In
statistics, probable error 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.
[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) ...
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 difference ...
, 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 o ...
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) fun ...
, 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 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 ...
this is
(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, m ...
* 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 o ...
References
{{reflist
Theory of probability distributions
Errors and residuals
Statistical deviation and dispersion
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