General Report on Tunny
25W Page 190
It is considered that for the significance of a single trial, a sufficient sigma-age is 2, i.e.
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| or approximately | ||
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(For this test R3 pp 5, 6, 7. A slight generalization R4 p 100. Controversy R4 pp 93, 100. Tests based on a letter count Χ2 test R4 p 54; R5 pp 3, 8, 11, 37, 113: Σn log n R4 pp 56, 70, 121; R5 pp 1, 3, 7.)
(b) Fundamental decibanning formula.
Let δ be the true proportional bulge of the run.
Suppose that a particular place in the cipher is observed to favour a dot in ΔΧ.
If the corresponding character of ΔΧ is a dot, the probability of the observation is 
If the corresponding character of ΔΧ is a cross, the probability of the observation is 
Whence the factor in favour of a dot is 
Hence if the double bulge in favour of a dot, i.e. the excess of places favouring dot over places favouring cross, is xi, the total factor is
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whose decibanage is  |
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i.e.  decibans per pip |
W9 |
N.B. This is the decibanage for dot rather than cross, not for dot rather than random.
The major problem is to find δ.
(c) Impractibility of exact formulae.
An exact formula is impracticable, for it must use not only the evidence of the run, but also general fish evidence. Even if it could be evaluated, its precision would be largely illusory. The formula is
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