Select the method or formula of your choice.

Minitab calculates the average ranks as follows:

- Ranks the combined samples, 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 both observations.
- Calculates the average of the ranks of each sample.

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 |

n_{j} | number of observations for the j^{th} group |

Tied values occur when two or more observations are equal. If your data has tied values, Minitab ranks the data as follows:

- Sort the observations in ascending order.
- Assign ranks to each observation as if there were no ties.
- For a tied set, take the average of the corresponding ranks and assign this value as the new rank to each tied value in that set.

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 |

The following information is also used when calculating the test statistics:

- The number of sets of ties = 2
- The number of tied values in the first set = 2
- The number of tied values in the second set = 3

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 (χ^{2}_{H, df})

P-value = 1 – CDF (χ^{2}_{H(adj), df})

For small samples, Minitab recommends that you use exact tables. For more details, see Hollander and Wolfe (1973)^{2}.

Term | Description |
---|---|

n_{j} | number of observations in group j |

N | total sample size |

average of the ranks in group j | |

average of all of the ranks | |

t_{i} | number of tied values in the i^{th} set of ties |