# Kruskal-Wallis Test

The Kruskal-Wallis Test was developed by Kruskal and Wallis (1952) jointly and is named after them. The Kruskal-Wallis test is a nonparametric (distribution free) test, and is used when the assumptions of ANOVA are not met. They both assess for significant differences on a continuous dependent variable by a grouping independent variable (with three or more groups). In the ANOVA, we assume that distribution of each group is normally distributed and there is approximately equal variance on the scores for each group. However, in the Kruskal-Wallis Test, we do not have any of these assumptions. Like all non-parametric tests, the Kruskal-Wallis Test is not as powerful as the ANOVA.

**Hypothesis:**

**Null hypothesis:** Null hypothesis assumes that the samples are from identical populations.

**Alternative hypothesis:** Alternative hypothesis assumes that the samples come from different populations.

**Questions answered:**

How do test scores differ between the different grade levels in elementary school?

Does a marketing strategy impact intent to purchase scores among different races?

**Procedure:**

1. Arrange the data of both samples in a single series in ascending order.

2. Assign rank to them in ascending order. In the case of a repeated value, or a tie, assign ranks to them by averaging their rank position.

3. Then sum up the different ranks, e.g. R_{1} R_{2} R_{3….}, for each of the different groups..

4. To calculate the value, apply the following formula:

Where,

H = Kruskal-Wallis Test statistic

N = total number of observations in all samples

T_{i} = Sum of the ranks assigned

The Kruskal-Wallis test statistic is approximately a chi-square distribution, with k-1 degrees of freedom where n_{i }should be greater than 5. If the calculated value of the Kruskal-Wallis test is less than the critical chi-square value, then the null hypothesis cannot be reject. If the calculated value of Kruskal-Wallis test is greater than the critical chi-square value, then we can reject the null hypothesis and say that the sample comes from a different population.

**Assumptions**

1. We assume that the samples drawn from the population are random.

2. We also assume that the cases of each group are independent.

3. The measurement scale for should be at least ordinal.

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