Probit

It has applications in data analysis and machine learning, in particular exploratory statistical graphics and specialized regression modeling of binary response variables.

, so the probit is defined as Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics.

Continuing the example, In general, The idea of the probit function was published by Chester Ittner Bliss in a 1934 article in Science on how to treat data such as the percentage of a pest killed by a pesticide.

[1] Bliss proposed transforming the percentage killed into a "probability unit" (or "probit") which was linearly related to the modern definition (he defined it arbitrarily as equal to 0 for 0.0001 and 1 for 0.9999):[2] These arbitrary probability units have been termed "probits" ...He included a table to aid other researchers to convert their kill percentages to his probit, which they could then plot against the logarithm of the dose and thereby, it was hoped, obtain a more or less straight line.

The method introduced by Bliss was carried forward in Probit Analysis, an important text on toxicological applications by D. J.

In addition to providing a basis for important types of regression, the probit function is useful in statistical analysis for diagnosing deviation from normality, according to the method of Q–Q plotting.

If a set of data is actually a sample of a normal distribution, a plot of the values against their probit scores will be approximately linear.

The normal distribution CDF and its inverse are not available in closed form, and computation requires careful use of numerical procedures.

Other environments directly implement the probit function as is shown in the following session in the R programming language.

Wichura gives a fast algorithm for computing the probit function to 16 decimal places; this is used in R to generate random variates for the normal distribution.

[6] Another means of computation is based on forming a non-linear ordinary differential equation (ODE) for probit, as per the Steinbrecher and Shaw method.

is the probability density function of w. In the case of the Gaussian: Differentiating again: with the centre (initial) conditions This equation may be solved by several methods, including the classical power series approach.

From this, solutions of arbitrarily high accuracy may be developed based on Steinbrecher's approach to the series for the inverse error function.