H. E. SopER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 373 
deviation from normality for high values of p in the sampled population, and 
the usual "probable error of r" must be treated with caution in these cases. 
Pearson's curves would give better results, as is indicated by the agreement of 
r to five figures. 
(11) Table for determining the Mode r of the Frequency Distribution of con- 
siderable Size n when the Correlation in the sampled Population is known to be p. 
The required value of the mode is 
- = 4- -1 ^'"^ -4- 4- ^3 4 "4 
' ^ ^ w - i ^ - 1)2 ^ (n - 1)3 ^ (n - 1)4 ' 
when p is positive ; if p be negative r has the same value as for p positive, but with 
opposite sign. 
TABLE VIII. Functions required in determining the Mode 
of a large or fairly large Sample. 
p 
"1 (p) 
(P) 
J-:! (P) 
"i (p) 
•00 
0 
0 
0 
0 
•05 
■12468,75000 
+ ^37872,26953 
+ 1-1.3249,90199 
+ 3^3691.3,85919 
•10 
■24750,00000 
+ •74237,62.500 
+ 2-18002,51688 
+ 6-33154,83878 
■15 
■36656,25000 
+ 1-07636,49609 
+ 3-06395,34936 
+ 8^53428,44284 
•20 
■48000,00000 
+ 1-36704,00000 
+ 3-71804,16000 
+ 9-72192,80640 
•25 
•58593,75000 
+ r60217,28516 
+ 4-09383,77380 
+ 9-77504,47690 
•30 
•68250,00000 
+ 1-77142,87500 
+ 4-16514,90563 
+ 8-72821,01391 
•35 
•76781,25000 
+ r86684,01172 
+ 3-9312.5,67461 
+ 6-7704.5,11170 
•40 
•84000,00000 
+ 1-88328,00000 
+ 3-418.56,48000 
+ 4-22414,17680 
•45 
•89718,75000 
+ 1-81893,55078 
+ 2-68036,91046 
+ l-50278,.32653 
•50 
•93750,00000 
+ 1-67578,12.500 
+ 1-79443,35938 
- ^94981,38428 
•55 
•95906,25000 
+ 1-4600.5,27734 
+ -85806,01815 
- 2-73364,05345 
•60 
•96000,00000 
+ 1-18272,00000 
- -01966,08000 
-3-57140,42880 
•65 
•93843,75000 
+ -85996,06641 
- -72873,29366 
-3-38017,64160 
■70 
•89250,00000 
+ -51363,37500 
-1-17258,21188 
-2-31971,93665 
•75 
•82031,25000 
+ -17175,29297 
- 1-28614,42566 
- -79706,28440 
■80 
•72000,00000 
- -13104,00000 
- 1-05802,56000 
+ -59787,92160 
•85 
•58968,75000 
- -35300,16797 
- •.5.5704,89418 
+ 1-25603,36730 
•90 
•42750,00000 
- -44481,37500 
+ ^03650, 10188 
+ -84819,10191 
•95 
•23156,25000 
- ^34910,94141 
+ ^39461,98756 
- -19874,71356 
1-00 
0 
0 
0 
0 
The above Table will give the value of r correctly to about the sixth figure if 
n = 100 or more, to about the fourth figure if w = 25 or more. Below 25 it can 
only serve as a "taking off point" for more accurate approximations, and these 
are fairly troublesome if n be very low. It will be found best to interpolate for 
the expression to be added to p. 
Illustration. To find the modal /" for samples of 9 when p = -2852. 
r = p + -106,272, for p = -25, 
r = p + -121,126, for p = -30. 
