Data Distributions and Random Numbers

Overview SPSS can generate new variables reflecting the random variate of any of over a dozen specific distribution. Different distributions require different parameters with the syntax code parentheses, as explained below. The researcher may wish to repeat sequences of pseudorandom numbers by setting the the same Random Number Seed before each sequence. This is done in the SPSS menu system under Transform, Random Number Generators, as illustrated below. By default, however, SPSS automaticaly changes the random number seed whenever a random number is generated for use in transformations such as functions listed below. Initializing the seed is only necessary when it is desired to replicate a sequence of random numbers.

Contents Key concepts and terms Frequently asked questions Bibliography

Key Concepts and Terms

• SPSS Random Number Generator dialog:

• Random variates for specified distributions
  • NORMAL(stddev). For example, NORMAL(5) returns a normally distributed pseudorandom number froom a distribution with mean 0 and a standard deviation of 10.
  • RV.NORMAL(mean, stddev). For example, RV.NORMAL(10,4) returns a normally distributed pseudorandom number from a distribution with a mean of 10 and a standard deviation of 4.
  • UNIFORM(max). For example, UNIFORM(100) returns a uniformly distributed pseudorandom number between 0 and 100.
  • UNIFORM(min, max). UNIFORM(100, 200) returns a random value from a uniform distribution with a minimum of 100 and maximum of 200.
  • RV.BERNOULLI(prob). RV,BERNOULLI(.5) returns a Bernoulli distributed random variate with a .5 probability parameter.
  • RV.BETA(shape1, shape2). RV.BETA(m,n) returns a random value from a Beta distribution with specified shape parameters.
  • RV.BINOM(n, prob). RV.BINOM(100,.25) returns a binomially distributed random variate with 100 trials, each wtih .25 probability.
  • RV.CAUCHY(loc, scale). RV.CAUCHY(loc, scale) returns a random value from a Cauchy distribution with specified location and scale parameters.
  • RV.CHISQ(df). RV.CHISQ(45) returns a random value from a chi-square distribution with 45 degrees of freedom.
  • RV.EXP(scale). RV.EXP(m) returns a random value from an exponential distribution with specified scale parameter of m.
  • RV.F(df1, df2). RV.F(2, 23) returns a random value from an F distribution with 2 and 23 degrees of freedom.
  • RV.GAMMA(shape, scale). RV.GAMMA(m, n) returns a random value from a Gamma distribution with specified m shape and n scale parameters.
  • RV.GEOM(prob). RV.GEOM(.667) returns a random value from a geometric distribution with a probability of .667.
  • RV.HALFNRM(mean, stddev). RV.HALFNRM(100, 10) returns a random value from a half normal distribution with a mean of 100 and a standard deviation of 10
  • RV.HYPER(total, sample, hits). RV.HYPER(500, 100, 25) returns a random value from a hypergeometric distribution with 500 total, 100 sample, and 25 hits.
  • RV.IGAUSS(loc, scale). RV.IGAUSS(m, n) returns a random value from an inverse Gaussian distribution with m location and n scale parameters.
  • RV.LAPLACE(mean, scale). RV.LAPLACE(100, m) returns a random value from a Laplace distribution with a mean of 100 and a scale parameter of m.
  • RV.LOGISTIC(mean, scale). RV.LOGISTIC(100, m) returns a random value from a logistic distribution with a mean of 100 and a scale parameter of m.
  • RV.LNORMAL(a, b). RV.LNORMAL(a, b) returns a random value from a log-normal distribution with specified parameters.
  • RV.NEGBIN(threshold, prob). RV.NEGBIN(m, .5) returns a random value from a negative binomial distribution with a threshold of m and a probability of .5.
  • RV.PARETO(threshold, shape). RV.PARETO(m, n) returns a random Pareto-distributed value with a threshold of m and shape parameter of n.
  • RV.POISSON(mean). RV.POISSON(.5) returns a Poisson-distributed random value with a mean or rate of .5.
  • RV.T(df). RV.T(99) returns a random value from a Student’s t distribution with 99 degrees of freedom.
  • RV.WEIBULL(a, b). RV.WEIBULL(a, b) returns a random value from a Weibull distribution with the specified parameters.

• SPSS syntax to create individual random variables with given distributions

  * Create 1,000 cases:
  * Return normally distributed values with a mean of 100 and a standard deviation of 20:
  * Return Poisson distributed  values with a mean of 100.
  * Return random value from a geometric distribution with a probability of .667:
  NEW FILE.
  INPUT PROGRAM.
     LOOP #I=1 TO 1000.
        COMPUTE Normal = RV.NORMAL(100,20).
        COMPUTE Poisson = RV.POISSON(100).
        COMPUTE Geometric  = RV.GEOM(.667).
        END CASE.
     END LOOP.
     END FILE.
  END INPUT PROGRAM.
  EXECUTE.
  
  * Print out all three histograms but no tables. 
  FREQUENCIES VARIABLES = ALL
     /HISTOGRAM 
     /FORMAT = NOTABLE.
  

  This yields results randomly similar to those below:

  

• Using SPSS syntax to create multiple random numbers in a given distribution
  The following syntax generates 10 random number variables with 1,000 cases each in Poisson distributions, with means from 1 to 10. To change the number of variables, change the "10" in the VECTOR statement and the #J loop. To change the number of cases, change the "1000" in the #I loop. The "#J" in the COMPUTE statement increases the mean by 1 for each #J loop. To change to a specific mean, replace #J in the COMPUTE statement with a number. The one-line middle DESCRIPTIVES code creates a table of the first five results. The lower "Chart Builder" syntax then creates a histogram of the first result.

  NEW FILE.
  INPUT PROGRAM.
     VECTOR X(10).
         LOOP #I = 1 TO 1000.
             LOOP #J = 1 TO 10.
               COMPUTE X(#J) = RV.POISSON(#J).
             END LOOP.
             END CASE.
         END LOOP.
         END FILE.
  END INPUT PROGRAM.
  EXECUTE.
  
  DESCRIPTIVES X1 X2 X3 X4 X5.
  
  * Chart Builder.
  GGRAPH
    /GRAPHDATASET NAME="graphdataset" VARIABLES=X1 MISSING=LISTWISE REPORTMISSING=NO
    /GRAPHSPEC SOURCE=INLINE.
  BEGIN GPL
    SOURCE: s=userSource(id("graphdataset"))
    DATA: X1=col(source(s), name("X1"))
    GUIDE: axis(dim(1), label("X1"))
    GUIDE: axis(dim(2), label("Frequency"))
    ELEMENT: area(position(summary.count(bin.rect(X1))), missing.wings())
  END GPL.
  

  Here is the resulting output for the first Poisson variable (mean of 1):

  

  
  

Frequently Asked Questions

• Can I generate cumulative, inverse, and other functions of distributions as well as random variates?
  Yes. SPSS supports the following functions. All operate as prefixes, similar to RV (ex., RV.Poisson, CDF.Poisson, IDF.Poisson, etc.). See SPSS (2007) for the specific parameters required for each distribution.
  • CDF Cumulative distribution function.
  • CDF.d_spec(x,a,...) returns a probability p that a variate with the specified distribution (d_spec) falls below x for continuous functions and at or below x for discrete functions.
  • IDF Inverse distribution function. Inverse distribution functions are not available for discrete distributions. An inverse distribution function IDF.d_spec(p,a,...) returns a value x such that CDF.d_spec(x,a,...)=p with the specified distribution (d_spec).
  • PDF Probability density function.
  • PDF.d_spec(x,a,...) returns the density of the specified distribution (d_spec) at x for continuous functions and the probability that a random variable with the specified distribution equals x for discrete functions.
  • RV Random number generation function.
  • RV.d_spec(a,...) generates an independent observation with the specified distribution (d_spec).
  • NCDF Noncentral cumulative distribution function.
  • NCDF.d_spec(x,a,b,...) returns a probability p that a variate with the specified noncentral distribution falls below x. It is available only for beta, chi-square, F, and Student’s t.
  • NPDF Noncentral probability density function.
  • NCDF.d_spec(x,a,...) returns the density of the specified distribution (d_spec) at x. It is available only for beta, chi-square, F, and Student’s t.
  • SIG Tail probability function. A tail probability function SIG.d_spec(x,a,...) returns a probability p that a variate with the specified distribution (d_spec) is larger than x. The tail probability function is equal to 1 minus the cumulative distribution function.

• How can I generate an ID variable?
  For existing data, run this in syntax to add an ID variable:

  COMPUTE ID=$CASENUM.
  EXECUTE.
  

  For a blank data sheet, where you wish to have 100 cases, run this syntax:

  NEW FILE.
  INPUT PROGRAM.
     LOOP #I=1 TO 100.
        COMPUTE ID=$CASENUM.
        END CASE.
     END LOOP.
     END FILE.
  END INPUT PROGRAM.
  EXECUTE.
  

Bibliography

• Abramowitz, M. & Stegun, I. A., eds. (1970). Handbook of mathematical functions. NY: Dover Publications.
• SPSS, Inc. (2007). SPSS 16 Command Syntax Manual. Chicago, SPSS. See the section on "Random Variable and Distribution Functions," pp. 69-93.