.
H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce level.
 = 0.05
Rejection Region  /2 = 0.025
Rejection Region  /2 = 0.025
Since z= > and < 1.96, we do not reject H0,

13. Mann-Whitney U Test • The two samples are independent and random.
The Mann Whitney U test can be used to compare two samples from two populations if the following assumptions are satisfied:
• The two samples are independent and random.
• The value measured is a continuous variable.
• The measurement scale used is at least ordinal.
• If they differ, the distributions of the two populations will differ only with respect to the central location.

14. Mann-Whitney U Test U-STATISTICS where:
n1 and n2 are the two sample sizes
R1 and R2 = Sum of ranks for samples 1 and 2

15. Mann-Whitney U Test - Large Samples -
MEAN AND STANDARD DEVIATION FOR THE U-STATISTIC
where:
n1 and n2 = Sample sizes from populations 1 and 2

16. Mann-Whitney U Test - Large Samples -
MANN-WHITNEY U-TEST STATISTIC

17. Mann-Whitney U Test (Example 15-4)
Rejection Region  = 0.05
Since z= > , we do not reject H0,

18. Wilcoxon Matched-Pairs Test
The Wilcoxon matched pairs signed rank test can be used in those cases where the following assumptions are satisfied:
• The differences are measured on a continuous variable.
• The measurement scale used is at least interval.
• The distribution of the population differences is symmetric about their median.

19. Wilcoxon Matched-Pairs Test
WILCOXON MEAN AND STANDARD DEVIATION
where:
n = Number of paired values

20. Wilcoxon Matched-Pairs Test
WILCOXON TEST STATISTIC

21. Kruskal-Wallis One-Way Analysis of Variance
Kruskal-Wallis one-way analysis of variance can be used in one-way analysis of variance if the variables satisfy the following:
• They have a continuous distribution.
• The data are at least ordinal.
• The samples are independent.
• The samples come from populations whose only possible difference is that at least one may have a different central location than the others.

22. Kruskal-Wallis One-Way Analysis of Variance
H-STATISTIC
where:
N = Sum of sample sizes in all samples
k = Number of samples
Ri = Sum of ranks in the ith sample
ni = Size of the ith sample

23. Kruskal-Wallis One-Way Analysis of Variance
CORRECTION FOR TIED RANKINGS
where:
g = Number of different groups of ties
ti = Number of tied observations in the ith tied group of scores
N = Total number of observations

24. Kruskal-Wallis One-Way Analysis of Variance
H-STATISTIC CORRECTED FOR TIED RANKINGS

25. Key Terms • Kruskal-Wallis One-Way Analysis of Variance • Run
• Mann-Whitney U Test
• Nonparametric Statistical Procedure
• Run
• Runs Test
• Wilcoxon Test