\vi xvil] 



The mean uri^li! (' the " Unsatisfactory " health class is 



y,= 17-027,473 Ibs. 



Accordingly ^ " ^ = -601,0054, 



<Ty 



and, if rt, be the biserial coefficient of correlation, 



r b = ^ "^ x * (1 " g) = -C01.0G54 x -890,0825 

 ** 



= -5350. 



We now turn to Table XVI and interpolate for (1 -a) = -.317,073 linearly 

 will suffice in order to find \i, X 2 2 and X 3 . We obtain : 



Xj = -0008, X 2 2 = 1 -7065, X 3 = 2-6024. 

 Applying first formula (iii) to correct r b we have 



r b = -5350/(1 + -fa (-0008 + -1431)) 



= -5349, 



or, we see that the correction is not in this case worth making, the total of 574 

 observations rendering it insignificant. 



Next the probable error of r b is given by (iv), or: 



P.E. of r b = ' {1-7065 - 2-6024 (-5350) 2 + (-5350) 4 )* 



= -02815 x 1-021,544 = -0288, 

 so that our answer is r b '5349 '0288. 



The probable error of a correlation coefficient of intensity "5349, obtained from 

 a full table by the product-moment method, would be "0204, so that there is an 

 increase in inaccuracy of about 41 / , which would be greater if \(\a) were 

 smaller. 



TABLE XVII. 



Distribution of Standard Deviations of Small Samples drawn from an Univariate 

 Normal Population. (Biometrika, Vol. x. pp. 522 529, and Vol. XL pp. 277 280.) 



Let <r be the standard deviation in the sampled or parent population, n the size 

 of the samples, and 2 the standard deviation in any one sample. Then we consider 

 the curve of frequency of 2 in M samples 



- 2 



= 



2 



2 taking values from to oo . 



This is a skew curve, only approaching a normal distribution as n increases in- 

 definitely. It is desirable to know the deviations from normality in the distributions 

 of 2 for small samples. Let 2 equal the mean, 2 the modal S, er 2 the standard 

 deviation of 2. Then it is very usual to make use of the limiting values (;i-*-oc ) 



