The Friedman Non Parametric hypothesis test is an alternative to the one-way ANOVA with repeated measures. Friedman test was developed by the 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 is preferred when compared to other non parametric tests in a situation where the 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 the sign test. This test is the 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 whether 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 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 column (treatments) and then sum the squared columns’ total
  • Compute the test statistic
  • Determine critical value from the 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

Friedman Non Parametric Hypothesis Test

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 whether the measures taken to clean up drinking water were effective. Trihalomethanes (THMs) in 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

Friedman Non Parametric Hypothesis Test


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

The 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 Non Parametric Test Questions:

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

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


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D:  The Friedman test is to test for differences between groups when the dependent variable is at least ordinal. It is used to compare three or more matched or paired groups.

Comments (7)

Hi Abdul,
When ties occur in the Friedman test, one common approach is to assign tied observations the average rank of the tied ranks. This is known as the average rank method.
In the above example, counties 3 and 6 have ties, we took the average ranks.


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