XIII,—SIMPLE SAMPLING OF ATTRIBUTES. Co 
“standard-deviation of simple sampling” may be regarded as a 
measure of the magnitude of such errors, and may be called ac- 
cordingly the standard error, 
Three principal cases of comparison may be distinguished. 
Case I.—I¢t is desired to know whether the deviation of a certain 
observed number or proportion from an expected theoretical value 
is possibly due to errors of sampling. 
In this case the observed difference is to be compared with the 
standard error of the theoretical number or proportion, for the 
number of observations contained in the sample. 
Example i.—In the first illustration of § 7, 25,145 throws of a 4, 
5, or 6 were made in lieu of the 24,576 expected (out of 49,152 
throws altogether). The excess is 569 throws. Is this excess 
possibly due to mere fluctuations of sampling ? 
The standard error is 
o= 3x4 x49152 
= 1109, 
The deviation observed is 5°1 times the standard error, and, 
practically speaking, could not occur as a fluctuation of simple 
sampling. It may perhaps indicate a slight bias in the dice. 
The problem might, of course, have been attacked equally well 
from the standpoint of the proportion in lieu of the absolute 
number of 4’s, 5s, or 6’s thrown. This proportion is 0-5116 instead 
of the theoretical 05000, difference in excess 0:0116. The 
standard error of the proportion is 
L 
sent X 4 X 49159 =0-00226, 
and the difference observed bears the same ratio to the standard 
error as before, as of course it must. 
Example ii.—(Data from the Second Report of the Evolution 
Committee of the Royal Society, 1905, p. 72.) 
Certain crosses of Pisum sativum gave 5321 yellow and 1804 
green seeds. The expectation is 25 per cent. of green seeds, or 
1781. Can the divergence from the exact theoretical result have 
arisen owing to errors of sampling only? 
The numerical difference from the expected result is 23. The 
standard error is 
o= 025x075 x T125 = 36-8. 
Hence the divergence from theory is only some 3/5 of the 
standard error, and may very well have arisen owing simply to 
fluctuations of sampling. 
26"