cxl Tables for Statisticians and Biometriciam [XXV 



It appears to me that in applying his test " Student " has really to face two 

 problems, which cannot be solved by a single investigation in the manner he 

 proposes : 



(i) If we take two wholly independent sets of individuals, and administer the 

 laevo- form of the soporific to one and the dextro- to the other, is there a prob- 

 ability, and what value has it, that the two means differ, and can we thus determine 

 which is the more efficient ? 



(ii) If we administer both soporifics to the same set of individuals, i.e. allowing 

 for the individual reactions to the two forms of the drug, will the data indicate 

 that the one is more effective than the other ? 



Now (i) can be answered by " Student's " test, because he can suppose the 

 samples drawn from the same population, and thus see how improbable the results are. 

 Or, the (e') test may be used, if the samples are of different sizes. 



But (i) must be answered before (ii). If (i) show there to be no substantial 

 difference in the hours of sleep of the two sets, then u may be put = v in (e) for (ii). 

 But if the answer to (i) is that u and v in all probability differ, then it does not 

 seem valid to put u = v in (e) for (ii). It is clear that if u be not equal to v, then a 

 very different value and a much smaller value will be obtained for x' than that 

 given by " Student." The problem thus raised appears to repeat itself in others of 

 " Student's " illustrations, and my object is to press for caution in the application 

 of his test, and indeed in other tests similar to it. 



NOTE TO THIS SECTION. 

 On Interpolation into Table XXV for small Values of q = %(n 1). 



Interpolation for q %(n 1) is bound to be laborious, even if it be straight- 

 forward. In interpolating for q into Table XXV, it will be found best, particularly 

 in the earlier part of the table, to use a forward difference formula, e.g. 



If we use the tabled value P x (\, q), we may have to find, even at x '25, eight or nine 

 differences to get the correct result to seven decimal places. But if we reduce the 

 P x (, q) to B x (i q) by the relation B x (%, q) = \2P X (%,q)-I}xB (, q) four or five 

 differences will suffice for 7-figure accuracy when x = '25. For x = '50, the seventh 

 difference is required for the B x (%, q)'s. The I x (^, g)'s would need far more. In 

 order that Ag may not exceed '25, we can use when it appears desirable a negative 

 interpolation. 



The value of B (\, q) is given at the top of each column to assist the reader in 

 reducing the tabled entries to B x (|, q). From the interpolated value of the latter we 

 find Pa (\, q} by determining from a table of the complete T-functions the complete 

 B-function corresponding to the interpolated value. 



