RACES OF HERRING, SOUTHEASTERN ALASKA 
127 
in 20 times in the former case and 1 in 100 times in the latter. The main advantage of using vari- 
ance as a measure of variability lies in the additive property of the sums of squares, and of the 
degrees of freedom. Thus we may add together S(x m — x m ) 2 and S(x n — x n ) 2 and obtain an efficient 
estimate of <r 2 founded on n m — n n degrees of freedom, or we may divide our whole sample of counts 
into n' , sets, with n\, n'% . . n' , counts in the different sets, as we did when considering weighted 
means and obtain 
< S(a?i — xi) 2 4-5(a; 2 — x 2 ) 2 + . . . . + S(x r —x r ) 2 
n-\-n 2 -\-. . . n r 
or as an estimate of a 2 based on N' —r degrees of freedom. This last estimate is usually 
N — r 
called the variance “within classes” while that based on weighted means is called the variance 
“between classes.” It should be mentioned that, in adding the sums of squares about the group 
means and dividing by the total number of degrees of freedom involved, we are assuming that the 
variances within the different groups are not significantly different. If significant differences are 
suspected, the 2 -test should be applied, since the Analysis of Variance is not sound if such differences 
are found. 6 
To clarify the exact steps in testing the data on the hypothesis of homogeneity, 
the following table has been inserted: 
Table 2.— Analysis of vertebral counts of samples of the 1926 class year from Noyes Island (see 23, 
fig. 2) to illustrate method of testing for homogeneity of the population 
A 
B 
C 
D 
E 
F 
G 
June 21, 1930 
51. 786 
14 
—0. 402 
0. 161604 
2. 262456 
6. 357 
Do. 
52. 250 
24 
.062 
. 003844 
. 092256 
12. 500 
52. 348 
23 
. 160 
. 025600 
. 58S800 
15. 217 
52. 226 
31 
.038 
. 001444 
. 044764 
13. 419 
Do. 
52. 000 
9 
— . 188 
. 035344 
. 318096 
2. 000 
June 27, 1930 
51. 857 
7 
—.331 
. 109561 
. 766927 
2. 857 
52. 240 
25 
.052 
. 002704 
. 067600 
10. 560 
July 18,' 1930 
52. 231 
39 
.043 
. 001521 
. 059319 
16. 923 
July 28| 1930 
52. 214 
14 
.026 
. 000676 
. 009464 
4. 357 
186 
4. 209682 
84. 190 
Note.— General mean, 52.188; column A, date of sampling; column B, mean of sample; column C, number in sample; column D, 
column B minus general mean; column E, column D squared; column F, column C times column E; and column G, sum of squared 
deviations from mean of sample. 
Variance 
Degrees of 
freedom 
Sum of 
squares 
Mean 
square 
Loge mean 
square 
8 
4. 209682 
0. 5262 
-0. 6420 
177 
84. 190 
.4756 
-. 7433 
Difference.-. .. - 
Observed z (half of difference) 8 . 
Calculated z for probability of 0.05 6 
0. 1013 
.0507 
. 3322 
» The variance “within classes” is the weighted mean of all the sample variances, the weights being the numbers of counts 
in the samples, less one in each case. 
6 The value of observed z being less than that of calculated z (methods of calculating z are given by Fisher, 1930, p. 201) at a 
probability of 0.05 shows the population to be homogeneous. 
