## 1 Introduction  
As is well-known, the residual magnetism in a rock sample can only be recovered with some uncertainty, commonly referred to as \"error\". This is attributable to inhomogeneities that result during its formation as well as specific aspects that are not fully controllable during the laboratory measurement process. Information about the direction of the residual magnetization vector is important, and can be determined more reliably than its length (i.e., the magnetization intensity). The technique for determining the remanence vector has been well-described (see for example [1, 2]) and is based on a series of successive partial demagnetizations and accompanying intermediate measurements. The direction of the ancient field is established by taking both the entire series and possible subseries into account. The available databases of paleomagnetic directions have been compiled over decades from scientific publications that do not report intermediate details, but instead report only the accuracy of the final direction as calculated using the accepted standardized approaches. The natural way to present the directional precision is to set the cone within which the true paleomagnetic direction is expected to lie with high confidence (0.95 percent or so). Here, it is necessary to note the difference in the work with sedimentary rock samples and samples of erupted rocks: in the first case, the exact mutual synchronization of specimen ages is hardly possible. Therefore, the specificity of sedimentary rocks in terms of meaning gives only the averaging of the ancient magnetic field over a certain period of deposition, much longer than the period of fixation of the ancient magnetic field in the lava flow. Partly for this reason, for sedimentary rocks, the characteristic of the principal component analysis MAD (Maximal Angular Deviation) [3] was given in the publications instead of the confidence cone of the recovered direction; at the intuitive level, the MAD indicates the \"noisiness\" properties of the studied intermediate demagnetization stages. Direct recalculation of the MAD into a confidence cone is not possible [4] without knowing all the details of demagnetization (usually inaccessible from early publications). On the other hand, within a sufficiently simple and intuitive probabilistic description of the demagnetization process, the necessary recalculation is quite possible, see [5]. Turning now to methods for estimating the accuracy of directions determined from erupted rocks, we have a completely different picture. It is possible to select several specimens from the same lava flow so that the implied vector of ancient magnetization in them will share the same direction. The apparent inconsistencies in the responses in different specimens should be attributed to uncontrolled errors of measurements in the demagnetization process. Because of this consideration, it is worth seeking a method to compensate for these errors by means of averaging all the values. Thus, the corresponding method is based on some assumptions about the statistical nature of directions. In a nutshell, it is assumed that a sample of all possible estimates of ancient magnetic fields that can be extracted from samples belonging to a fixed lava flow at a given location (=the site) corresponds to an axisymmetric continuous distribution of points on the sphere [6]. Specifically, the scatter corresponds to the Fisher probability density \(f_{K}(\theta)\) with some concentration parameter \(K\):  
\[f_{K}(\theta)=\frac{Ke^{K\cos\theta}}{2\pi\Big{(}e^{K}-e^{-K}\Big{)}} \tag{1}\]  
The practical a95 method mimics the classical statistical method of estimating the parameters of a one-dimensional Gaussian law from a sample of values. It consists of an approximate calculation of the average direction of \(n\) unit vectors, the 0.95-confidence cone axis and a solid angle at its apex. This sheds light on the question of where the true direction of the supposed Fisher axis of distribution and the ancient field lie: both belong to this confidence cone.