# Interpret all statistics for Kruskal-Wallis Test

Find definitions and interpretation guidance for every statistic that is provided with Kruskal-Wallis.

## N

The sample size (N) is the total number of observations in each group.

### Interpretation

The sample size affects the confidence interval and the power of the test.

Usually, a larger sample yields a narrower confidence interval. A larger sample size also gives the test more power to detect a difference. For more information, go to What is power?.

## Median

The median is the midpoint of the data set. This midpoint value is the point at which half of the observations are above the value and half of the observations are below the value. The median is determined by ranking the observations and finding the observation at the number [N + 1] / 2 in the ranked order. If your data contain an even number of observations, the median is the average value of the observations that are ranked at numbers N / 2 and [N / 2] + 1.

### Interpretation

The sample median is an estimate of the population median of each group. The overall median is the median of all observations.

## Mean rank

The mean rank is the average of the ranks for all observations within each sample. Minitab uses the mean rank to calculate the H-value, which is the test statistic for the Kruskal-Wallis test.

To calculate the mean rank, Minitab ranks the combined samples. Minitab assigns the smallest observation a rank of 1, the second smallest observation a rank of 2, and so on. If two or more observations are tied, Minitab assigns the average rank to each tied observation. Minitab calculates the mean rank for each sample.

### Interpretation

When a group's mean rank is higher than the overall average rank, the observation values in that group tend to be higher than those of the other groups.

## Z-Value

The z-value indicates how the average rank for each group compares to the average rank of all observations.

### Interpretation

Interpret the z-values for each group as follows:
• The higher the absolute value, the further a group's average rank is from the overall average rank.
• A negative z-value indicates that a group's average rank is less than the overall average rank.
• A positive z-value indicates that a group's average rank is greater than the overall average rank.

## Null hypothesis and alternative hypothesis

The null and alternative hypotheses are two mutually exclusive statements about a population. A hypothesis test uses sample data to determine whether to reject the null hypothesis.
Null hypothesis
The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value. The null hypothesis is often an initial claim that is based on previous analyses or specialized knowledge.
Alternative hypothesis
The alternative hypothesis states that a population parameter is smaller, larger, or different from the hypothesized value in the null hypothesis. The alternative hypothesis is what you might believe to be true or hope to prove true.

## DF

The degrees of freedom (DF) equals the number of groups in your data minus 1. Under the null hypothesis, chi-square distribution approximates the distribution of the test statistic, with the specified degrees of freedom. Minitab uses the chi-square distribution to estimate the p-value for this test.

## H-Value

H is the test statistic for the Kruskal-Wallis test. Under the null hypothesis, the chi-square distribution approximates the distribution of H. The approximation is reasonably accurate when no group has fewer than five observations.

### Interpretation

Minitab uses the test statistic to calculate the p-value, which you use to make a decision about the statistical significance of the terms and the model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

A sufficiently high test statistic indicates that at least one difference between the medians is statistically significant.

You can use the test statistic to determine whether to reject the null hypothesis. However, using the p-value of the test to make the same determination is usually more practical and convenient.

## P-Value

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

### Interpretation

Use the p-value to determine whether any of the differences between the medians are statistically significant.

To determine whether any of the differences between the medians are statistically significant, compare the p-value to your significance level to assess the null hypothesis. The null hypothesis states that the population medians are all equal. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
P-value ≤ α: The differences between some of the medians are statistically significant
If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that not all the population medians are equal. Use your specialized knowledge to determine whether the differences are practically significant. For more information, go to Statistical and practical significance.
P-value > α: The differences between the medians are not statistically significant
If the p-value is greater than the significance level, you do not have enough evidence to reject the null hypothesis that the population medians are all equal. Verify that your test has enough power to detect a difference that is practically significant. For more information, go to Increase the power of a hypothesis test.
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