# Notes

## Section 9: Statistical Analysis

Tests of randomness

Statistical analysis has in practice been much more concerned with finding regularities in data than in testing for randomness. But over the course of the past century a variety of tests of randomness have been proposed, especially in the context of games of chance and their government regulation. Most often the tests are applied not directly to sequences of 0's and 1's, but instead say to numbers obtained from blocks of 8 elements. A typical collection of tests described by Donald Knuth in 1968 includes: (1) frequency or equidistribution test (possible elements should occur with equal frequency); (2) serial test (pairs of elements should be equally likely to be in descending and ascending order); (3) gap test (runs of elements all greater or less than some fixed value should have lengths that follow a binomial distribution); (4) poker test (blocks corresponding to possible poker hands should occur with appropriate frequencies); (5) coupon collector's test (runs before complete sets of values are found should have lengths that follow a definite distribution); (6) permutation test (in blocks of elements possible orderings of values should occur equally often); (7) runs up test (runs of monotonically increasing elements should have lengths that follow a definite distribution); (8) maximum-of-t test (maximum values in blocks of elements should follow a power-law distribution). With appropriate values of parameters, these tests in practice tend to be at least somewhat independent, although in principle, if sufficient data were available, they could all be subsumed into basic block frequency and run-length tests. Of the sequences on page 594, (a) through (d) as well as (f) fail every single one of the tests, (e) fails only the serial test, while (g) and (h) pass all the tests. (Failure is defined as a value that is as large or small as that obtained from the data occurring below a specified probability in the set of all possible sequences.) Widespread use of tests like these on pseudorandom generators (see page 974) began in the late 1970s, with discoveries of defects in common generators being announced every few years.

In the 1980s simulations in physics had begun to use pseudorandom generators to produce sequences with billions of elements, and by the late 1980s evidence had developed that a few common generators gave incorrect results in such cases as phase transition properties of the 3D Ising model and shapes of diffusion-limited aggregates. (These difficulties provided yet more support for my contention that models with intrinsic randomness are more reliable than those with external randomness.) In the 1990s various idealizations of physics simulations—based on random walks, correlation functions, localization of eigenstates, and so on—were used as tests of pseudorandom generators. These tests mostly seem simpler than those shown on page 597 obtained by running a cellular automaton rule on the data.

Over the years, essentially every proposed statistical test of randomness has been applied to the center column of rule 30. And occasionally people have told me that their tests have found deviations from randomness. But in every single case further investigation showed that the results were somehow incorrect. So as of now, the center column of rule 30 appears to pass every single proposed statistical test of randomness.

From Stephen Wolfram: A New Kind of Science [citation]