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This distribution applies also to the sum of the squares of (n+1) such variables whose sum is fixed.

(m) __Some simple formulae of a non-analytic type concerning proportional bulges__.

If P and Q are independent propositions

P.B.(P.Q.) = P.B.(P) P.B.(Q) + P.B.(P) + P.B.(Q)

If P_{i}(i = 1, 2, ...) are mutually exclusive and exhaustive propositions each with the same 'random probability' then

If P_{i} (i = 1, 2, ...) are mutually exclusive propositions with the same random probability,

If P, Q, Φ, Θ are teleprinter letters which have the same number of components, then

(n) __The general formula for sigma in Tunny work__.

Let two tapes be compared, one with a proportion p_{i} of letters A_{i} and the other with a proportion q_{i} of letters B_{i} (i = 1, 2, ... r). Let the overlap of the two tapes be N**.**. Let the number of times A_{i} comes opposite B_{i} be υ_{i}. Let . Then the average of υ is and .

In particular, if r = 1 | |

σ^{2} = Np(1-p)q(1-q). |

The proof of the general formula is best done by the method of characteristic functions. We do not describe this method here, but instead refer the reader to R4, 105-108.

(o) __The statistician's fallacy__.

A standard type of statistical experiment is exemplified by the following. A new fertiliser is tried and the amount of the crop produced is increased by 2σ. A deviation 2σ above the mean occurs about once in 40 experiments at random, assuming a normal distribution, and the result would probably be regarded as significantly good. As a conventional test of significance this is a useful method and one which is used in Tunny breaking also (as in the significance test for a short wheelbreaking run).

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