Standard deviation: (3.2.19b)
The angle i(/63 of (3.2.19b) is often referred to as the angular standard deviation or angular dispersion and vj;95 represents the angle from the mean direction beyond which only 5% of the directions lie.
Fisher (1953) has shown that if N observations are drawn from a Fisher distribution with precision parameter k and the length of the resultant vector is R, then the direction of this resultant vector (i.e., the mean direction) will also be Fisher distributed about the true mean direction but with precision kR. From this he showed that for k > 3 the true mean direction of the population will, with probability (1-P), lie within a circular cone of semi-angle aabout the resultant vector R, where
Normally, P is taken as 0.05 to give a circle of 95% confidence about the mean. When a is small the following approximate relations may be used
Note that previous texts (e.g., Irving, 1964; McElhinny, 1973a; Butler, 1992) have used ~JkN instead of \lkR in (3.2.21). However, since the mean is distributed with precision kR (Fisher, 1953), -JkR is a more appropriate choice and gives a better approximation to the correct value given by (3.2.20). For large values of k (e.g., >50), R approaches N and it matters little which is used.
If some known direction, such as that of the present Earth's field at the sampling site, falls within a95, then statistically there is no reason to suppose that the observations were drawn from a distribution with a true mean direction that differs from the known direction. However, if the known direction falls outside a95, then the hypothesis that the known direction is the true mean direction of the observations can be rejected at the 95% confidence level.
When designing sampling schemes it is important that different statistical properties of results from different rock types be taken into account when calculating a mean formation direction. Suppose that samples are collected at the ;th site leading to an estimate kwi of the within-site precision. The site mean direction will then be Fisher distributed with a precision of about kVJInl (which is easier to use here than k^jR,), so it is important to keep the product kwln, roughly constant at each site, thereby maintaining an approximately constant error in each site mean direction (3.2.20). Thus, if the rock types being collected are much the same with similar within-site precisions (kwi constant), the number of
samples being collected at each site can be the same. If kwl is not constant at each site because different rock types are being sampled, then the number of samples at each site needs to be varied to accommodate the varying kwi. Under these conditions the mean formation direction can be calculated in the usual way as the mean of the site mean directions.
One problem that arises in paleomagnetic investigations is determining whether an apparent outlier is truly discordant with other observations and, if so, whether it should be rejected. Consider a set of (jV+1) observations in which n of the observations seem to group together but the (AM-l)4 observation is an outlier. McFadden (1982) has shown that, given the observed grouping of the n concordant observations, with resultant vector of length R, there is a probability P that an outlier from the same distribution will exceed an angle Ja-p) from the mean of the concordant group, where cosy (,.,,)
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