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In statistics, probable error defines the
half-range In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: :M=\frac. The mid-range is closely related to the range, a measure of st ...
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.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', OUP. Thus for a symmetric distribution it is equivalent to half the interquartile range, or the
median absolute deviation In statistics, the median absolute deviation (MAD) is a Robust statistics, robust measure of the statistical dispersion, variability of a univariate sample of quantitative data. It can also refer to the statistical population, population paramete ...
. One such use of the term ''probable error'' in this sense is as the name for the scale parameter of the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) func ...
, which does not have a standard deviation. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', Chapman & Hall/CRC. (Section 2.2.13) which requires that at least the second
statistical moment In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total m ...
of the distribution should exist, whereas the other definition does not. For a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
this is \gamma = 0.6745 \times \sigma (see details)


See also

*
Average absolute deviation The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, ...
* Circular error probable * Confidence interval *
Standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of ...


References

{{reflist Theory of probability distributions Errors and residuals Statistical deviation and dispersion ru:Срединное отклонение