108 Franas B. Sumner 
only to cases where we are dealing with large numbers of indi- 
viduals. Davenport suggests 200 as the minimum number of 
individuals to be gathered for statistical treatment when the 
material is available; though he grants that much smaller num- 
bers may be employed to advantage, where we are restricted by 
circumstances. But it must be borne in mind, he says” “that 
the rules for determination of averages, probable errors, standard 
deviations and all the rest become less and less significant as the 
number of variants becomes smaller. Finally, in the region of 
twenty or so, the results can no longer be treated by mass statis- 
tics; twenty hardly makes a mass.” ‘To the experimentalist it 
must often happen, as in the present work, that the use of a 
large number of individuals, in any single series, is excluded by 
reason of the laboriousness of the methods employed. In such 
cases, we are told, no exact mathematical equivalent can be 
offered for the probability of a given result, even though the 
frequency distributions afford strong evidence on the subject. 
Of course the cumulative testimony of several independentseries 
of experiments is of highvalue. Ingeneral, it would seem that the 
experimentalist demands a somewhat different statistical tech- 
nique from the student of variation per se, and it is to be regarded 
as unfortunate that the methods at our disposal, have, thus far 
been developed mainly by the latter type of investigator, and 
with very little reference to the special needs of the former. To 
the experimentalist, who is studying the effects of artificial con- 
ditions, it is the significance of differences, and scarcely anything- 
else in the whole field of statistical theory, that is likely to be of 
interest. 
Regarding the accuracy of my computations, I can only say 
that every step has been gone over at least twice, and that, wherever 
possible, a different method of calculation has been employed in 
the repetition. 
RESULTS IN DETAIL 
Series of 1G00-1907 
Owing to the small number of individuals used and the tentative 
character of my methods at the outset, this first series will not be 
16 Tn a letter to the author 
