ON THE APPLICATION OF "GOODNESS OF FIT" TABLES 
TO TEST REGRESSION CURVES AND THEORETICAL 
CURVES USED TO DESCRIBE OBSERVATIONAL OR 
EXPERIMENTAL DATA. 
By KARL PEARSON, F.R.S. 
Let us suppose that a sample of size N with class groups n^j, is taken out cf an 
indefinitely large population of size M with class groups v^^,, these classes being 
arranged according to two variates x and y. Then the mean of any array of .t's 
for a given range of y variates connoted by the centre, y^, of this (usually small) 
range will be 
'"p = 7^ 1 • 
Here the number in the , class, and Uj,, the total number in the pth array 
of s's, will vary from sample to sample. But Xg and will remain of course the 
same. Now let mj, be the mean value of Wj, found from a large number A of 
samples and let us measure = nip + 8)Hj, from m„ and n^p from v^p = Nv^p/M, 
and Up from NvJM, or take = Tip^ + Svpg, and np^Vp + Sn.p. Here the 
differentials are statistical differences and do not at present denote that we are 
going to neglect their higher powers. From (i) we have 
+ s 
\ rip j\ np \fip) \T}pJ 
Now we shall sum this for all A samples (dividing by A) and suppose that third 
Sit S7} 
order powers and products of — and are negligible as compared with lower 
11 p >i p 
order powers and products*, li S denote a summation for all A samples 
= 0, 
S {Snip) _ S (Siigp) _ S (8n„) 
AAA 
since all these quantities are measured from their mean values. Thus we find 
* Actually terms of the third order also vanish. 1 have not investigated whether this be true for 
terms of the fourth and higher orders. 
