XXV] liitrnilnrti,,,, CXXV 



(a) After n = 8, interpolations for n lying between tabled values are successful. 

 if we use 8 a and occasionally &*. Neither Table XXV, nor the supplementary 

 Table XXV bis, will give satisfactory results with brief interpolations f.,r leas 

 thun H. It may even be doubted, if the argument n were tabled by (M instead of 

 I'O, whether satisfactory brief interpolation could be achieved. Although the graph* 

 of the function for constant # give very simple smooth curves, aft< r many trials no 

 short interpolation process has been yet discovered. Luckily the chief use of the 

 present tables is their application to small samples, and in such cases n is a whole- 

 number. For interpolation by the forward difference formulae, see the Note ap- 

 pended to this section (pp. cxl cxliii). 



(6) With regard to direct interpolation for x, this is feasible for x = '11 onward 

 throughout the table using 8 a , or occasionally if greater accuracy be required 8* 

 and S 4 . But from x = *00 to *10, ordinary interpolation formulae cannot be applied, 

 owing to the infinite differential coefficients appearing with the factor x~l in the 

 integral. Accordingly an auxiliary table Table XXV bis has been formed which 

 gives the function 



and also its S 2 *. This will suffice to ascertain ^ x (n) for any value of x from '00 

 to '10, and then 



P x (n) = 9 x (n)*/x + Q'b. 

 The user of Table XXV bis must therefore find the square root of the argument"!* 



with which he enters it, as the multiplier for ^ x (n). 







4. Illustrations of the use of the Tables. 



Illustration (i). The frequency curve for the distribution of the correlation 

 coefficient r in samples of size p taken from a parent population in which the cor- 

 relation is zero is given by the curve 



2\i (/' 4) 



rr . 



where the mean, r, = 0, and since a*= 1, cr= - - . What is the chance that in 



Vp-1 

 a sample of 20, 



(a) r will lie outside twice its standard deviation ? 



(b) r will lie outside the limits ^ '50 ? 



The above curve is our Type (ii), and therefore m t = $ (p 4) = 8 for this special 

 case. Now w 2 = (n 3) = 8, and accordingly n = 19. The proper transformation 



is r 2 = x. We have <r = - - = '229,4157. 



* Determined from the nine-figure B-f nnotion Table. For 5 2 c/ , (n) we used the formula 



+ x will not generally exceed four decimals. BO that any table of square roots will provide what u 

 required. 



