**ERIC Number:**ED064395

**Record Type:**RIE

**Publication Date:**1972-Apr

**Pages:**7

**Abstractor:**N/A

**Reference Count:**0

**ISBN:**N/A

**ISSN:**N/A

Rational Approximations of the Inverse Gaussian Function.

Byars, Jackson A.; Roscoe, John T.

There are at least two situations in which the behavioral scientist wishes to transform uniformly distributed data into normally distributed data: (1) In studies of sampling distributions where uniformly distributed pseudo-random numbers are generated by a computer but normally distributed numbers are desired; and (2) In measurement applications where standardization of an instrument requires that percentile ranks be transformed into normally distributed standard scores. The problem investigaged in this study is find z when given P(z). The difficulty is that expressions which approximate the integral from minus infinity to z are not readily solvable for z. A number of investigators have derived algebraic approximations to the inverse Gaussian. The most widely used algebraic approximations of the inverse Gaussian function are those derived by Hastings. The Hastings approximations are valid only for values of P(z) greater than 0.50, and a computer program must make logical provisions for the situation where P(z)0.50. Burr approached the problem through the use of a cumulative moment theory and also derived two approximations. Burr's approximations have the advantage that they are valid for all values of P0. They are also conveniently expressed in one FORTRAN statement. It was the objective of Byars and Roscoe to develop an approximation of the inverse Gaussian which was both more accurate and more efficient than previous transformations. A final expression was obtained from the solution of approximately 4300 equations in six unknowns. The three sets of approximations were compared on accuracy and computational efficiency, and the Byars-Roscoe approximation was found to be superior to the others. (CK)

**Publication Type:**N/A

**Education Level:**N/A

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**Sponsor:**N/A

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**Note:**Paper presented at the annual convention of the AERA (Chicago, Ill., April 1972)