Assessing the error of linear and planar field data using Fisher statistics

Vince Cronin, Baylor University
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Summary

picture of slickenlines
This resource provides geoscientists with the tools necessary to determine the average and 95% confidence interval in the orientation of a set of near-parallel planes (e.g., bedding surface, joints, faults) or near-colinear vectors (e.g., slip vectors, glacial striae, paleomagnetic vectors) using Fisher statistics. Included is descriptive material about the method, an Excel spreadsheet, worked examples, and a sample problem set.

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Learning Goals

The goals are
  • to reinforce the idea that numbers that are used in science should carry with them a valid assessment of their associated error or uncertainty,
  • to reinforce the practice of collecting multiple observations associated with a given phenomenon, such as the orientation of a bed at a given outcrop, and
  • to teach geoscientists how to assess the error associated with the collection of field data related to the orientation of near-parallel lines, vectors and planes.

Context for Use

picture of jointed rock
This resource could be used in any course where field data concerning the orientation of near-parallel geological vectors, lines or planes are collected, including courses in field geology, structural geology, paleomagnetism and stratigraphy/sedimentation. It involves mathematical operations that are within the national standards for graduating high school students (http://www.nctm.org/standards). It would be helpful if students had a very basic understanding of Excel spreadsheet operations; a good source of information for modeling with Excel is available at serc.carleton.edu/introgeo/mathstatmodels/UsingXL.html.

Description and Teaching Materials

A manuscript with a fuller explanation of this application of Fisher statistics has been submitted for publication, and is currently available upon request from Vince Cronin.

Explanation of Fisher statistics for students, in MS Word (Microsoft Word 1.2MB Jul17 04)
Explanation of Fisher statistics for students, in a PDF file (Acrobat (PDF) 1.5MB Jul17 04)
Excel spreadsheet to compute Fisher statistics of 3 to 5 observations (Excel 16kB Jul16 04)
Problem set in MS Word (Microsoft Word 315kB Jul17 04)
Problem set in a PDF file (Acrobat (PDF) 658kB Jul17 04)

Teaching Notes and Tips

It is very useful to have students apply this analysis to data that they have collected in the field, probably after they have worked through some prepared problems like the ones in the sample problem set. Have students guess what the average and 95% confidence interval (CI) will be while they are actually looking at the surface or linear features in the field. Take digital photographs of the surfaces, and bring them back to the lab to review after the analysis with Fisher statistics. What is the 95% CI for an apparently smooth surface? ...for an apparently rough surface?
Once students have acquired these statistical skills, it would be good to encourage them to make the collection of multiple observations and subsequent statistical analysis a routine part of their field work.

References and Resources

Borradaile, G., 2003, Statistics of Earth science data -- Their distribution in time, space and orientation: Berlin, Springer, 351 p.
Fisher, N.I., Lewis, T., and Embleton, B.J.J., 1987, Statistical analysis of spherical data: Cambridge University Press, Cambridge, 329 p.
Fisher, R.A., 1953, Dispersion on a sphere: Proceedings of the Royal Society, London, v. A17, pp. 295-305.
Irving, E., 1964, Paleomagnetism and its application to geological and geophysical problems: John Wiley & Sons, New York, 399 p.
Mardia, K.V., 1972, Statistics of directional data: Academic Press, London, 357 p.
Opdyke, N.D., and Channell, J.E.T., 1996, Magnetic stratigraphy: Academic Press, San Diego, 346 p.
Tarling, D.H., 1971, Principles and applications of palaeomagnetism: Chapman and Hall, London, 164 p.
Tauxe, L., 1998, Paleomagnetic principles and practices: Kluwer Academic Publishers, Dordrecht, 299 p.