# 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.

## Individual evidence

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.
2. ^ G. Blom: Statistical Estimates and Transformed Beta Variables. John Wiley, New York 1958.