XI.—CORRELATION : MISCELLANEOUS THEOREMS, 227 
found for 36 small registration districts in which the number of births in a 
decade ranged between 1500 and 2500 :— 
Proportion of Male Births 
per 1000 of all Births. 
Decade, 
Mean Standard- 
: deviation. 
1881-1890 . : 508-1 12°80 
1891-1900 ; 5084 10-37 
Both decades 50825 1165 
It is believed, however, that a great part of the observed standard-deviation 
is due to mere ‘‘ fluctuations of sampling ” of no real significance. 
Given that the correlation between the proportions of male births in a 
district in the two decades is+ 036, estimate (1) the true standard-deviation 
freed from such fluctuations of sampling ; (2) the standard-deviation of fluctua- 
tions of sampling, 4.e, of the errors produced by such fluctuations in the observed 
proportions of male births, 
4. (Data from Pearson, ref. 11.) The coefficients of variation for breadth, 
height, and length of certain skulls are 3°89, 3:50, and 324 per cent. respec- 
tively. Find the ‘spurious correlation ” between the breadth/length and 
height/length indices, absolute measures being combined at random so that 
they are uncorrelated. 
5. (Data from Boas, communicated to Pearson : ¢f. Fawcett and Pearson, 
Proc. Roy. Soc., vol. Ixii. p. 413.) From short series of measurements on 
American Indians the mean coefficient of correlation found between father and 
son, and father and daughter, for cephalic index, is 0°14 ; between mother and 
son, and mother and daughter 0-33. Assuming these coefficients should be 
the same if it were not for the looseness of family relations, find the proportion 
of children not due to the reputed father. 
6. Find the correlation between X, + X; and X,+ X;; X,, X, and X, being 
uncorrelated. 
7. Find the correlation between X, and aX, +bX, X, and X, being 
uncorrelated. 
8. (Referring to illustration iv., § 14, Chap. X.) Use the answer to 
question 7 to estimate, very roughly, the correlation that would be found 
between annual movements in infantile and general mortality if the mortality 
of those under and over 1 year of age were uncorrelated. Note that— 
pesos) brs } =infantile mortality per 1000 births x a 
+ deaths over one year per 1000 of population. 
and treat the ratio of births to population as if it were constant at a rough 
average value, say 0°033. The standard-deviation of annual movements in 
infantile mortality is (loc. cit.) 9'6, and that of annual movements in mortality 
other than infantile may be taken as sensibly the same as that of general 
mortality, or say 1 unit. 
9. If the relation 
az) +b.ag+ecxg=0