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:

data1data2data3data4
3.163.063.313.31
3.593.243.543.11
3.942.893.753.15
3.863.154.012.63
2.93.583.843.04
3.773.673.593.59
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.

Samples2
Total_Points13
R12.1775
Pop_Mean_Angle3.42966
rw0.941734
K1.04234
F_Statistic0.984252
Critical_F4.84434
T_Statistic0.992095
Critical_T2.20099

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 H0 (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.

Samples3
Total_Points19
R17.9096
Pop_Mean_Angle3.50861
rw0.952209
K1.03494
F_Statistic1.66266
Critical_F3.63372
T_Statistic1.82355
Critical_T2.11991

Here, again, H0 can't be rejected. By contrast, we have to reject H0 in the following test:

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

Samples4
Total_Points26
R24.1286
Pop_Mean_Angle3.3923
rw0.952457
K1.03476
F_Statistic3.89983
Critical_F3.04912
T_Statistic3.42045
Critical_T2.07387

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.