Circular Two Sample Tests
This example covers four tests that are available in the StatsCircularTwoSampleTest operation. All tests are used to compare two samples of circular data which represent either raw data or means of data (second order analysis).
For the first two examples we use the waves Sample1, Sample2 and Sample3:
Sample1

Sample2

Sample3

NonParametric PairedSample Test
To test if the pairs represented by corresponding rows in Sample1 and Sample2 are the same (H_{0}) execute the command:
StatsCircularTwoSampleTest/Q/T=1/NPR Sample1,Sample2
The reusults of the test are summarized in the table:
numPairs  11 
Rp  0.350962 
Critical  1.04402 
PValue  0.739203 
Since the statistic Rp is smaller than the critical value we can't reject H_{0} that the two paired distributions are the same.
Parametric PairedSample Test
To test if the pairs represented by corresponding rows in Sample1 and Sample2 are the same (H_{0}) execute the command:
StatsCircularTwoSampleTest/Q/T=1/PPR Sample1,Sample2
The reusults of the test are summarized in the table:
numPairs  11 
xBar  0.0520224 
yBar  0.0247836 
F  0.246806 
Critical  4.25649 
PValue  0.786411 
As expected the parameteric test results in the same conclusion that H_{0} can't be rejected. To repeat the test and compare Sample1 to Sample3 execute the command:
StatsCircularTwoSampleTest/Q/T=1/PPR Sample1,Sample3
The reusults of the test are summarized in the table:
numPairs  11 
xBar  0.170992 
yBar  0.0770423 
F  4.62463 
Critical  4.25649 
PValue  0.0415419 
In this case the pairwise equality of the samples is rejected.
TwoSample Parametric Second Order Test
Suppose you have two samples of means of circular data contained in Sample4 and Sample5 as shown below.
Sample4

Sample5

To test if the means of the populations from which the two samples were taken are equal (H_{0}), execute the following command:
StatsCircularTwoSampleTest/Q/T=1/PSOA Sample4,Sample5
The reusults of the test are summarized in the table:
Samples1  8 
xBar1  0.398434 
yBar1  0.872187 
r1  0.958885 
a1  1.99931 
Samples2  11 
xBar2  0.25697 
yBar2  0.935076 
r2  0.969743 
a2  1.83899 
F  4.03807 
Critical  3.63372 
PValue  0.0380423 
Since the F statistic is greater than the critical value we reject H_{0}.
TwoSample NonParametric Second Order Test
To test the same data using a nonparametric test, execute the command:
StatsCircularTwoSampleTest/Q/T=1/NSOA Sample4,Sample5
The reusults of the test are summarized in the table:
Total_Points  19 
Watson_U2  0.202153 
Critical_Tiku  0.184557 
Approx_P  0.0345285 
Critical  0.184103 
Clearly the Watson_U2 statistic is greater than the critical value and H_{0} (equality of the means) must be rejected.