The Chemistry Maths Book, Second Edition

616 Chapter 21Probability and statistics

The function is symmetric aboutx 1 = 1 μ. It is broad when σis large and narrow when

σis small, and it falls to about 0.6 of its maximum value whenx 1 = 1 ±σ, the points of

inflection of the function.

Errors

The Gaussian distribution is important in the sciences because it describes the

distribution of errors in a sequence of random experiments.

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Measurements, however

accurately made, are always accompanied by errors. They are of two kinds: determinate

and random. Determinate errors may be simply mistakes or they may be systematic

errors due to faulty or incorrectly calibrated apparatus. It is usually possible to discover

the causes of such errors, and either to eliminate them or make corrections for them.

Random errors, on the other hand, are indeterminate. They may be due to the ‘shaky

hand’, the imperfect resolution of an instrument, the unpredictable fluctuations of

temperature or voltage.

Errors are in general due to several or many causes. The central limit theoremof

probability theory tells us that a variable produced by the cumulative effect of several

independent variables is approximately Gaussian or ‘normal’. Many observed quantities

are of this kind; for example, anatomical measurements such as height or length of

nose are due to the combined effects of many genetic and environmental factors, and

their distributions are Gaussian. In addition, many theoretical distributions behave

like Gaussian distributions when the numbers are large. The binomial distribution

tends to a Gaussian withμ 1 = 1 npand asn 1 → 1 ∞. Random numbers

have a uniform distribution when considered singly. If taken nat a time, however,

they have a distribution that becomes Gaussian as nbecomes large.

The distribution function

The probability distribution functionF(x)of the Gaussian distribution is defined by

(21.40)

Fx xdx e dx

xx

x

() ()

()

==

−−

−

ZZ

∞∞

ρ

σ

μσ

1

2

22

2

π

σ=−np p()1

σ= 0. 5

σ=1. 0

σ=2. 5

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x

ρ(x)

μ− 3 μ− 2 μ− 1 μ μ+1 μ+2 μ+3

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Figure 21.7

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The first description of the normal law of errors was by de Moivre in 1738. It was derived by Gauss in his

Theory of motion of the heavenly bodiesof 1809 from a principle of maximum probability: ‘if any quantity has been

determined by several observations, made under the same circumstances and with equal care, the arithmetical

mean of the observed values affords the most probable value’. Gauss considered the function (21.39) to be the

‘correct’ error function because he was able to derive from it the principle of least squares. Laplace gave an

alternative derivation of the normal law in 1810, and included it in his 1812 Analytical theory of probability.