XIL.—PARTIAL CORRELATION. 
Illustrative Applications of Economic Interest, 
(9) YULE, G. U., “ An Investigation into the Causes of Changes in Pauperism in England, 
ete.,” Jour. Roy. Stat. Soc., vol. Ixii., 1899, p. 249. 
(10) HOOKER, R. H., ‘The Correlation of the Weather and the Crops,” Jour. Roy. Stat. 
Soc., vol. Ixx., 1907, p. 1. 
(11) SNOW, E. C.. “The Application of the Method of Multiple Correlation to the Estima- 
tion of Post-censal Populations,” Jour. Roy. Stat. Soc., vol. 1xxiv., 1911, p. 575. 
EXERCISES. 
1. (Ref. 10.) The following means, standard-deviations, and correlations are 
found for 
X, =seed-hay crops in cwts. per acre, 
X, = spring rainfall in inches, 
X;=accumulated temperature above 42° F. in spring, 
in a certain district of England during 20 years. 
M,=2802 op =4"42 T19= +080 
M,= 4-91 a,=1"10 ri3= —-040 
M;=594 o3=385 Tog = -0-56 
Find the partial correlations and the regression-equation for hay-crop on spring 
rainfall and accumulated temperature, 
2. (The following figures must be taken as an illustration only : the data 
on which they were based do not refer to uniform times or areas.) 
X, =deaths of infants under 1 year per 1000 births in same year (infantile 
mortality). 
X,=proportion per thousand of married women occupied for gain, 
Xy=death-rate of persons over 5 years of age per 10,000. 
X,=proportion per thousand of population living 2 or more to a room 
(overcrowding). 
Taking the figures below for 30 urban areas in England and Wales, find the 
partial correlations and the regression-equation for infantile mortality on the 
other factors. 
M, =164 n= 20°0 T= + 0°49 Tos= +015 
My= 158 09= 74 ‘9 T3= +78 Toy= -037 
3=M3  gy= ld ry =4020 r= +023 
1=205 os=1300 
3. If all the correlations of order zero are equal, say =r, what are the values 
of the partial correlations of successive orders ? 
Under the same condition, what is the limiting value of » if all the equal 
correlations are negative and n variables have been observed ? 
4. What is the correlation between z, ,and I 
5. Write down from inspection the values of the partial correlations for the 
three variables 
x), Xy, and Xy=0a.X, +0. X,. 
Check the answer to Qu. 7, Chap. XI, by working out the partial 
correlations. 
6. If the relation 
az, +b.xy+exy=0 
holds for all sets of values of z;, ,, and x, what must the partial correlations 
be? 
Check the answer to Qu. 9, Chap. XI, by working out the partial 
correlations. 
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