272 On 'Best' Values of Constants in Frequency Distributions 
where m, is the mean of the fth array of the sample of size M from a population 
of size N , while Wi^ is the theoretical mean as found from the regression curve, 
J = ^vM-jN , is the mean frequency and a,-^^ the mean standard deviation of the 
^th array in the samples. The difficulty in applying the 'goodness of fit' test 
lies in finding adequate values for and a^^. Let us assume them to be found. 
The 'best' values of the constants fx, f^, ■■■ of the regression curve, i.e. the values 
which make a minimum, will then be found from equations of the type 
0 = - 2,S 1^ {m,, - m,) -^-^| + 8 \{>n.,- mj^-^j (H). 
As will be seen these equations fall into the equations resulting from using 
11 . . 
the method of least squares if is independent of the constants of the regression 
curve and at the same time for the different arrays proportional to the of the 
sample. Even if our sample be derived from truly Gaussian data, these conditions 
will only approximately be satisfied, the ct^^, although constant, being dependent 
upon the constants of the regression curve and the of the formula not being 
really the sample value. 
71 
Supposing to be independent of the constants of the regression line 
mj, = ax+ b, the equations (11) take the form 
S {Vj, {nij, — ax — b) x} = 0, 
S {v,^ {m.jj — ax — b)} = 0, 
when we put Vp for 
S 
From these equations we find 
S {VpmpX) . S (Vp) — S {Vpnij) S (x) 
a = 
S {VpX^) . S {Vp) - {S {VpX)Y 
^ , S ivp?n.^) S (VpX) 
and 0 = — r a — r- , 
formulae agreeing with those derived from the method of least squares if equals 
the marginal frequencies of the sample. But not agreeing with them if, for example, 
the material be heteroscedastic. 
(7) Illustration VI. Auricular Height of School Girls. 
This example was first used by Pearson in the memoir on skew correlation* 
and later as an illustration of the test of 'goodness of fit' of regression curves f. 
* Drapers^ Company Research Memoirs, Biometric Series ii. p. 34. 
t Biomeirika, Vol. xi. p. 253. 
