392 | 36 INTRODUCING TURING’S mATHEmATICS
his powers, and was certainly influential in his approach to codebreaking, so it makes a fitting
first topic for this chapter.
Turing’s single paper on numerical analysis, published in 1948, is not described in detail
here. It concerned the potential for errors to propagate and accumulate during large-scale com-
putations; as with everything that Turing wrote in relation to computation it was pioneering,
forward-looking, and conceptually sound. There was also, incidentally, an appreciation in this
paper of the need for statistical analysis, again harking back to Turing’s earliest work.
The central limit theorem
The idea of a ‘bell-shaped’ data distribution is a familiar one: that female human beings, say,
have an average height (the ‘mean’ of the distribution) and that roughly as many are shorter than
this height as are taller, tailing off in both directions. The measure of ‘standard deviation’ is a
way of making precise this ‘tailing off ’: to say that a woman taller than 2 m is extremely unlikely
is to say that this part of the height distribution is, perhaps, four standard deviations away from
the mean. To be even more precise, this tailing off can be drawn as a mathematical curve. A
curve that has been observed to describe the bell-shaped data distribution particularly well is
the ‘normal distribution’, also known as the ‘Gaussian distribution’, although it was analysed
by Pierre-Simon Laplace in 1783 when Carl Friedrich Gauss was only six years old (Fig. 36.1).
Laplace observed that the normal distribution is prevalent in astronomical and probability
calculations. He began the process of tabulating its values; to this day such tables are generally
included in textbooks on statistics and probability. Some standardization is essential and we
use the ‘standard normal distribution’, in which the mean is 0, the standard deviation is 1, and
the height of the distribution at the mean is also 1. The values that are tabulated are areas,
because these correspond to probabilities: given any non-negative value X, the probability that
any given measurement falls between 0 and X is the area lying between the standard normal
curve and the horizontal axis between 0 and X.
So we now have a beautifully simple way of making predictions! What is the probability that
the next woman I pass in the street is taller than 2 m? I shift the average observed height to the
l–3 –2 –1 0 1 2 3figure 36.1 The normal
distribution, standardized to
place its mean at 0.