The Kruskal-Wallis test is a non-parametric statistical test used to compare three or more independent groups. It measures whether there are any differences between the medians of these groups. It is an extension of the Mann-Whitney U test, which is used to compare two independent groups.

The Kruskal-Wallis test is appropriate when the assumptions of normality and equal variances are violated. This makes it a useful alternative to parametric tests, such as the one-way ANOVA, when these assumptions cannot be met.

To perform the Kruskal-Wallis test, the data should be ranked and summed within each group. The total sum of ranks for each group is then calculated. These sums are used to compute the test statistic, which follows a chi-square distribution with degrees of freedom equal to the number of groups minus one. If the test statistic is greater than the critical value, it indicates that there are significant differences between the groups.

The Kruskal-Wallis test is commonly used in various fields, including social sciences, healthcare, and market research. It allows researchers to analyze data that cannot be assumed to be normally distributed. By using ranks instead of actual values, it provides a reliable method for comparing groups without relying on specific distributional assumptions.

In conclusion, the Kruskal-Wallis test is a powerful tool for comparing three or more independent groups when the assumptions of normality and equal variances cannot be met. Its non-parametric nature makes it robust and flexible, allowing researchers to draw meaningful conclusions even with skewed or non-normal data.