The Friedman nonparametric hypothesis test is an alternative to the one-way ANOVA with repeated measures. Friedman test was developed by an American economist Milton Friedman.

The Friedman non parametric hypothesis test is to test for differences between groups (three or more paired groups) when the dependent variable is at least ordinal. Friedman test to be preferred when compared to other non parametric test in a situation where same parameter has been measured under different conditions on the same subject. Example: Patient Serum content monitoring before treatment, after one month and after three months of treatment.

Friedman’s test is similar to the Kruskal-Wallis Test and also an extension of sign test. This test is best statistic to use for a repeated measures type of experiment to determine if a particular factor also has an effect.

The Friedman test is to test the k paired samples (k>2) of n size, are from the same population or the samples from populations having similar properties, considering the position parameter.

## Assumptions of the Friedman Test

• The group is a random sample from the population
• No interaction between blocks (rows) and treatment levels (columns)
• The one group that is measured on three or more different occasions
• Data should be at least an ordinal or continuous
• The samples are do not need to be normally distributed

## Procedure to conduct Friedman Test

• Rank the each row (block) together and independently of the other rows. When there are ties, the average ranks of the observations.
• Sum the ranks for each columns (treatments) and then sum the squared columns total
• Compute the test statistic
• Determine critical value from Chi-Square distribution table with k-1 degrees of freedom
• Formulate decision and conclusion

## Calculation of Friedman Non Parametric Hypothesis Test

The test statics of Friedman’s test is

where Rj is the sum of the ranks for sample j.

n is the number of independent blocks

k is the number of groups or treatment levels

DF= number of groups -1 (k-1)

• Null Hypotheses H0: Median treatment effects of the population are all the same
• Alternative Hypotheses H1: There is a difference in treatment effects.

## Example of Friedman Non Parametric Hypothesis Test

Department of Public health and safety monitors the measures taken to cleanup drinking water were effective. Trihalomethanes (THMs) at 12 counties drinking water compared before cleanup, 1 week later and 2 weeks after cleanup.

• Null Hypothesis H0 = the cleanup system had no effect on the THMs
• Alternative Hypothesis H1= the cleanup system effected the THMs

Significance level α=0.05

Calculate the Rj

Q=20.16

For the values of independent blocks (n) greater than 20 and/or values of groups (k) greater than 6, use χ2 table with k-1 degrees of freedom otherwise use the Friedman table

Calculated Q value is greater than the critical value of Q for a 0.05 significance level. Qcalculated >Qcritical hence reject the null hypotheses.

So, it is concluded that the cleanup system effected the THMs of drinking water

## Six Sigma Black Belt Certification Friedman Test Questions:

Question: A psychologist monitored the same group of student’s reactions while watching comic films and compared the student’s reaction when watching political and drama films. The tabulated data skewed. Which of the following test to be used to analyse the data?

(A) Mood’Median test
(B) Kruskal-Wallis test
(C) Mann-Whitney U test
(D) Friedman Test

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