Probability Calculator ............................. Created August 1992
by
Michael E. Miller
This shareware allows the user to calculate the probability of obtaining a value from a Normal, Students t, Chi-Square, F, or binomial distribution that is as extreme or more extreme than the value entered by the user. This application was developed for use on a Macintosh using some form of System 7.0. However, it may operate under other versions of the Macintosh Operating System and has been shown to operate under System 6.
This software may be used to circumvent the use of lookup tables. However, it was originally written to allow the calculation of probabilities for the F distribution when the degrees of freedom are non-integer values. The calculation of these probabilities for noninteger degrees of freedom may be especially useful to the experimenter who wishes to obtain probability values after applying the Greenhouse-Geisser correction during an analysis of variance. It should be noted that all distributions, except the binomial, will accept and calculate probabilities for distributions using non-integer degrees of freedom.
GENERAL INFORMATION:
This program was originally developed for the author's own use. However, some colleagues have expressed interest in this application. This interest has prompted me to offer it as shareware. Please feel free to copy and share this mini-application with friends and colleagues free of charge for 30 days. If you find it useful, please send $5.00 to the address in the application's ABOUT box. I believe that the code is relatively stable, at least it is on my machine.
I would like to hear any suggestions for improvement that you may have. If you have comments, you may send them to the address in the ABOUT box or you may send me a note via Email (mikem@vtvm1.cc.vt.edu).
INSTRUCTIONS FOR USE:
1) Choose the desired distribution by selecting it from the distribution menu.
2) Enter the value, and the appropriate degrees of freedom (Note: you may use the tab keys, left and down arrow keys, or mouse to move among the input fields).
3) If you are calculating a probability for a Normal or t distribution, click on the appropriate radio button to indicate if the probability should be calculated for a one or two-tailed test. Two tailed tests are not appropriate for the Chi-squared, F, or binomial distributions.
4) Use the mouse or press RETURN or ENTER to select the CALCULATE NOW button.
5) When the program has completed all calculations it will return the calculated probability near the bottom of the window.
RESTRICTIONS TO USE:
This program will not calculate exact values for the Chi-Square Distribution if the degrees of freedom is less than 2 (values are estimated if the 1 df is input). Additionally, it is incapable of integrating values for the F-Distribution if the numerator degrees of freedom is a non-integer value below 2.0 (integer values are calculated).
OPTIMAL ACCURACY:
In general, this program should be accurate to within plus or minus about 0.0001. For certain values of df (particularly small values of df) a significant amount of computation time may be required. Do not be alarmed this time is needed for the algorithms that were implemented to maintain the desired accuracy level.
GENERAL APPROACH:
The general approach taken by this program to calculate the desired probability values for the Normal, t, Chi-Squared, and F Distribution is to numerically integrate the probability density functions to the desired value. This integration is completed using Simpson's method. The integration is conducted, successively taking smaller and smaller steps, until the error between any two integration passes is 1E-7. Many of these PDFs also require the area under the Gamma Function as an input. This area is calculated using a variety of methods, including brut integration for very small values of N (requiring significant times for integration), the use of a factorial expression for integer numbers greater than 1, and the Lanczos approximation for noninteger, larger values.
The basic PDFs used by this program are given by:
Hines, W. W. and Montgomery, D. C. (1980). Probability and Statistics in Engineering and Management Science. New York: John Wiley and Sons.
The general approach taken for the calculation of the Binomial Distribution is to calculate the exact value from an equation given by:
Siegel, S. Castelian, Jr., N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill.
Notice that this formula contains N! (where N is the number of degrees of freedom). To avoid overflows for low values of N, the log transform is used during the calculation of the number of possible combinations. This allows the calculation of the Binomial for large values of N.
