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to extract certain features that are relevant for drawing specific conclusions about the data.

And a fundamental example is to try to determine whether a given sequence can be considered perfectly random—or whether instead it contains obvious regularities of some kind.

From the point of view of statistical analysis, a sequence is perfectly random if it is somehow consistent with a model in which all possible sequences occur with equal probability.

But how can one tell if this is so? What is typically done in practice is to take a sequence that is given and compute from it the values of various specific quantities, and then to compare these values with averages obtained by looking at all possible sequences.

Thus, for example, one might compute the fraction of squares in a given sequence that are black, and compare this to 1/2. Or one might compute the frequency with which more than two consecutive black squares occur together, and compare this with the value 1/4 obtained by averaging over all possible sequences.

And if one finds that a value computed from a particular sequence lies close to the average for all possible sequences then one can take this as evidence that the sequence is indeed random. But if one finds that the value lies far from the average then one can take this as evidence that the sequence is not random.

The pictures at the top of the next page show the results of computing the frequencies of different blocks in various sequences, and in each case each successive row shows results for all possible blocks of a given length. The gray levels on every row are set up so that the average of all possible sequences corresponds to the pattern of uniform gray shown below. So any deviation from such uniform gray potentially provides evidence for a deviation from randomness.

And what we see is that in the first three pictures, there are many obvious such deviations, while in the remaining pictures there are no obvious deviations. So from this it is fairly easy to conclude that the first three sequences are definitely not random, while the remaining sequences could still be random.

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