1. Chapter 18 – Nonparametric Statistics
Introduction to Business Statistics, 6e
Kvanli, Pavur, Keeling
Chapter 18 – Nonparametric Statistics
Slides prepared by Jeff Heyl, Lincoln University
©2003 South-Western/Thomson Learning™

2. Nonparametric Statistics
Many of the nonparametric statistical tests answer the same sorts of questions as the parametric tests. With nonparametric tests the assumptions can be relaxed considerably.
Consequently, nonparametric methods are used for situations that violate the assumptions of parametric procedures.

3. The Runs Test (Small Samples)
Sequence 1: HHHHH TTTTT
Run 1 Run 2
Sequence 2: H T H T H T H T
Run 1 Run 10
Sequence 3: T H H H T H T H T T
Run 1 Run 7
R = number of runs

4. Test for Randomness; The Runs Test
Arrangement Number Arrangement Number of Runs
1 HHHHHTTTTT 2  sequence 1
2 HHHHTHTTTT 4
3 HHHHTTHTTT 4
4 HHHHTTTHTT 4
5 HHHHTTTTHT 4 .
130 THHHHTTTHT 5
131 THHHHTTTTH 4
132 THHHTHHTTT 5
133 THHHTHTHTT 7  sequence 3
134 THHHTHTTTT 7
248 TTTTHHHHTH 4
249 TTTTHHHTHH 4
250 TTTTHHTHHH 4
251 TTTTHTHHHH 4
252 TTTTTHHHHH 2
Table 18.1

5. Test for Randomness; The Runs Test
Number of Number of Times Relative Cumulative
Runs (R) R Occurred Frequency Relative Frequency
Table 18.2

6. Test for Randomness; The Runs Test
Number of Number of Times Relative Cumulative
Runs (R) R Occurred Frequency Relative Frequency
Number of arrangements = A = n!
n1!n2!
Table 18.2

7. The Runs Test (Small Samples)
Ho: The sequence was generated in a random manner
Ha: The sequence was not generated in a random manner
Reject Ho if R ≤ k1 or R ≥ k2
P(R ≤ k1) ≤ = .025  2
Largest value
Smallest value
P(R ≥ k1) ≤ = .025  2

8. Runs Test (Large Samples)
2n1n2
n1 + n2
R =
2n1n2(2n1n2 - n1 - n2)
(n1 + n2)2(n1 + n2 - 1)
Ho: The sequence was generated in a random manner
Ha: The sequence was not generated in a random manner
Reject Ho if |Z| > Z/2
Z =
R - µR R

9. Excel Runs Test
Figure 18.1

10. Regression and Runs Test
Figure 18.2

11. Residual Plot • Residuals Time 100 – 50 – 0 – -50 – -100 – | 10 20 30| 10 20 30 40
Figure 18.3

12. Nonparametric Test Central Tendency: Two Populations| µ1 µ2
Females
Males
Ho: µ2 ≤ µ1
Ha: µ2 > µ1
Height
Figure 18.4

13. Dependent (Paired) Samples
Couple 1
Couple 2
Couple 3
(etc.)
Data
Sample 1
(women)
Sample 2
(men) X .
Figure 18.5

14. Mann-Whitney U Test for Independent Samples
Ho: The two populations have identical probability distributions
Ha: The two populations differ in location
T1 = sum of the ranks of the observations from the first sample in this pooled sample
T2 = sum of the ranks of the observations from the second sample

15. Mann-Whitney U Small Samples
U1 = n1n T1
n1(n1 + 1) 2
U2 = n1n T2
n2(n2 + 1) 2

16. Mann-Whitney U Small Samples
Procedure:
1. Assume that n1 ≤ n2 (reverse the samples if necessary)
2. Determine U1 and U2
3. Use the value from Table A.10 to test Ho versus Ha
Two-Sided Test
Ha: the two populations differ in location
Reject Ho if Table A.10 value for U < /2, where U = the minimum of U1 and U2

17. Mann-Whitney U Small Samples
Procedure:
1. Assume that n1 ≤ n2 (reverse the samples if necessary)
2. Determine U1 and U2
3. Use the value from Table A.10 to test Ho versus Ha
One-Sided Test
Ha: population 1 is shifted to the right of population 2
Reject Ho if Table A.10 value for U < , where U = U1
One-Sided Test
Ha: population 1 is shifted to the left of population 2
Reject Ho if Table A.10 value for U < , where U = U2

18. Mann-Whitney U Large Samples2
µU =
n1n2(n1 + n2 + 1) 12
U =
Z =
U2 - µU U 2

19. Mann-Whitney U Large Samples
Ho: The two populations have identical probability distributions
Determine U2 = n1n T2
n2(n2 + 1) 2
Reject Ho if |Z| > Z/2
Ha: The two populations differ in location
Two-Sided Test

20. Mann-Whitney U Large Samples
Ho: The two populations have identical probability distributions
Determine U2 = n1n T2
n2(n2 + 1) 2
Reject Ho if Z > Z
Ha: Population 1 is shifted to the right of population 2
One-Sided Test
Reject Ho if Z < -Z
Ha: Population 1 is shifted to the left of population 2

21. Mann-Whitney Test
Figure 18.6

22. Wilcoxon Signed Rank Test for Paired Samples
When small samples from suspected nonnormal populations are used, a nonparametric technique is required.
The Wilcoxon test is used for such situations.

23. Wilcoxon Signed Rank Test for Paired Samples
Determine the difference for each sample pair
Arrange the absolute value of these differences in order, assigning a rank to each
Let T+ = sum of ranks having a positive value and T- = sum of ranks for the negative values
T+, T-, or T = the minimum of T+ and T- is used to define a test of Ho versus Ha

24. Wilcoxon Signed Rank Test for Small Paired Samples
Ho: The population differences are centered at 0
Two-Sided Test
Ha: the population differences are not centered at 0
Reject Ho if T ≤ table value, where T = the minimum of T+ and T-

25. Wilcoxon Signed Rank Test for Small Paired Samples
Ho: The population differences are centered at 0
One-Sided Test
Reject Ho if T- ≤ table value
Ha: the population differences are centered at a value > 0
Reject Ho if T+ ≤ table value
Ha: the population differences are centered at a value < 0

26. Wilcoxon Signed Rank Test for Large Paired Samples
n(n + 1) 4
µT = +
n(n + 1)(2n + 1) 24
T =
Z =
T+ - µT T +

27. Wilcoxon Signed Rank Test for Large Paired Samples
Ho: The population differences are centered at 0
Two-Sided Test
Ha: the population differences are not centered at 0
Reject Ho if |Z| > Z/2

28. Wilcoxon Signed Rank Test for Large Paired Samples
Ho: The population differences are centered at 0
One-Sided Test
Reject Ho if Z > Z
Ha: the population differences are centered at a value > 0
Reject Ho if Z < -Z
Ha: the population differences are centered at a value < 0

29. Hardwood ConcentrationZ
-1.41
p value = area of
= .0793
Figure 18.7

30. Wilcoxon Test Solution
Figure 18.8

31. Kruskal-Wallis Test
The nonparametric counterpart to the one-way ANOVA test
Ho: the k populations have identical probability distributions
Ha: at least two of the populations differ in location
KW = ∑ (n + 1) 12
n(n + 1)
Ti2 ni k
i = 1
Reject Ho if KW is “large”

32. Kruskal-Wallis Statistic
Area = .005
12.84
13.83
p value < .005 2
Figure 18.9

33. Kruskal-Wallis Test
Figure 18.10

34. The Friedman Test
The nonparametric counterpart to the randomized block ANOVA test
Ho: the k populations have identical probability distributions
Ha: at least two of the populations differ in location
FR = ∑Ti2 - 3b(k + 1) 12
bk(k + 1) k
i = 1
Reject Ho if FR > 2.df

35. The Friedman Test
Figure 18.11

36. Spearman’s Rank Correlation
Figure 18.12

37. Spearman’s Rank Correlation
Spearman’s is the nonparametric counterpart to the Pearson Correlation
rs =
∑R(x)R(y) - [∑R(x)][∑R(y)] / n
∑R2(x) - [∑R(x)]2 / n ∑R(y) - [∑R(y)]2 / n
rs = 1 -
6∑d2
n(n2 - 1)

38. Measure of Association
15 –
12 –
9 –
7 –
5 – Y | 1 2 3 4 5 6 7 8 9 10 11 X
r < 1 From equation 19-15
Interval X
data Y
4 –
3 –
2 –
1 –
rs = 1 From equation 19-17
Ranks R(X)
R(Y)
Perfect agreement,
so rs = 1
Figure 18.13

39. Spearman’s Rank Correlation
Figure 18.14

40. Spearman’s Rank Correlation
Two-Sided Test
Ho: no association exists between X and Y
Ha: association does exist between X and Y
Reject Ho if |rs| > (table value)

41. Spearman’s Rank Correlation
One-Sided Test
Ho: no association exists between X and Y
Ha: a positive relationship exists between X and Y
Reject Ho if rs > (table value)
Ha: a negative relationship exists between X and Y
Reject Ho if rs < -(table value)