Statistics

In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the probable error of a quantity is a value describing the probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 of that quantity. It defines the half-range of an interval about a cental point for the distribution, such that half of the values from the distribution will lie within the interval and half outside. Thus it is equivalent to half the interquartile range

Interquartile range

In descriptive statistics, the interquartile range , also called the midspread or middle fifty, is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles...

  However the term also has an older meaning that is still currently repeated as the only meaning, but that has been deprecated for some time. This specifies the probable error γ as being fixed a multiple of the standard deviation

Standard deviation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

, σ, where the multiplying factor derives from the normal distribution. Specifically,
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

Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

 of the Cauchy distribution

Cauchy distribution

The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

.

Measurement

In the context of measurement

Measurement

Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

, 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. The implication would be that nearly all measurement would be in the range defined by this version of probable error.

Military and GPS

In assessing the accuracies of military weapons and of Global Positioning System

Global Positioning System

The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...

 applications, the term "circular error probable

Circular error probable

In the military science of ballistics, circular error probable is an intuitive measure of a weapon system's precision...

" (CEP) is used with the implication of a 50% inclusion rate.