## Introduction

Palaeomagnetism is basically concerned with the analysis of natural remanent magnetization (NRM) of rocks in bid to come up with a clear understanding of the crust’s magnetic field. Professor Blackett and Nagata have been credited for taking this study a notch higher since its establishment. Most of the discoveries that have been made in this study has facilitated in adding knowledge to what people knew about magnetic fields.

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In addition, palaeomagnetism has helped in coming up with credible information with respect to plate tectonics and continental drift. Of late, palaeomagnetism has been used in solving problems related to home geology. Once a data is collected in the field, it is analyzed statistically to come up with inferences (Tarling, 1983, p. 132). There are numerous statistical methods used in analyzing palaeomagnetic data. This paper aims at looking at some of these statistical methods; consider some of the methods used in paleomagnetic results and analyze the significance of paleomagnetic results.

## Statistical methods used in paleomagnetism

The need for measuring mean direction from a group of established directions in palaeomagnetic data has led to establishment of statistical methods of analysis. Additionally, there are times when collected data need to be tested on the magnitude of its palaeomagnetic stability. All these tests can not be successful without the use of statistical methods. Some of the statistical methods used in analyzing palaeomagnetic data include the normal distribution and the fisher distribution.

### The normal distribution

For any statistical method to be helpful in obtaining a mean of the collected data, it must have a unique probability density function. The function gives information with respect to the spread of the collected data. The normal distribution also referred to as Gaussian probability density function has a dome-shaped distribution curve. The function for the distribution is given by:

F (z) = 1/σ (2π)^{1/2} (exp (-z^{2 }/2))……………………………. (1)

Z = (x- μ)/σ………………………………………………………. (2)

Where x is the entity being measured, μ is the actual mean and σ is the standard deviation. When determining the mean direction of a set of observed directions in palaeomagnetic study, x stands for a sample of the directions observed. To obtain the mean direction of the observed directions, one integrates the function F (z) with respect to z. In most cases, the actual mean of the observed directions is not known (Fisher, Lewis & Embleton, 1987, pp. 323-337). To get it, one obtains the total summation of the individual observations and divides by the total number of observations. To determine the range of distribution of individual observation from the mean, one performs the following equation:

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Var (x) = ∑ (x_{i }– m)^{ 2}/ (n-1) = s^{2} ……………………………………… (3)

From i=1 to i= n and n is the number of observations while m stands for the actual mean. The distribution is then obtained by getting the square root of s^{2}. Error in mean direction of the data obtained is then determined by getting:

∆m =s/ n^{1/2} …………………. (4)

Where s is the distribution obtained after getting the square root of the variance. The obtained standard error of the sample mean facilitates in determining the confidence interval of the mean direction.

### The fisher distribution

This is another statistical method applied in determining the mean direction of observed means in palaeomagnetic analysis. Every observed direction in the data collected is assigned a unit weight and a single point used on its behalf in a circle of unit radius. The formula used in determining the distribution of the observed directions is given by:

P_{dA} (θ) = K exp (K cos θ)/ 4π sinh (K) ……………………………… (5)

Where q stands for the angle from the real mean direction and K is the accuracy factor. The above formula gives the likelihood of getting a direction lying within a given angular area represented by dA. From the equation, it is observed that the distribution of the directions is symmetrically distributed about the actual mean. Taking x as the relative angle about the actual mean direction, the likelihood of a direction lying within an angular area, dA, can be given by the formula:

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P_{dA} (θ) dA=P_{dA} (θ) sin (θ) d θ d ξ ………………………………… (6)

As the area of the bandwidth varies in the same manner with sin θ, it is used to represent it. Integrating the above equation with respect to θ and ξ respectively gives a unit area (Fisher, Lewis & Embleton, 1987, p. 342). This implies that the fisher distribution is normally distributed.

## Methods used in palaeomagnetic results

### The site-mean direction

The mean direction for observed directions in a palaeomagnetic analysis is calculated at different levels. On instances where an observer has come up with different samples of the observed directions, then the average ChRM directions of all samples have to be obtained. Fisher distribution is also used in determining the site-mean direction. This helps in identifying the direction of a geomagnetic field at any given time. It is imperative to ensure that site-mean direction is accurately obtained. However, it is vital to acknowledge the numerous values of site-mean direction observed during the calculation. An accurate site-mean direction is obtained where there is minimal site dispersion.

### Confidence interval

To precisely obtain the mean direction of a set of observed directions in a palaeomagnetic study, it is imperative to come up with a confidence interval for the obtained mean direction. This interval is similar to the standard error obtained using the normal distribution. In the case of Fisher distribution, the confidence interval is determined in terms of angular radius from the obtained mean direction. To completely define the confidence interval, one has to come up with a likelihood level. For a data consisting of N directions and whose actual mean is unknown but found to fall within a confidence interval given by (1-P), it angle α_{(1-P)} is given by:

Cos α_{(1-P)} = 1- (N – R)/R ((1/P)^{1/(N-1)}– 1) ………………………….. (7)

Generally, the interval (1-P) is given as 0.95 or 95% and is usually represented by α_{95}.

By obtaining the mean direction, confidence interval and the standard deviation, it means that the data used was randomly obtained from a set of directions with Fisher distribution. However, the actual mean and the accuracy factor K of the set of directions are not known (Fisher, Lewis & Embleton, 1987, p. 349). They can only be estimated. It follows that the calculated mean direction becomes the actual mean of the set of directions while the approximated value of K becomes the true value of the precision parameter K. α_{95} implies that a person is 95% confident that the approximated actual mean direction of the set of directions is equivalent to sample directional mean obtained from a sample of directions picked from the population.

## Significance of palaeomagnetic results

### Fold test

After determining palaeomagnetic stability, it is imperative to come up with methods of quantitatively analyzing the significance of the obtained results. For instance, it is crucial to determine if palaeomagnetic directions significantly differ from one another. One of the statistical methods used in determining the significance of the obtained results is the fold test. This is used in determining the coming together of directions before and after they have been structurally corrected. Where clustering is found to have changed positively after correction then one comes up with a judgment that ChRM was obtained before folding, hence satisfies the fold test (McElhinny, 1964, pp. 338-340). This test helps in determining if the development in coming together is statistically significant.

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### Reversal test

In paleomagnetism, there comes a time when one has more than one mean direction obtained from a set of directions. When one has two modes, the reversal test is used to establish a common mean for the two. In this test, the bootstrap test is applied to determine the common mean for the two modes. Generally, a 95% confidence interval is used when conducting the test. Cumulative distribution of the Cartesian coordinates of the test is used (Fisher, Lewis & Embleton, 1987, p.356). In case both the normal and reverse antipodes’ confidence intervals overlap, it implies that it is hard to distinguish the two means at the given confidence level (95%). Thus the data being tested satisfies the bootstrap reversal test.

## The state of knowledge

Statistical methods in paleomagnetism have been credited for the breakthrough that has been made in determining the mean direction of the magnetic fields. Methods such as the normal distribution and fisher distribution have been used in determining the direction of magnetic fields. Tests such as the reversal test and fold tests have added value to traditional methods used in testing the impact of palaeomagnetic stability.

## Reference List

Fisher, N. I., Lewis, T. & Embleton, B. J. (1987). *Statistical analysis of spherical data*. London: Cambridge.

McElhinny, M. W. (1964). Statistical significance of the fold test in paleomagnetism. *Goephys. J. Roy. Astron. Soc*., 8, pp. 338-340.

Tarling, D. H. (1983). *Palaeomagnetism: principles and applications in Geology, Geophysics and Archaeology*. New York: Chapman and Hall.