## Watson-Williams Test

The Watson-Williams test for the equality of the means of two or more samples. In this example we consider the following 3 samples where the numerical values represent angles in radians:

data1 | data2 | data3 | data4 |

3.16 | 3.06 | 3.31 | 3.31 |

3.59 | 3.24 | 3.54 | 3.11 |

3.94 | 2.89 | 3.75 | 3.15 |

3.86 | 3.15 | 4.01 | 2.63 |

2.9 | 3.58 | 3.84 | 3.04 |

3.77 | 3.67 | 3.59 | 3.59 |

3.76 | 2.7 |

First, we test the equality of the means of data1 and data2. To execute the test, select the blue line below and type Ctrl-Enter:

StatsWatsonWilliamsTest/T=1/Q data1,data2

The results are given in the Watson-Williams Test table.

Samples | 2 |

Total_Points | 13 |

R | 12.1775 |

Pop_Mean_Angle | 3.42966 |

rw | 0.941734 |

K | 1.04234 |

F_Statistic | 0.984252 |

Critical_F | 4.84434 |

T_Statistic | 0.992095 |

Critical_T | 2.20099 |

In this case the test provides both the F and the T statistics together with
their critical values. It is evident that the critical values are much larger than the two test statistics
so H_{0} (equality of means) can't be rejected. The remaining test results, include the population mean
angle (in radians) as well as the weighted value rw and the correction factor K used in both the F and
T statistics calculations.

You can use this operation with more than two waves as in the following example. To execute the test, select the blue line below and type Ctrl-Enter:

StatsWatsonWilliamsTest/T=1/Q data1,data2,data3

The results are given in the Watson-Williams Test table.

Samples | 3 |

Total_Points | 19 |

R | 17.9096 |

Pop_Mean_Angle | 3.50861 |

rw | 0.952209 |

K | 1.03494 |

F_Statistic | 1.66266 |

Critical_F | 3.63372 |

T_Statistic | 1.82355 |

Critical_T | 2.11991 |

Here, again, H_{0} can't be rejected. By contrast, we have to reject H_{0} in the
following test:

StatsWatsonWilliamsTest/T=1/Q data1,data2,data3,data4

Samples | 4 |

Total_Points | 26 |

R | 24.1286 |

Pop_Mean_Angle | 3.3923 |

rw | 0.952457 |

K | 1.03476 |

F_Statistic | 3.89983 |

Critical_F | 3.04912 |

T_Statistic | 3.42045 |

Critical_T | 2.07387 |

Note: the Watson-Williams test applies to data from a von Mises distribution where the different samples have the same dispersions. If these assumptions are invalid, you should consider using one of the non-parametric tests. See, for example Wheeler-Watson Test.