Minitab displays the values for each group under Ave Rank in the output.
Minitab calculates the z-value for each group as follows:
Term | Description |
---|---|
average rank for group j | |
average rank for all observations | |
N | number of observations |
nj | number of observations for the jth group |
A sample has 9 observations: 2.4, 5.3, 2.4, 4.0, 1.2, 3.6, 4.0, 4.3, and 4.0
Observation | Rank (assuming no ties) | Rank |
---|---|---|
1.2 | 1 | 1 |
2.4 | 2 | 2.5 |
2.4 | 3 | 2.5 |
3.6 | 4 | 4 |
4.0 | 5 | 6 |
4.0 | 6 | 6 |
4.0 | 7 | 6 |
4.3 | 8 | 8 |
5.3 | 9 | 9 |
Under the null hypothesis, the chi-square distribution with k – 1 degrees of freedom approximates the distribution of H. The approximation is reasonably accurate when no group has fewer than five observations. A higher H value provides stronger evidence for the null hypothesis that the difference between some of the medians is statistically significant.
Some authors, such as Lehmann (1975)1, suggest adjusting H when the data have ties. Minitab displays H(adj) when the data have ties.
Under the null hypothesis, the chi-square distribution with k – 1 degrees of freedom approximates the distribution of H and H(adj).
P-value = 1 – CDF (χ2H, df)
P-value = 1 – CDF (χ2H(adj), df)
For small samples, Minitab recommends that you use exact tables. For more details, see Hollander and Wolfe (1973)2.
Term | Description |
---|---|
nj | number of observations in group j |
N | total sample size |
average of the ranks in group j | |
average of all of the ranks | |
ti | number of tied values in the ith set of ties |