Mann-Whitney U Test (Nonparametric Test).pptx

1. Mann-Whitney
U Test

2. LEARNING OBJECTIVES
● Determine when to use parametric to nonparametric
tests
● Highlight the history and the assumptions of the Mann-
Whitney U test.
● Differentiate t-test of independence to Mann-Whitney U
test.
● Identify the critical values and hypothesis for Mann-
Whitney U test.
● Calculate the U statistic and infer it.

3. What is U Test
01

4. What can you say about the shape of
the curves of the two figures above?

5. What can you say about the shape of the curves of the two figures above?
bell-shaped curve off-centered curve
normal distribution
skewed distribution or
not normal distribution

6. off-centered curve
skewed distribution or
not normal distribution
Use
NONPARAMETRI
C TESTS
analyze statistical
data
infer findings
to
in order to

7. Wilcoxon Signed
Rank Test
Spearman’s
rho
NONPARAMETRIC TESTS
Mann-
Whitney U
Test
Kruskal Wallis
Test
Friedman Test

8. NONPARAMETRIC TESTS
Mann-Whitney
U Test
What is
?

9. Mann-Whitney U
Test

10. Mann-Whitney U
Test
Both the one-sample signed rank and
the two-sample rank sum test were
devised by me as part of a significance
test that opposed a point null hypothesis
against its complementary alternative, or
equal vs not equal. However, I only
tabulated a few points for the equal-
sample size situation in that work
(although I provided larger tables in a
later publication).

11. Mann-Whitney U
Test
In 1947, my student
Donald Ransom Whitney
and I published a paper
that featured a recurrence
that allowed us to compute
tail probabilities for arbitrary
sample sizes as well as
tables for sample sizes of
eight or fewer.

12. Mann-Whitney U
Test

13. Mann-Whitney U
Test
The Mann-Whitney U test is the nonparametric
counterpart to the t-test for independent samples
Group A Group B

14. Assumptions,Critic
al Values and
Hypothesis
02

15. ASSUMPTION
S
two independent samples
at least ordinal scaled
characteristic of groups
not normally distributed

16. ASSUMPTION
S
two independent samples
Group A Group B

17. ASSUMPTION
S
at least ordinal scaled characteristic of groups

18. at least ordinal scaled characteristic of groups
ASSUMPTION
S
Group A Group B
comparing gender to salary

19. ASSUMPTION
S
not normally distributed

20. FORMULA (if n ≤ 20 for both
groups)
𝑼𝟏 = 𝒏𝟏𝒏𝟐 +
𝒏𝟏(𝒏𝟏 + 𝟏)
𝟐
− 𝑹𝟏
U statistic for the first group
𝑼𝟐 = 𝒏𝟏𝒏𝟐 +
𝒏𝟐(𝒏𝟐 + 𝟏)
𝟐
− 𝑹𝟐
U statistic for the second group
final U statistic
𝑼 = 𝒎𝒊𝒏 (𝑼𝟏, 𝑼𝟐)
Where:𝑹𝟏 is the sum of the ranks of
the first group, 𝒏𝟏 is the sample size
of the first group, and 𝒏𝟐 is the
sample size of the second group
Where:𝑹𝟐 is the sum of the ranks of
the second group, 𝒏𝟏 is the sample
size of the first group, and 𝒏𝟐 is the
sample size of the second group
the lower U value is the U statistic

21. CRITICAL
VALUES
for α = 0.05

22. HYPOTHESIS
H0: There is no difference (in terms of
central tendency) between the two
groups in the population.
H1: There is a difference (in terms of
central tendency) between the two
groups in the population.
if U > critical
value
if U ≤ critical
value

23. Computing the U
Statistic
03

24. EXAMPLE Gender Response time
female 34
male 33
male 35
female 37
female 44
male 45
female 36
male 39
female 41
female 43
male 42
A research was
conducted to see if
males and females have
different response time
(in seconds) when it
comes to problems.

25. EXAMPLE Gender Response time
female 34
male 33
male 35
female 37
female 44
male 45
female 36
male 39
female 41
female 43
male 42
Do a normality test
not normally
distributed

26. EXAMPLE Gender Response time
female 34
male 33
male 35
female 37
female 44
male 45
female 36
male 39
female 41
female 43
male 42
Group the data
according to
gender group

27. EXAMPLE
Gender
Response
time Rank
female 34
female 36
female 41
female 43
female 44
female 37
male 45
male 33
male 35
male 39
male 42
Assign rank to
each data
starting from
lowest to
highest 1
2
3
4
5
6
7
8
9
10
11

28. EXAMPLE
Gender
Response
time Rank
female 34
female 36
female 41
female 43
female 44
female 37
male 45
male 33
male 35
male 39
male 42
Calculate
the rank
sums of
each
group and
find the
number of
samples
per group
1
2
3
4
5
6
7
8
9
10
11
R1=37
2+4+7+9+10+5=37
R2=29
11+1+3+6+8=29
n1=6
n2=5

29. EXAMPLE
Find the
critical
value (CV)
at 0.05
significance
level
n1=6
n2=5
CV = 3

30. EXAMPLE
Find the U
statistic
for each
group
R1=37
R2=29
n1=6
n2=5
𝑼𝟏 = 𝒏𝟏𝒏𝟐 +
𝒏𝟏(𝒏𝟏 + 𝟏)
𝟐
− 𝑹𝟏
𝑼𝟐 = 𝒏𝟏𝒏𝟐 +
𝒏𝟐(𝒏𝟐 + 𝟏)
𝟐
− 𝑹𝟐
CV = 3
𝑼𝟏 = (𝟔)(𝟓) +
𝟔 (𝟔 + 𝟏)
𝟐
− 𝟑𝟕
𝑼𝟏 = 𝟏𝟒
𝑼𝟐 = (𝟔)(𝟓) +
𝟓(𝟓 + 𝟏)
𝟐
− 𝟐𝟗
𝑼𝟐 = 𝟏𝟔

31. EXAMPLE
Find the U
statistic
and
compare
to CV
CV = 3
𝑼𝟏 = 𝟏𝟒
𝑼𝟐 = 𝟏𝟔
𝑼 = 𝒎𝒊𝒏 (𝑼𝟏, 𝑼𝟐)
𝑼 = 𝒎𝒊𝒏 (𝟏𝟒, 𝟏𝟔)
𝑼 = 𝟏𝟒
H0: if U >
critical value
H1: if U ≤
critical value

32. EXAMPLE
Find the U
statistic
and
compare
to CV
CV = 3
𝑼𝟏 = 𝟏𝟒
𝑼𝟐 = 𝟏𝟔
𝑼 = 𝒎𝒊𝒏 (𝑼𝟏, 𝑼𝟐)
𝑼 = 𝒎𝒊𝒏 (𝟏𝟒, 𝟏𝟔)
𝑼 = 𝟏𝟒
H0: if U >
critical value
H1: if U ≤
critical value
𝟏𝟒 > 𝟑

33. EXAMPLE
Make
conclusions
based on
the test
statistic
H0: if U >
critical value
Accept H0
There is no difference between the
males and females towards the
response time when it comes to
problems.