1. I conclude, then, that the application of the mathematical methods, discussed in the preceding chapter, to the general problem of statistical inference is invalid. Our state of knowledge about our material must be positive, not negative, before we can proceed to such definite conclusions as they purport to justify. To apply these methods to material, unanalysed in respect of the circumstances of its origin, and without reference to our general body of knowledge, merely on the basis of arithmetic and of those of the characteristics of our material with which the methods of descriptive statistics are competent to deal, can only lead to error and to delusion.

But I go further than this in my opposition to them. Not only are they the children of loose thinking, and the parents of charlatanry. Even when they are employed by wise and competent hands, I doubt whether they represent the most fruitful form in which to apply technical and mathematical methods to statistical problems, except in a limited class of special cases. The methods associated with the names of Lexis, Von Bortkiewicz, and Tschuprow (of whom the last named forms a link, to some extent, between the two schools), which will be briefly described in the next chapter, seem to me to be much more clearly consonant with the principles of sound induction.

2. Nevertheless it is natural to suppose that the fundamental ideas, from which these methods have sprung, are not wholly égarés. It is reasonable to presume that, subject to suitable conditions and qualifications, an inversion of Bernoulli's theorem must have validity.