FROM JOURNAL OF CLASSIFICATION
                             VOLUME 9, 1992 PP 141-142

               A Computer Program to Construct Distributions with
                      Specific Degrees of Skew and Kurtosis

               Robert Saltstone                         Ken Stange
     
   Much has been written about the importance of measures of skew and kurtosis in
research and in research reporting (e.g., Hopkins and Weeks 1990).  Recent books on
hypothesis testing using computer intensive methods (Noreen 1989) and computer simulations
(Whicker and Sigelman 1991) describe methods for testing hypotheses from distributions with
known parameters, but do not provide computer code to build distributions with specific first four
moment characteristics.  A method for constructing multivariate non-normal distributions is
described by Vale and Maurelli (1983), but most researchers probably would not want to specify
inter-element correlations for distributions.  Various other algorithms to create non-normal
distributions are discussed by Tadikamalla (1980).
   The program DISTRIB.EXE builds and provides graphic displays of distributions with a
selected mean and standard deviation for any desired degree of skew and/or kurtosis.  The
authors created the program to examine estimates of classification consistency obtained by
Subkoviak's method (1976) under varying degrees of skew and kurtosis.  (This method is used to
establish cutoff points which are optimally reliable in distributions of test scores where pass/fail
classifications must be established.)  Researchers interested in such other classification issues
as the effect of distribution shape on a particular clustering algorithm or distance measure may
find the program similarly useful.
   DISTRIB.EXE is a 'brute force' program which runs on a microcomputer under MS-DOS. 
It is a structured program written in Microsoft QuickBASIC.  The routines could easily be
converted to another structured language.  Options for making changes to the code for such
purposes as re-running previous data, setting timers, and changing array cell replacement
heuristics are clearly documented in the code.
   A 486 or 386 processor and math co-processor are recommended but are not essential. 
The program uses the compiler's pseudo-random number generator to create a distribution of
elements (invariably resulting in a rectangular distribution when a distribution of any size is
constructed) and then randomly substitutes a new pseudo-random number for each element until
the desired distribution characteristics are obtained.  Microsoft compilers use the linear
congruential method of pseudo-random number generation (Knuth 1981); seed values are
obtained from the system clock.
   This program uses efficient computational formulae to build distributions of any size and,
within reason, any desired degree of skew and/or kurtosis.  The user sets an array size, desired
mean and standard deviation, and an acceptable error range for skew and kurtosis.  (The more
latitude the user allows for the values of skew and kurtosis, the faster DISTRIB.EXE will be able
to construct an array.)  DISTRIB.EXE begins with an array of random numbers of the desired
size and randomly substitutes for each value in the array until the desired values of skew and
kurtosis are obtained.  As indicated, the source code is available and the program is easily
modified (alternate code is included) to repeatedly run identical sets of pseudo-random arrays.
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   DISTRIB.EXE and source code can be obtained by sending an address label and
MS-DOS formatted 3.5 inch diskette to: Robert Saltstone, Research Services, Ontario Ministry of
Correctional Services, 200 First Avenue West, North Bay, Ontario, Canada, P1B 9M3.
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                                   References

Hopkins, K.D., and Weeks, D.L. (1990), "Tests for Normality and Measures of Skewness and Kurtosis:
    Their Place in Research Reporting,"  Educational and Psychological Measurement, 50,
    717-729.
Knuth, D.E. (1981), The Art of Computer Programming, Menlo Park, CA: Addison-Wesley.  Noreen, E.W.
    (1989), Computer Intensive Methods for Testing Hypotheses,  New York: Wiley.
Subkoviak, M.J. (1976), "Estimating Reliability from a Single Administration of a Criterion-Referenced
    Test," Journal of Educational Measurement, 13, 265-276.
Tadikamalla, P.R. (1980), "On Simulating Non-Normal Distributions," Psychometrika, 45, 273-279.
Whicker, M.L. and Sigelman, L. (1991),  Computer Simulation Applications,  Newbury Park, CA: Sage.
Vale, C.D. and Maurelli, V.A. (1983), "Simulating Multivariate Nonnormal Distributions," Psychometrika,
    48, 465-471.