### Klustermann's Statistics at Rectangle B6

#### Introduction

In the realm of data analysis and statistical modeling, understanding how to interpret results from various statistical tests is crucial for making informed decisions. One such test that has gained popularity in recent years is the Kruskal-Wallis H Test, which is used to compare the medians of three or more independent samples. This test is particularly useful when the assumptions of ANOVA (Analysis of Variance) are not met.

#### The Kruskal-Wallis H Test

The Kruskal-Wallis H Test is a non-parametric alternative to the one-way ANOVA. It does not assume that the data follow a normal distribution, making it suitable for use with ordinal or continuous data. The test works by ranking all the observations from all groups together and then calculating a test statistic based on these ranks.

#### Calculation Steps

1. **Rank the Data**: Combine all the observations into a single list and rank them from smallest to largest. Assign each observation its rank. If there are ties, assign the average rank to all tied values.

2. **Calculate the Sum of Ranks**: For each group, calculate the sum of the ranks assigned to the observations within that group.

3. **Compute the Kruskal-Wallis H Statistic**: Use the following formula to compute the H statistic:

\[

H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)

\]

where \( N \) is the total number of observations, \( k \) is the number of groups, \( R_i \) is the sum of ranks for the \( i \)-th group, and \( n_i \) is the number of observations in the \( i \)-th group.

4. **Determine Degrees of Freedom**: The degrees of freedom for the Kruskal-Wallis test is calculated as \( k-1 \).

5. **Compare with Critical Value**: Compare the computed H statistic with the critical value from the chi-square distribution table for the given degrees of freedom and significance level. If the computed H statistic exceeds the critical value, reject the null hypothesis and conclude that there is a significant difference between the groups.

#### Example Application

Suppose we have the following data representing the scores of four different classes on a math test:

| Class | Score |

|-------|--------|

| A | 85 |

| B | 90 |

| C | 78 |

| D | 88 |

We want to determine if there is a significant difference in the median scores among the four classes.

1. **Combine and Rank**: The combined data is [78, 85, 88,Ligue 1 Express 90], and the ranks are [1, 2, 3, 4].

2. **Sum of Ranks**:

- Class A: Sum of ranks = 1

- Class B: Sum of ranks = 4

- Class C: Sum of ranks = 2

- Class D: Sum of ranks = 3

3. **Compute the Kruskal-Wallis H Statistic**:

\[

H = \frac{12}{(4)(5)} \left( \frac{1^2}{1} + \frac{4^2}{1} + \frac{2^2}{1} + \frac{3^2}{1} \right) - 3(5) = \frac{12}{20} \times (1 + 16 + 4 + 9) - 15 = \frac{12}{20} \times 30 - 15 = 18 - 15 = 3

\]

4. **Degrees of Freedom**: \( k-1 = 4-1 = 3 \)

5. **Critical Value**: Using a chi-square table with 3 degrees of freedom and a significance level of 0.05, the critical value is approximately 7.815.

Since the computed H statistic (3) is less than the critical value (7.815), we fail to reject the null hypothesis. Therefore, there is no significant difference in the median scores among the four classes.

#### Conclusion

The Kruskal-Wallis H Test is a powerful tool for comparing the medians of three or more independent samples without assuming normality. By following the steps outlined above, researchers can effectively analyze their data and draw meaningful conclusions. This method is particularly valuable in fields such as psychology, biology, and social sciences where data may not meet the assumptions required for traditional parametric tests.

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• Klstermann
• B6