342 Distribution of Correlation Coefficient in Small Samples 
In order to make use of our ordinates for graduating frequency curves we must 
express the distance from our origin to our mean (i.e. from r = 0 to r = r„) in 
terms of the standard deviation, and further the unit of argument of the abscissae, 
i.e. -05 in r, also in terms of the standard deviation. Our interpolated frequency 
ordinates (reduced of course, to the size of the actual population) will then have 
to be plotted to intervals of -OSo-g/a,. , the origin being r^a^/a,. from the mean of 
the graduated data, where o„ is the standard deviation of the graduated data. 
Care must be taken to so choose the axis of abscissae of the graduated data that 
the sign of is the same in the graduated material and the graduating frequencies. 
Table C gives the distance from the mean to the origin of coordinates in each 
case and also the abscissal unit for plotting both in terms of the standard deviation. 
(4) On tlie Determination of the Mode. Differentiating (iv) we have 
7„ w-6 
f = I (n - 3) ! r " ^ ' dWr"^ - ' - - - dJrW- 
Hence the mode r is given by 
/ d"-'^TJ \ d^-m 
\d {;rpY-y ^ d {rpY 
or writing == rp, we have 
fln-\ If JJ 
ip'-p")d{^--^=^'''''^^p"iWr'- 
where {7 is t/ with p put for rp. 
Now (xxxv) is by no means easy to solve adequately, for if we solve it by 
approximation, r = p and p = p^ is not sufficiently close for an effective first 
approximation, especially when p differs considerably from zero. We have indeed 
from (v) the relation 
d^-^l7 d'^-'^ty d'^^-^lJ 
(1 - p') - ^' ~ iWr"' ~ ^ m^' ^ " 
and this might be combined with (xxxv) to deduce in succession relations between 
lower pairs of differential coefficients, till we ultimately reach a relation between 
d tfjdp and U, but the process is too laborious except for very low values of n. 
Fisher has outhned another method of approaching the mode*. It is easy to 
see that 
cos-i {-^) _ 2 / 1- ^ 
Vl - a;2 Vl 
r' ftan-i — — tan-i , ^ \ 
-x^\ Vl- Vl - xV 
2 
tan~^ 
$ — X 
Vl — X 
0 
_ jl dj j-l d$_ 
" hii - a;)2 + 1 - x^ ~ J 0 ^2 - 2x$ + 1 ' 
with a scries of Pearson-curves lias been long nnder consideration, but the immense labour of calculating 
the ordinates of 400 to 500 curves has so far prevented the aotualisation of this idea. The present 
ordinate-tables go some waj' to supply the need fxalton pointed out. 
* Biornctrika, Vol. x. p. 520. 
