- THEORY OF STATISTICS 
holds for all values of @;,, ®, and #; (which are, in our usual notation, 
deviations from their respective arithmetic means), find the correlations 
between @,, 2, and «3; in terms of their standard-deviations and the values of 
a, b and c. 
10. What is the effect on a weighted mean of errors in the weights or the 
quantities weighted, such errors being uncorrelated with each other, with the 
weights, or with the variables—(1) if the arithmetic mean values of the errors 
are zero ; (2) if the arithmetic mean values of the errors are not zero ? 
11. Cf. (Pearson, ‘‘On a Generalised Theory of Alternative Inheritance,” 
Phil. Trams., vol. cciii., A; 1904, p. 53). If we consider the correlation 
between number of recessive couplets in parent and in offspring, in a 
Mendelian population breeding at random (such as would ultimately result 
from an initial cross between a pure dominant and a pure recessive), the 
correlation is found to be 1/3 for a total number of couplets n. If n=1, the 
only possible numbers of recessive couplets are 0 and 1, and the correlation 
table between parent and offspring reduces to the form 
Parent. 
Offspring. . 
Total 
3 
Total 
Verify the correlation, and work out the association coefficient Q. 
12. (Cf. the above, and also Snow, Proc. Roy. Soc., vol. Ixxxiii., B, 1910, 
Table III, p. 42.) For a similar population the correlation between 
brothers, assuming a practically infinite size of family, is 5/12. The table is 
First Brother. 
Second 
Brother. | Total. 
0 18 
2 Tl ° 
Total NRO! : 
Verify the correlation, and work out the association coefficient Q. : 
18. Referring to the notation of § 10, show that we have the following 
expressions for the regressions in a fourfold table :— 
you Nb (4B) (48) 
op (B)B) (B) (B) 
yO V5 (4B) — (eB) 
gp (4)a) (4) (a) 
Verify on the tables of questions 11 and 12. 
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