Probability net

The probability net or probability paper is one of the mathematical papers . A probability paper can be used to examine the data of a statistical feature to determine whether they are based on a certain probability distribution . It is provided with a coordinate network in which the quantiles of the distribution are equidistantly plotted on the abscissa , while the associated functional values of the distribution are plotted on the ordinate in linearized form. When you enter the value pairs (quantile, distribution) you get a straight line.

The probability network is a conventional tool that was widely used, especially before the introduction of electronic data processing, in order to achieve a fairly quick and efficient distribution check of data. The probability network of normal distribution is generally known , but the Weibull distribution is also represented in probability networks .

The normal distribution in the probability plot

The quantiles x of a standard normally distributed random variable are plotted on the abscissa, as well as on the ordinate. The ordinate does not show the values of x, but rather their distribution function values

{\ displaystyle \ Phi (x) = {{1} \ over {\ sqrt {2 \ pi}}} \ int _ {- \ infty} ^ {x} e ^ {- {{t ^ {2}} \ over {2}}} \, \ mathrm {d} t}

displayed. If the scale lines are arranged on the ordinate in such a way that arithmetically the distances between the distribution function values are the same, the typical pattern of the Gaussian probability network is obtained.

This linearization results in a straight line for the value pairs . Probability paper enables a simple drawing of such a function or a simple check whether given pairs of values fit a normal distribution (they must then lie on a straight line). ${\ displaystyle (x; \ Phi (x))}$

As an example, the function for the mean value μ = 100 and the scatter σ = 20 is shown on probability paper.

Practical application using the example of normal distribution

In practice, the procedure is that the observed values are sorted according to size. The associated values of an empirical distribution function are then assigned to the ordered values ( ). There are various proposals or estimation formulas for determining the empirical distribution function, e.g. B. ${\ displaystyle n}$ ${\ displaystyle x_ {i}}$ ${\ displaystyle x _ {[i]}}$

{\ displaystyle F (x_ {i}) = {\ frac {3i-1} {3n + 1}}}

to Rossow

{\ displaystyle F (x_ {i}) = {\ frac {i-0 {,} 375} {n + 0 {,} 25}}}

to Blom

{\ displaystyle F (x_ {i}) = {\ frac {i-0 {,} 5} {n}}}

{\ displaystyle F (x_ {i}) = {\ frac {i} {n + 1}}}

after Weibull and Gumbel

If the value pairs result in an approximately straight line, a normal distribution can be assumed for the population of the data. Estimates for the parameters median and standard deviation can be read directly from the probability paper.

literature

Joachim Hartung, Bärbel Elpelt, Karl-Heinz Klösener: Statistics. Munich 2002, ISBN 3-486-25905-9 .
JM Chambers, WS Cleveland, Beat Kleiner, Paul A. Tukey: Graphical Methods for Data Analysis. Wadsworth, 1983, ISBN 0-534-98052-X .
Lothar Sachs , Jürgen Hedderich: Applied statistics. Toolkit with R . Springer 2009, ISBN 978-3-540-88901-4 .

Individual evidence

↑ E. Rossow: A simple slide rule approximation to the percentage points corresponding to the normal scores. In: Currently economic production. 59, issue 12, 1964.
^ G. Blom: Statistical Estimates and Transformed Beta Variables. John Wiley, New York 1958.

[Rossow19642-1] E. Rossow: A simple slide rule approximation to the percentage points corresponding to the normal scores. In: Currently economic production. 59, issue 12, 1964.

[Blom1958-2] G. Blom: Statistical Estimates and Transformed Beta Variables. John Wiley, New York 1958.