# Probable error

In statistics, the probable error of a quantity is a value describing the probability distribution of that 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 However, the term also has an older meaning that is still currently repeated as the only meaning,[2] but that has been deprecated for some time.[3] This specifies the probable error γ as being a fixed multiple of the standard deviation, σ, where the multiplying factor derives from the normal distribution. 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.

In the context of measurement, 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.

## References

1. ^ a b Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
2. ^ a b 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