Hypothesis test for difference in medians among more than two samples


The test scores of three groups of people are saved as separate vectors in R.

group1 <- rnorm(100, mean = 75, sd = 10)
group2 <- rnorm(100, mean = 85, sd = 10)
group3 <- rnorm(100, mean = 95, sd = 10)

I want to know if there is a significant difference in the medians between these groups. I know that I could test group 1 versus group 2 using the Wilcoxon test, like so.

wilcox.test(group1, group2)

However, this compares only two groups at a time, and I would like to compare all three simultaneously. I would like a statistical test that yields a p value at the 0.05 significance level. Could someone please help?

Edit #1 – Mood’s median test

Following user Hibernating’s suggested answer, I tried Mood’s median test.

median.test <- function(x, y){
    z <- c(x, y)
    g <- rep(1:2, c(length(x), length(y)))
    m <- median(z)
    fisher.test(z < m, g)$p.value

median.test(group1, group2)

However, this approach allows me to test for a significant difference between the medians of only two groups at a time. I am not sure of how to use it to compare the medians of all three simultaneously.

Edit #2 – Kruskal-Wallis test

User dmartin’s suggested answer appears to be more or less what I need, and allows me to test all three groups simultaneously.

kruskal.test(list(group1, group2, group3))

Edit #3

User Greg Snow helpfully notes in his answer that the Kruskal-Wallis test is appropriate as long as it makes strict assumptions that make it also a test of means.


The Kruskal-Wallis test could also be used, as it’s a non-parametric ANOVA. Additionally, it is often considered to be more powerful than Mood’s median test. It can be implemented in R using the kruskal.test function in the stats package in R.

To respond to your edit, interpreting K-W is similar to a one-way ANOVA. A significant p-value corresponds to rejected the null that all three means are equal. You must use a follow-up test (again, just like an ANOVA), to answer questions about specific groups. This typically follows specific research questions you may have. Just by looking at the parameters of the simulation, all three groups should be significantly different from one another if you do a follow-up test (as they’re all 1 SD apart with N = 100).

Source : Link , Question Author : Alexander , Answer Author : dmartin

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