Probable error

Probable error has two meanings in statistics.

The first meaning is a value describing the probability distribution of a given quantity. It defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.^[1] Thus it is equivalent to half the interquartile range, or the median absolute deviation.

The term also has an older meaning (sometimes stated as the only meaning),^[2] that has been deprecated for some time:^[3] it is denoted γ and defined as a fixed multiple of the standard deviation, σ, where the multiplying factor derives from the normal distribution, more specifically,^[1][2]

$\gamma = 0.6745 \times \sigma .$

Clearly this latter definition requires that at least the second moment of the distribution should exist, whereas the first definition does not. One use of the term probable error in statistics is as the name for the scale parameter of the Cauchy distribution.

A third meaning exists in the context of measurement theory and practice, where the probable error of a measurement made on an instrument having a scale, is defined as being one-half of the finest division on that scale.^[4] The implication would be that nearly all measurement would be in the range defined by this version of probable error.

See also

• Circular error probable

References

1. Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
2. Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. ISBN 1584880597 (Section 2.2.13)
3. Yule, G.U.; Kendall, M.G. (1950) An Introduction to the Theory of Statistics, 14th Edition, Griffin. ISBN 0-85264-140-0 (Section 17.9)
4. Principles of Measurement at Integrated Publishing: www.tpub.com/math1/7b.htm