statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, 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 In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) or pro ...

it is equivalent to half the

interquartile range In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the differen ...

, or the

median absolute deviation In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample. For a u ...

. One such use of the term ''probable error'' in this sense is as the name for the

scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family ...

of the

Cauchy distribution The Cauchy distribution, named after Augustin-Louis 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) ...

, 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 of the distribution should exist, whereas the other definition does not. For a

normal distribution In probability theory and 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 ...

this is

\gamma = 0.6745 \times \sigma

(see details)

References

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

See also

References