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 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.

 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, H0 can't be rejected. By contrast, we have to reject H0 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. Forum Support Gallery