General Report on Tunny
23L Page 99
the expected sigma-age would be 5.3. This would be more than sufficient to distinguish between the 2257 (= 37x61) different possible hypotheses about the possible motor settings. With D = 3/4 and the same value of rx and r. the expected sigma-age would be only 3.7.
The argument and formula for the expected sigma-age would be equally valid for any other letter or group of letters, instead of /'s. In particular it can be used for groups of weak letters instead of strong ones. The expected sigma-age is then negative and one has to look for low scores instead of high ones. The formula is not so reliable in this case, since the sampling numbers rx and r. are smaller.
(c) Expected sigma-age with limitation not Χ2.
When the limitation is not Χ2, a similar formula can be obtained equally easily if ΔD is assumed to be 'flat' against motor crosses.
If r is the number of occurrences of the letters in ΔD and if p, q, N, D have the same meanings as before, then, by equating the expected value of r to the observed value we have
| r |  |
|
| and E.S. |  |
|
| = r - (1 - D)Nq | ||
| Expected bulge | = (1-D)(r-Nq) | |
| = (1-D)(r-Nε) |
where υ is the number of letters of the alphabet being looked for. Expected sigma-age is
 |
||
| ≈ |  |
Sometimes the assumption of 'flatness' opposite motor crosses is quite wrong. For example, if / is a common P letter then 8 is a good motor cross ΔD letter and a motor run for 8's may be far less powerful than the preceding formula suggests.
(See operational log 01, pp. 32, 37, and R5 p. 32 etc.).
This difficulty does not arise when the limitation is Χ2.
(d) Complementary nature of machine and hand methods.
It is interesting to observe that, for given ΔD count, the expected sigma-age on any motor run is larger for smaller μ37 dottages d. This is what has been described as a 'swings and roundabouts' effect. When d is
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