To conduct a Kruskal-Wallis test, follow these steps carefully. This non-parametric test assesses whether there are significant differences among three or more independent groups. It serves as an alternative to the one-way ANOVA, particularly when the data do not fulfill the assumptions of normality or equal variances.

Steps to Perform the Kruskal-Wallis Test

1. Rank the Data: Start by ranking all data points from lowest to highest across all groups. Assign ranks such that the smallest value receives the rank of $1$, the second smallest receives $2$, and so on.

2. Calculate the Sum of Ranks: For each group, compute the sum of the ranks assigned to the observations within that group. For instance, suppose you have four groups with the following number of observations:

   • Group 1: $10$ observations
   • Group 2: $8$ observations
   • Group 3: $12$ observations
   • Group 4: $9$ observations

   The sums of ranks for each group would be calculated as follows:

   • Group 1: $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55$
   • Group 2: $11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 116$
   • Group 3: $19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = 271$
   • Group 4: $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 315$

3. Calculate the Test Statistic: Use the following formula to compute the Kruskal-Wallis test statistic, denoted as $H$:

   $$H = \left( \frac{12}{N(N + 1)} \sum (R_j - \frac{N + 1}{2})^2 \right) - 3(N + 1)$$

   In this formula:

   • $N$ represents the total number of observations.
   • $R_j$ denotes the sum of ranks for each group.

   The test statistic $H$ follows a chi-squared distribution, with degrees of freedom equal to the number of groups minus one.

4. Compare with Critical Value: Finally, compare the calculated test statistic $H$ to the critical value from the chi-squared distribution table. If $H$ exceeds the critical value, you can reject the null hypothesis, indicating that there is a statistically significant difference among the groups.

By following these steps, you can effectively use the Kruskal-Wallis test to analyze your data when the assumptions for parametric tests are not met.