Main article.
Of course, we cannot claim that our example satisfies fully all theoretical
assumptions; however, on the other hand, we can scarcely believe that the agreement of
the numbers we have discovered is pure coincidence; rather, that it is related to a certain
correspondence of the theoretical assumptions and the conditions of the example.
Now, we shall turn to the other arrangement of the 20,000 letters in hundreds that
we have made. We construct a table with the repetitions of individual numbers, like
the one before.
The arithmetic mean of these 200 new numbers is the same as before
However, the sum of the squares of their deviations from 43.2 is considerably greater than before; it is, namely,
Here, it is necessary to pay attention to the assumption of the independence of the quantities, which is usually connected with the method of least squares (see chapter VII of my book "Calculus of Probability"); let us recall why this assumption is necessary. It is necessary to determine the weight of the end result that is expressed by equation (21), and also to calculate the mathematical expectation W, which gives the approximate value k (see my book). However, this condition will prove to be superfluous, if first, we leave aside the question of the weight of equation (21), and second, replace ΞΎ in expression W with the number a, which we shall assume to be equal to a0 , in that we disregard the difference a - a0 . Then, the equations
and
form the basis of our deductions, not requiring any independence of the quantities
