where the symbol is to be read as "is distributed as" and F[v,,v2] is the F distribution with v, and v2 degrees of freedom as in (3.2.7). Values of h = k]/k2 far from unity strongly suggest the two populations do not have the same precision. The critical values of h at a given confidence level may be determined from tables (or computer code) of the F distribution.
If more than two populations are involved and the sample sizes are all the same, then the test may be performed as follows. The largest observed value of k is tested against the smallest value and if these two values could have been obtained by random sampling from populations having a common precision then so could all the intermediate values. If the result is marginal, or if the sample sizes vary, the test is not quite so simple. Bartlett (1937) presented a test for multiple observed variance estimates and its application to the Fisher distribution has been discussed by Stephens (1969), Mardia (1972), and McFadden and Lowes (1981).
It is often necessary or desirable to determine whether two or more sets of paleomagnetic observations could have been drawn from a common distribution; that is, do they have a common true mean direction and a common precision k. A criterion sometimes used is that if cones of confidence do not intersect then the samples do not share a common mean direction. This is often phrased as saying that the mean directions are discernibly different. Although this criterion is certainly correct, it is extremely conservative; cones of confidence may overlap even though the mean directions are discernibly different. An appropriate test was devised by Watson (1956a) and re-examined by McFadden and Lowes (1981).
Suppose there are m samples, each having N, observations giving resultant vectors of length /?,-. If the samples were all drawn from a common population (which carries with it the assumption of a common k), then where the summations are for i from 1 to m and N = Z/V,.
The term (N-1.R,) = £(/V,-/?,) is the sum of the within-sample dispersions and does not vary as the sample mean directions change. However, if the sample mean directions are more dispersed than is to be expected from the within-sample dispersion, then the algebraic sum of the sample resultants Z7?, will be very much greater than their vector sum R and so / will be large compared with the relevant F statistic. This would suggest that the hypothesis of a common true mean direction is false.
For m = 2 (two samples) / is distributed as F[2,2(jV-2)], which has a particularly simple representation (McFadden and Lowes, 1981). The null hypothesis that two samples share a common mean direction may be rejected at the level of significance P if
With P = 0.05, the test is performed at the 95% level of confidence. Test for Uniform Randomness
In some cases the directions of magnetization may be widely scattered, and the question then arises as to whether these directions could have been obtained by sampling from a uniform random population. For a truly uniform random (i.e., isotropic) population k = 0 so that the population has no preferred, or true mean, direction. In practice, however, the observed R, and therefore the observed k (the best estimate of k), is never zero. Watson (1956b) has devised the following test. For a sample of size N, the length of the resultant vector R will be large if a preferred direction exists, or small if it does not. Assuming that no preferred direction exists, a value R0 may be calculated that will be exceeded by R with any stated probability. Watson (1956b) calculated R0 for various sample sizes for probabilities of 0.05 and 0.01. To carry out the test one merely enters Watson's table at the row corresponding to the sample size N in order to find the value of R0 which would be exceeded with given probability. For N greater than about 10, 3F/N is approximately distributed as a chi-square variate with 3 degrees of freedom. Consequently, N%1(0.05) at the 95% confidence level for large
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