The law of error and the combination of observations

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  • HJHarold Jeffreys
    Harold Jeffreys
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Abstract
The normal or Gaussian law of error rests partly on a particular hypothesis about the nature of error, that the error of any individual observation is the resultant of a large number of comparable and independent components; and partly on comparison with frequencies in actual series of observations. Both arguments are defective. The theory does not prove that the law is true for errors of any magnitude, even if its fundamental hypothesis is accepted. It involves a number of approximations, and when the effects of these are examined it is found that the law should hold only up to a moderate multiple of the standard error. This is obvious in the simplest case of the binomial distribution, since the normal law predicts a finite though small chance of an error of any amount, but the binomial law is rigidly limited at each end. The assertion of the normal law for errors of any magnitude as the limit of the binomial really assumes that the number of components is infinite, and that all have the same infinitesimal range. This is a very remarkable hypothesis. There may be some inductive reason, in cases where many sources of variation have already been determined and allowed for, to suppose that there are several others just below the magnitude that can be detected separately; but there is every reason to suppose that these are not infinite in number, but merely the largest members of a convergent series, and in such a case there is no reason to suppose that their resultant will tend to the normal law. Whittaker and Robinson give a striking example to the contrary (1924, p. 178), where the components all follow the median law and their magnitudes diminish. Indeed, if there is one dominant component the law for the resultant will approximate to the law for that component separately. Even if there are several equal components the proof may fail. Fisher has shown (1922, p. 321) that if several errors follow the law P (dx/h) (1+x2)-1dx the mean of any number will follow the same law with the same constants.

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