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, execute the following command:

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, execute the following command:

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

Igor Pro 9

Learn More

Igor XOP Toolkit

Learn More

Igor NIDAQ Tools MX

Learn More