Kruskal Wallis Test with Minitab

Kruskal Wallis One-Way Analysis of Variance

The Kruskal Wallis one-way analysis of variance is a statistical hypothesis test to compare the medians among more than two groups.

  • Null Hypothesis (H0): η1 = η2 = … = ηk
  • Alternative Hypothesis (Ha): at least one of the medians is different from others
  • ηi is the median of population i
  • k is the number of groups of our interest

It is an extension of Mann–Whitney test. While the Mann–Whitney test allows us to compare the samples of two populations, the Kruskal–Wallis test allows us to compare the samples of more than two populations.
One key difference between this test and the Mann–Whitney test is the robustness of the test when the populations are not identically shaped. If this is the case, there is a different test, called Mood’s median, which is more appropriate.

Kruskal Wallis One-Way Analysis of Variance: Assumptions

  • The sample data drawn from the populations of interest are unbiased and representative.
  • The data of k populations are continuous or ordinal when the spacing between adjacent values is not constant.
  • The k populations are independent to each other.
  • The Kruskal–Wallis test is robust for the non-normally distributed population.

How Kruskal Wallis One-Way ANOVA Works

The Kruskal–Wallis test works very similarly to the Mann–Whitney Test.
Step 1: Group the k samples from k populations (sample i is from population i) into one single data set and then sort the data in ascending order ranked from 1 to N, where N is the total number of observations across k groups.
Step 2: Add up the ranks for all the observations from sample i and call it ri, where i can be any integer between 1 and k.

Step 3: Calculate the test statistic

kruskal Wallis equation MTB

  • k is the number of groups
  • ni is the sample size of sample i
  • N is the total number of all the observations across k groups
  • rij is the rank (among all the observations) of observation j from group i

Step 4: Make a decision of whether to reject the null hypothesis.

  • Null Hypothesis (H0): η1 = η2 =…= ηk
  • Alternative Hypothesis (Ha): at least one of the medians is different from others

The test statistic follows chi-square distribution when the null hypothesis is true. If T is greater thanKruskal Wallis MTB(the critical chi-square statistic), we reject the null and claim there is at least one median statistically different from other medians. If T is smaller thanKruskal Wallis MTB, we fail to reject the null and claim the medians of k groups are equal.

Use Minitab to Run a Kruskal–Wallis One-Way ANOVA

Case study: We are interested in comparing customer satisfaction among three types of customers using a nonparametric (i.e. distribution-free) hypothesis test: Kruskal–Wallis one-way ANOVA.
Data File: “Kruskal–Wallis” tab in “Sample Data.xlsx”

Kruskal Wallis MTB_00

  • Null Hypothesis (H0): η1 = η2 = η3
  • Alternative Hypothesis(Ha): at least one of the customer types has different overall satisfaction levels from the others

Steps to run a Kruskal–Wallis One-Way ANOVA in Minitab

  1. Click Stat → Nonparametrics → Kruskal–Wallis.
  2. A new window named “Kruskal–Wallis” pops up.
  3. Select “Overall Satisfaction” as the “Response.”
  4. Select “Customer Type” as the “Factor.”
  5. Click “OK.”
  6. The Kruskal–Wallis test results appear in the session window.

Model summary: The p-value of the test is lower than alpha level (0.05), and we reject the null hypothesis and conclude that at least the overall satisfaction median of one customer type is statistically different from the others.

About Michael Parker

Michael Parker is the President and CEO of the Lean Sigma Corporation, a management consulting firm and online six sigma training, certification, and courseware provider. Michael has over 25 years of experience leading and executing lean six sigma programs and projects. As a Fortune 50 senior executive, Michael led oversight of project portfolios as large as 150 concurrent projects exceeding $100 million in annual capital expenditures. Michael has also managed multi-site operations with the accountability of over 250 quality assurance managers, analysts, and consultants. He is an economist by education, earning his Bachelor of Science degree from Radford University while also lettering four years as an NCAA Division I scholarship athlete. Michael earned his Six Sigma Master Black Belt certification from Bank of America and his Black Belt certification from R.R. Donnelley & Sons.

1 Comment

  1. Ammar Alhmedi on October 7, 2019 at 8:41 am

    But why two values for P ? and which one we should consider as p-value to conclude the data comparison outcome ? Thanks in advance,

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