Given an ordered sequence consisting of fixed numbers of A's and B's, it is sometimes necessary to test whether these letters occur in a random order. In particular situations A and B may represent positive and negative signs, heads and tails of a coin, defective and effective parts, wet and dry years, and so on. Many tests of randomness are available, but this note only deals with two of the possible tests. One of these tests is described by Johnson & Tetley in their Statistics text-book (vol. 2, p. 238), where the sequence of positive and negative signs of the deviations between actual and expected deaths in a mortality table graduation is investigated for randomness. However, it is not made plain in the official text-book exactly what aspect of randomness is being tested; it is hoped that this note will make the matter clear.

The situation to which both the tests described below are to be applied is as follows. Given fixed numbers n1 of A's and n2 of B's arranged in an ordered sequence, it is found that the particular sequence observed contains g1 groups of consecutive A's and g2 groups of consecutive B's.