176 
Miscellanea 
III. On certain Points connected with scale Order in the Case of 
the Correlation of two characters which for some arrangement 
give a Linear Regression Line. 
By KAEL PEARSON, F.R.S. 
In a recent memoir on contingency*, I have considered the problem of what alterations can 
be made in scale order without sensibly modifying the value of the correlation. The problem as 
I there state it is as follows : To find tinder ivhat other condition than normal correlation small 
changes in the order of grouping will not affect the value of the correlation (p. 19). The wording 
requires some explanation. If for any arrangement of the scales of the two variables there be 
normal correlation, then my memoir shows that the method of contingency gives the value of the 
correlation, even if the order of the scales be any whatever, in fact if the normal correlation order 
be absolutely unknown. Of course, if we proceed in any such case by the usual product method 
of determining the correlation we shall reach absolutely different results when the scale order of 
grouping is largely changed. My object in .stating the above problem was to determine, if possi- 
ble, whetlier any and if so what changes in the scale orders would not sensibly modify the 
correlation, when we still endeavom-ed to determine it, not by contingency, but by the method 
of products. The conclusion I came to was as follows — that with any distribution with linear 
regression "small changes (i.e. such that the sum of their squares may be neglected as compared 
with the square of mean or standard deviation) may be made in the order of grouping without 
aftecting the correlation coefficient " (p. 35). I think this conclusion is quite sound, and deserves 
further consideration. Although in the statement of the proposition I have used the word 
"small changes" in scale order (p. 19) and in the summary of my memoir (p. 35) stated what is 
to be understood by small, in this case, I think, as Mr G. U. Yule points out to me, that the 
wording on p. 20 is too unguarded, if the reader has not been sufficiently impressed with the 
wording on p. 19, or reached the sunmiary on p. 35. It will not be without value possibly to 
give the actual algebraical result on which the statement on p. 35 is based, for it has some 
importance for the general philosophical idea of correlation. 
Let .4' and y represent the two variable characters and let uKv be the frequency of the 
character between x and x-^-hx ; vhy that of the character between y and y+hy ; u and v being 
functions of x and y respectively and the distribution of the frequencies being of any nature. 
Now suppose the array Vs^ys of frequency between y^ and ys + ^ys to be bodily interchanged in 
position with the array rg'8//s' between y^, and ys' + dyg'. Let JV be the total frequency, and 
suppose the mean y to become y + 8y, the standard deviation of the y character to become 
o-j, + S(ry. Then we have : 
^ (J/ + Sj/) = 'S' {yv8y) - v,, dy.^ {y,. - y,) - v,by, {y, -y,,) 
sj^=(y.-.y.O^^^^^^^P^^ (i), 
M{cr„ + §a,)2 = ^ {y^v8y) - *v Sys- Ly/ - y:-) - 8y, {y,^ - y,,^) -N{y + 8yf 
= Na,/ + by,, - v,8y,) {y,^ - y,:^) - 2y (y, -y,') {v,' hy^ - v,8ys), 
iV(So-;,)2 + 2iVa-y8a„ = {v,'8y,. - rs8ys) {ys-y^') OA-y+«/s' -y)- 
* "Mathematical Contributions to the Theory of Evolution, III. On the Theory of Contingency 
and its Halation to Association and Normal Correhition. " Drapers' Research Memoirs (Dulau and Co, 
London). 
