If you have activated the Statistics option in the Stereonet Options window, the program will compute a variety of statistics. A summary of these will be included with the stereonet plot in a legend. In the topics below is a summary of the methods used to compute these directional statistics.
Best-Fit Directional Vectors, Eigenvalues, Eigenvectors
To find the best-fit directional vectors for the mean lineation vector and the best-fit great circle, the program calculates the eigenvalues and eigenvectors for the following matrix of direction cosine values:
l2 lm ln
ml m2 mn
nl nm n2
The variables l , m, and n represent the direction cosine values for the x (north), y (east), and z (vertical) axis respectively. For each data set, these values are summed up and the eigenvalues and eigenvectors calculated. For the best-fit great circle, the eigenvector associated with the third (least) eigenvalue gives the pole to this great circle. The eigenvector associated with the first (greatest) eigenvalue gives the mean lineation direction for the data set.
Six other statistical values associated with the eigenvector analysis are calculated and plotted as well. These are the three eigenvalues r 1, r 2, and K. The eigenvalues of the matrix will be displayed in descending order on the plot. These values are normalized between 0 and 1. The eigenvalues provide direct information about the distribution and uniformity of the data set.
Basically there are three types of point distributions:
The next three numbers involve further manipulation of the eigenvalues. r 1 is the ratio LN ( E1 / E2 ). r 2 is the ratio LN ( E2 / E3 ). K is the ratio r 1 / r 2. (Don't confuse this with the k parameter from the Spherical Gaussian density function.)
These three values can help to further elucidate the distribution of points. If r 1 and r 2 are both small, the data set has a random distribution. A large value for r 1 and a small value for r 2 indicates a grouping around a single point maximum. A large r 2 and small r 1 indicates a girdle distribution.
Spherical Variance, Rbar
These are two additional values that are computed for your 3-D orientation data. Rbar is the normalized value of R:
R = ( ( Sl)2 + ( Sm )2 + ( Sn )2 ) 1/2
or
Rbar = R / n
where n equals the number of data points.
The spherical variance is found as follows:
sv = ( n - R ) / n
or
1 - Rbar
The spherical variance will range between 0 and 1. The larger the dispersion of vectors about the mean direction, the large the value of the spherical variance.
Rbar can be used to test the uniformity of the data set. The value of Rbar calculated by the program can be compared to the following table of value for cutoffs at certain levels of significance. If the calculated value exceeds the table value, the hypothesis that the observations are uniformly distributed can be rejected at the specified level of significance.
# points Significance Level %
N 10 5 2 1
5 0.637 0.700 0.765 0.805
6 0.583 0.642 0.707 0.747
7 0.541 0.597 0.659 0.698
8 0.506 0.560 0.619 0.658
9 0.478 0.529 0.586 0.624
10 0.454 0.503 0.558 0.594
11 0.433 0.480 0.533 0.568
12 0.415 0.460 0.512 0.546
13 0.398 0.442 0.492 0.526
14 0.384 0.427 0.475 0.507
15 0.371 0.413 0.460 0.491
16 0.359 0.400 0.446 0.476
17 0.349 0.388 0.443 0.463
18 0.339 0.377 0.421 0.450
19 0.330 0.367 0.410 0.438
20 0.322 0.358 0.399 0.428
21 0.314 0.350 0.390 0.418
22 0.307 0.342 0.382 0.408
23 0.300 0.334 0.374 0.400
24 0.294 0.328 0.366 0.392
25 0.288 0.321 0.359 0.384
30 0.26 0.29 0.33 0.36
35 0.24 0.27 0.31 0.33
40 0.23 0.26 0.29 0.31
45 0.22 0.24 0.27 0.29
50 0.20 0.23 0.26 0.28
100 0.14 0.16 0.18 0.19
Table taken from Davis (1986)
This table can be used in the following manner: Suppose that you have run a linear data set with 55 data points and the program calculates an Rbar of 0.25. For a data set of 50 points, we have critical values of 0.2, 0.23, 0.26, and 0.28. Because 0.25 exceeds the first two values, we could reject the hypothesis of uniformity with a 95% confidence rating. If the calculated value for Rbar were greater than 0.26, then our confidence rating would be 98%; greater than 0.28, 99%.