1. Lecture Slides Elementary Statistics Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

2. Chapter 13 Nonparametric Statistics
13-1 Review and Preview
13-2 Sign Test
13-3 Wilcoxon Signed-Ranks Test for Matched Pairs
13-4 Wilcoxon Rank-Sum Test for Two Independent Samples
13-5 Kruskal-Wallis Test
13-6 Rank Correction
13-7 Runs Test for Randomness

3. Key Concept
This section introduces the runs test for randomness, which can be used to determine whether the sample data in a sequence are in a random order.
This test is based on sample data that have two characteristics, and it analyzes runs of those characteristics to determine whether the runs appear to result from some random process, or whether the runs suggest that the order of the data is not random.

4. Definition
A run is a sequence of data having the same characteristic; the sequence is preceded and followed by data with a different characteristic or by no data at all.
The runs test uses the number of runs in a sequence of sample data to test for randomness in the order of the data.

5. Fundamental Principles of the Run Test
Reject randomness if the number of runs is very low or very high.
Example: The sequence of genders FFFFFMMMMM is not random because it has only 2 runs, so the number of runs is very low.
Example: The sequence of genders FMFMFMFMFM is not random because there are 10 runs, which is very high.

6. Caution
The runs test for randomness is based on the order in which the data occur; it is not based on the frequency of the data.
For example, a sequence of 3 men and 20 women might appear to be random, but the issue of whether 3 men and 20 women constitute a biased sample (with disproportionately more women) is not addressed by the runs test.

7. Objective
Apply the runs test for randomness to a sequence of sample data to test for randomness in the order of the data.
Use the following null and alternative hypotheses:

8. Notation
n1 = number of elements in the sequence that have one particular characteristic (The characteristic chosen for n1 is arbitrary.)
n2 = number of elements in the sequence that have the other characteristic
G = number of runs

9. Requirements
1. The sample data are arranged according to some ordering scheme, such as the order in which the sample values were obtained.
2. Each data value can be categorized into one of two separate categories (such as male/female).

10. Test Statistic and Critical Values
For Small Samples and α = 0.05:
If n1 ≤ 20 and n2 ≤ 20, the test statistic is G and the critical values are found using Table A-10.
Decision Criterion: Reject randomness if the number of runs G is:
less than or equal to the smaller critical value found in Table A-10
greater than or equal to the larger critical value found in Table A-1

11. Test Statistic and Critical Values
For Large Samples or α ≠ 0.05:
If n1 > 20 or n2 > 20 or α ≠ 0.05:
Test Statistic:

12. Test Statistic and Critical Values
For Large Samples or α ≠ 0.05:
If n1 > 20 or n2 > 20 or α ≠ 0.05:
Critical values of z: Use Table A-2.
Decision Criterion: Reject randomness if the test statistic z is less than or equal to the negative critical z score or greater than or equal to the positive critical z score.

13. Runs Test for Randomness

14. Runs Test for Randomness

15. Example
Listed below are the most recent (as of this writing) winners of the NBA basketball championship game. Let W denote a winner from the Western Conference, E for the Eastern Conference.
Use a 0.05 significance level to test for randomness in the sequence:
E E W W W W W E W E W E W W W

16. Example - Continued
Requirement Check: The data are arranged in order, and each data value is categorized into one of two separate categories.
We must find the values of n1, n2, and G.
n1 = number of Eastern Conference Winners = 5
n2 = number of Western Conference Winners = 10
G = number of runs = 8

17. Example - Continued
Because n1 ≤ 20 and n2 ≤ 20 and α = 0.05, the test statistic is G = 8.
In Table A-10, the critical values are 3 and 12.
Because G = 8 is neither less than or equal to 3 nor greater than or equal to 12, we do not reject randomness.
There is not sufficient evidence to reject randomness in the sequence of winners.

18. Example
Data Set 3 in Appendix B lists data from 107 study subjects. Let’s consider the sequence of listed genders indicated below.
Test the claim that the sequence is random using a 0.05 significance level.

19. Example - Continued
Requirement Check: The data are arranged in order, and each data value is categorized into one of two separate categories (male / female).
We must find the values of n1, n2, and G.
Examination of the sequence of 107 genders gives:
n1 = number of males = 92
n2 = number of females = 15
G = number of runs = 25

20. Example - Continued
Since n1 > 20, we need to calculate the test statistic G:

21. Example - Continued
Since n1 > 20, we need to calculate the test statistic G:
Because the significance level is 0.05, the critical values are z = ±1.96.
The test statistic does not fall within the critical regions, so we fail to reject the null hypothesis of randomness.
The given sequence appears to be random.