176 SECTIONAL ADDRESSES. 
such as we have described. Should anyone ask why this particular 
equation should have the virtue of necessitating such a divisibility, I can 
only answer that it is but one out of all the miracles of mathematics. 
I never cease to be astonished at it myself. For further elucidation, 
reference must be made to the mathematical proof." 
So far we remain in comparatively smooth waters. The chief 
difficulty arises when we turn to what are known as the errors of sampling. 
Suppose you wanted to know whether a field of potatoes was bearing 
a good crop. You walk about, pulling up a plant here and there. This 
gives you some approximate knowledge, but not an exact one. For all 
you can tell, you may have been exceptionally lucky or unlucky in your 
selection. The degree of discrepancy between the average size of the 
potatoes actually pulled and that of the whole field is your error of 
sampling. Now, just the same befalls any co-efficient of correlation 
between two abilities. You want to know how closely these two go 
together with people of some general class. You pick out, say, 100 of 
these people. But just in this 100 the correspondence between the two 
abilities may happen to be rather higher or lower than on an average 
throughout the entire class. Your co-efficient of correlation will have an 
error of sampling. And our preceding tetrad-difference, being made up 
of correlational coefficients, will have one also. 
Now, the latest advance in the theory of g consists in showing the 
general magnitude of the tetrad-differences that will arise from the 
sampling errors alone, even when the true magnitude is always zero. 
This value of the tetrad-differences to be expected merely from sampling 
was published last year (Brit. J. Psychol.). 
But, having got this theoretical value, there remains the momentous 
step of comparing it with the median value which is actually observed. 
The theory of g stands or falls according as these two values are or are 
not found to agree. 
This step so fraught with fate has now been taken. To avoid all 
danger of personal bias, no work of my own was chosen for this crucial 
decision, but that of an investigator who, more than all others, had shown 
himself unsympathetic with the doctrine of g. Here is his table of 
correlations as published by himself : 
Test Lind oSric ined 6261) 7, 6813 59 ol0ndd da ash 14 
1. Completion .. 98 94 79 62 91 71 54 78 88 55 42 33 25 
2. Hard Opposites 98 84 80 64 81 79 70 73 74 52 43 26 25 
3. Memory words 94 84 62 55 82 49 56 73 71 53 40 28 21 
4. Easy Opposites... 79 80 62 57 52 68 53 42 56 45 29 38 48 
Be Aa Test® .< -- 62 64 55 57 55 54 73 39 51 39 59 25 22 
6. Memory pass. .. 91 81 82 52 55 53 57 59 66 54 31 28 19 
7. Adding .. -. 71 79 49 68 54 53 45 39 47 51 57 17 25 
8. Geomet. forms.. 54 70 56 53 73 57 45 35 49 34 56 25 25 
9. Learn. pairs .. 78 73 73 42 39 59 39 35 69 36 29 26 09 
10. Recog. forms .. 88 74 71 56 51 66 47 49 69 44 37 34 28 
TI) Serolls ser -- 55 52 53 45 39 54 51 34 36 44 31 19 27 
12. Compl. words .. 42 43 40 29 59 31 57 56 29 37 31 21 07 
13. Estimat. length 33 26 28 38 25 28 17 25 26 34 19 21 24 
14. Drawing length 25 25 21 48 22 19 25 25 09 28 27 O7 24 
To work out all the tetrad-differences was no light undertaking, since 
they run to the number of 3,003. The calculation of these was entrusted 
1 Proc. Roy. Soc., 1923. 
