12 JOURNAL OF CONCHOLOGY, VOL. l6, NO. I, JANUARY, 1919. 



much u.se ; if you can conveniently get more than loo so much 

 the better. For some purposes thousands may be desirable, but 

 they are seldom practicable, and the advance in precision by sub- 

 stituting 1,000 specimens for loo is the same as by taking loo 

 instead of lo. 



(2). The collection and measurement having been achieved, 

 the question of interpretation arises. If nine specimens of biden- 



tata from locus A measure in length lo'o, lo'i io"8 m.m., the 



average length is io"4 m.m. ; if nine specimens from locus B 



measure io"i, io'2 io'q m.m., the average is io"5 m.m. It is 



natural to conclude that bidentata from locus B is longer than from 

 locus A, but since the size is so variable and the whole snail popu- 

 lation of each locus much larger than the sample which has been 

 examined, it is a fairly open possibility that a second sample of nine 

 from locus A might give the same measurements as those from 

 locus B, and vice-versa. The question as to whether the species 

 differs in size in the two loci is really one of the probability of our 

 samples being really representative of the two populations ; for its 

 solution we have to proceed by statistical methods of numerical 

 analysis.-' In a case such as the -present, the recognised procedure 

 is as follows, using the examples already imagined. For locus A 

 take the mean length = io"4 m.m. ; for each specimen take the 

 difference (it does not matter whether it is in excess or in defect) 

 between its length and the mean, i.e., o'4, 0-3, 0*2, o-i, o"o, o"i, 

 o'2, o'3, o'4; square the differences, i.e., o'i6, o'og, 0-04, o'oi, o'oi, 

 o"04, o'09, o'i6; divide the sum of the squares by the number of 



specimens, i.e., =o"o667 ; take the square root of the quo- 

 tient, i,e., o'258 m.m., which is the standard deviation (shortly o-). 

 The standard error of the mean is the standard deviation divided by 



o"258 

 the square root of the number of observations, i.e., =o'o86 



m.m. For locus B the mean is io'5, its standard error and the 

 standard deviation being the same as for locus A. Take the square 

 root of the sum of the squares of the standard errors of the two 

 means, i.e. (o"oo74 -I- o'oo74) °'5 ^ o'i22 m.m., which is the sta7i- 

 dard error of the difference. The difference between the two means 

 must be at least three times the standard error of the difference to 

 be regarded as " significant." If it is less, there is quite a good 

 chance that we are merely dealing with two samples of the same 



I For an account of this important subject of more than ordinary clarity, see G. U. Yule, 

 I?itroduction to the Theory of Statistics (Griffin & Co.), a book to which biologists of all sorts 

 ought to be deeply indebted. The particular topics dealt with here will be found in chapters vi. 

 — viii. and xiii. 



