The Mann-Whitney U The Mann-Whitney U TestTest

What you need to knowWhat you need to know

When Should I Use the When Should I Use the Mann-Whitney U Test?Mann-Whitney U Test?

• Non-parametric distributionNon-parametric distribution• Independent MeasuresIndependent Measures• Ratio or interval data, but must be Ratio or interval data, but must be

reduced to ordinal data because it is non-reduced to ordinal data because it is non-parametricparametric

• Different variances for two groupsDifferent variances for two groups• It assumes Random assignments to It assumes Random assignments to

groupsgroups• Test statistic: UTest statistic: U

Mann-Whitney U TestMann-Whitney U Test

What is it for? A non-parametric test to compare What is it for? A non-parametric test to compare the central tendencies of two groupsthe central tendencies of two groups

It assumes Random assignments to groupsIt assumes Random assignments to groups Test statistic: UTest statistic: U

SampleNull hypothesisThe two groupsHave the same

median

Null distributionU with n1, n2

compare

How unusual is this test statistic?

P < 0.05 P > 0.05

Reject Ho Fail to reject Ho

Mann-Whitney U test

Test statisticU1 or U2

Null HypothesisNull Hypothesis

Involves the creation of two competing Involves the creation of two competing explanations for the data recorded.explanations for the data recorded.• Idea 1:These are pattern-less random data. Idea 1:These are pattern-less random data.

Any observed patterns are due to chance. This Any observed patterns are due to chance. This is the null hypothesis is the null hypothesis H0H0

• Idea 2: There is a defined pattern in the data. Idea 2: There is a defined pattern in the data.

This is the alternative hypothesis This is the alternative hypothesis H1H1 Without the statement of the competing Without the statement of the competing

hypotheses, no meaning test can be run.hypotheses, no meaning test can be run.

Step OneStep One

Arrange all the observations into a Arrange all the observations into a single ranked series. That is, rank all single ranked series. That is, rank all the observations without regard to the observations without regard to which sample they are in. which sample they are in.

In other words, combine all of the In other words, combine all of the data from both groups into a single data from both groups into a single column, in order, but keep track of column, in order, but keep track of what group they came from.what group they came from.

Mann-Whitney U TestMann-Whitney U Test

If you have ties:If you have ties:• Rank them anyway, pretending they Rank them anyway, pretending they

were slightly differentwere slightly different• Find the average of the ranks for the Find the average of the ranks for the

identical values, and give them all that identical values, and give them all that rankrank

• Carry on as if all the whole-number Carry on as if all the whole-number ranks have been used upranks have been used up

ExampleExampleData

142542141814

ExampleExampleSortedData

2 G12 G24 G25 G114 G114 G214 G118 G2

Data

142542141814

ExampleExampleSortedData

2 G12 G24 G25 G114 G114 G214 G118 G2

Data

142542141814

TIES

ExampleExampleSortedData

2 G12 G24 G25 G114 G114 G214 G118 G2

Data

142542141814

TIES

Rank them anyway, pretending they were slightly different

ExampleExampleRank A

2 G12 G24 G25 G114 G114 G214 G118 G2

12345678

SortedDataData

142542141814

ExampleExampleRank A

2 G12 G24 G25 G114 G114 G214 G118 G2

12345678

SortedDataData

142542141814

Find the average of the ranks for the identical values, and give them all that rank

ExampleExampleRank A

2 G12 G24 G25 G114 G114 G214 G118 G2

12345678

SortedDataData

142542141814

Average = 1.5

Average = 6

ExampleExampleRank A

2 G12 G24 G25 G114 G114 G214 G118 G2

12345678

SortedDataData

142542141814

1.51.5346668

Rank

ExampleExampleRank A

2 G12 G24 G25 G114 G114 G214 G118 G2

12345678

SortedDataData

142542141814

1.51.5346668

Rank

First, sort them back into the two groups,Then these can now be used for the Mann-Whitney U test

Step TwoStep Two

Add up the ranks for the Add up the ranks for the observations which came from observations which came from sample 1(the smaller group, fewer sample 1(the smaller group, fewer participants). participants).

Then add up the sum of ranks in Then add up the sum of ranks in sample 2 (Larger group)sample 2 (Larger group)

Step ThreeStep Three

UU is then given by: is then given by:

where where nn1 is the sample size for 1 is the sample size for sample 1(smaller group), and sample 1(smaller group), and RR1 is 1 is the sum of the ranks in sample 1the sum of the ranks in sample 1

Step FourStep Four

Step FiveStep Five

The smaller value of The smaller value of UU1 and 1 and UU2 is the 2 is the one used when consulting one used when consulting significance tables. significance tables.

CompareCompare

If the smaller value of U1 or U2 is If the smaller value of U1 or U2 is smaller than the critical value in the smaller than the critical value in the chart, then the probability that the chart, then the probability that the differences in the groups is obtained differences in the groups is obtained by chance is less than 0.05, and you by chance is less than 0.05, and you may reject the null hypothesis.may reject the null hypothesis.