Some simple randomness tests for a sequence of numbers.


Ivan Raikov




(require-extension random-test)




The random-test library provides a procedure that applies various statistical tests to a sequence of random numerical values, and a procedure to reports the results of those tests in convenient form.

The library is useful for evaluating pseudorandom number generators for statistical sampling applications, compression algorithms, and other applications where the properties of a random sequence are of interest. The code in the library is based on the ent program by John Walker.


procedure: make-random-test:: [CAR CDR NULL?] -> (SEQ -> RANDOM-STATS)

This procedure creates a procedure that reads in a sequence of numerical values, and performs statistical tests to tests the randomness of the elements of the sequence.

By default, the sequence is expected to be a list; however, if a different sequential data structure is used (e.g. a stream), the optional arguments CAR, CDR, NULL? may be used to specify procedures that perform the corresponding operations on the input sequence.

The returned procedure is of the form SEQ -> RANDOM-STATS, where SEQ is the sequence and the returned value is an alist with the following fields:

'ent entropy measure
'chisq the result of the Chi-Square test (see next section)
'pochisq the calculated probability of the Chi-Square test (see next section)
'mean the mean of the values in the input sequence
'min the minimum of the values in the input sequence
'max the maximum of the values in the input sequence
'montepi Monte Carlo value of pi (see next section)
'scc the serial correlation coefficient (see next section)

procedure: format-random-stats:: OUT * RANDOM-STATS -> UNDEFINED

Given an output port, and the value returned by the random test procedure, this procedure outputs a human readable interpretation of the test results.


  1. Chi-Square Test

    In general, the Chi-Square distribution for an experiment with k possible outcomes, performed n times, in which Y1, Y2,... Yk are the number of experiments which resulted in each possible outcome, and probabilities of each outcome p1, p2,... pk, is given as: \chi^{2} = \sum_{1 <= i <= k}{\frac{(Y_{i} - m  p_{i})^{2}}{np_{i}}}

    \Chi^2 will grow larger as the measured results diverge from those expected by pure chance. The probability Q that a Chi-Square value calculated for an experiment with d degrees of freedom (where d=k-1, one less the number of possible outcomes) is due to chance is: Q(\Chi^2,d) = [2^{d/2} * \Gamma(d/2)]^{-1} * \int_{\Chi^2}^{\infty}(t)^{d/2-1} * exp(-t/2) * dt

    Where Gamma is the generalization of the factorial function to real and complex arguments: \Gamma(x) = \int_{0}^{\infty} t^{x-1} * exp(-t) * dt

    There is no closed form solution for Q, so it must be evaluated numerically. Note that the probability calculated from the \Chi^2 is an approximation which is valid only for large values of n, and is therefore only meaningful when calculated from a large number of independent experiments.

    In this implementation, the Chi-Square distribution is calculated for the list of values given as argument to the random-test procedure and expressed as an absolute number and a percentage which indicates how frequently a truly random sequence would exceed the value calculated.

    The percentage can be interpreted as the degree to which the sequence tested is suspected of being non-random. If the percentage is greater than 99% or less than 1%, the sequence is almost certainly not random. If the percentage is between 99% and 95% or between 1% and 5%, the sequence is suspect. Percentages between 90% and 95% and 5% and 10% indicate the sequence is almost suspect.

  2. Arithmetic Mean

    This is simply the result of summing the all the values in the sequence and dividing by the sequence length. If the data are close to random, the mean should be about (2^b - 1)/2 where b is the number of bits used to represent a value. If the mean departs from this value, the values are consistently high or low.

  3. Monte Carlo Value for Pi

    Each pair of two values in the input sequence is used as X and Y coordinates within a square with side N (the length of the input sequence). If the distance of the randomly-generated point is less than the radius of a circle inscribed within the square, the pair of values is considered a hit. The percentage of hits can be used to calculate the value of pi:

     # points within circle     1/4 * pi * r^2 
     ----------------------  =  -------------- = 1/4 * pi
     # points within square          r^2
              # points within circle
     pi = 4 * ----------------------
              # points within square

    For very long sequences (this approximation converges very slowly), the value will approach the correct value of Pi if the sequence is close to random.

  4. Serial Correlation Coefficient

    This quantity measures the extent to which each value in the sequence depends upon the previous value. For random sequences, this metric (which can be positive or negative) will be close to zero. A non-random sequence such as a text file will yield a serial correlation coefficient of about 0.5. Predictable data will exhibit serial correlation coefficients approaching 1.


(require-extension random-test)
(require-extension srfi-1)

(randomize (current-milliseconds))

(define random-test (make-random-test))
(define lst (list-tabulate 1000000 (lambda (x) (random 1000000))))

(random-test lst 1000000)

(define stats (random-test lst 1000000))

(format-random-stats stats)


Copyright 2007 Ivan Raikov. 

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or (at
your option) any later version.

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
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