306 
Probable Error of a Correlation Coefficient 
to come too low, so that there is a deficiency of cases towards 1 and — 1. This 
introduces an error into the Standard Deviation of all the series to some extent, 
but of course the mean is unaltered when there is no correlation. The series for 
samples of 4 are affected more than those from samples of 8, as the mean Standard 
Deviation of samples of 4 is the smaller, so that the unit of grouping is compara- 
tively larger. 
The moment coefficients of the five distributions were determined, and the 
following values found*: — 
Mean 
S.D. 
Ml! 
M4 
^1 
/32 
Samples of 4 (r= 0 ) 
•5512 
•3038 
•1768 
r9i8 
Samples of 8 (r= 0 ) 
•3731 
•1392 
•0454 
2 336 
Samples of 4(r=-66) 
■5609 
•4680 
•2190 
- ^1570 
•2152 
2^245 
4^489 
Samples of 8(7'='66) 
.6139 
•2684 
•07202 
- ^02634 
•02714 
1^857 
5^232 
Samples of 30 (r=-66) 
•661 
•1001 
•01003 
- ^000882 
•000461 
•7713 
4^580 
Considering first the "no correlation" distributions I attempted to fit a Pearson 
curve to the first of them. As might be expected, the range proved limited and 
as symmetry had been assumed in calculating the moments, a Type II curve 
resulted. The equation was y = yo(^l— j^yj^ > range of which is 2^074. 
Now the real range is clearly 2, and only a very small alteration in is 
required to make the value of the index zero. Consequently the equation 
2/ = i/o (1 — oc'^y was suggested. This means an even distribution of r between 1 
and -1, with S.D. = ^5774 + •OlO vice •5512 actual, = -3333 + -0116 vice •3038, 
fji,= ^2000 ± -016 vice •1768 and ^83= 1^800 + -12 vice 1^918, all values as close as 
could perhaps be expected considering that the grouping must make both 
fi2 and /jbi too low. 
Working from y = y^(\ — x")" for samples of 4 I guessed the formula 
w - 4 
y = y„{\—w'^) 2 and proceeded to calculate the moments. 
B)^ using the transformation cc = sin 6 we get y = yo cos"""^ d, 
dx = cos fldfl, 
TT 
2 r ydx = 2?/o f ^ COS"-' edO, 
J 0 J 0 
2 ( ' af'ydw = 2y„ cos"-^ 0dB - 2y„ j" cos""' OdO, 
and SO on. 
Whence 
^^~(«-l)(7* + l)' H + l ~ n+l- 
* In the cases of no correlation the moments were taken about zero, the known centroid of the 
distribution. 
