632 MEMOIRS NATIONAL ACADEMY OF SCIENCES. tvouxv, 



frequencies in the lowest three or four class intervals for many of the tests, due to accidental 



scores, as pointed out above, seems to warrant cutting them off along with the zero-arrays. 



This procedure is especially valid in the case of alpha tests 3, 6, and 7. On the other hand, 



owing to the fact that the score in alpha test 4 is obtained by subtracting the number of wrong 



items from the number of right ones, accidental scores practically do not exist, and the lowest 



array only needs to be cut off. In no case have we based our constants upon less than half of 



the total population. 



In order to make the above procedure perfectly clear the arithmetic involved in applying 



the method to a sample table is here given. Take for an example the contingency table of 



alpha test 4 and alpha test 8, Table 52. It is obvious from the table that there is an enormous 



piling up of cases in the zero class interval on the alpha test 4 variable. So we eliminate this 



first array entirely and treat the remainder of the table as though it had never been there. 



We have now cut out 393 cases, leaving 654 of the 1,047, or 62.46 per cent. This 62.46 per 



cent is the truncated surface, from which we calculate p, s n , and s t . In this case alpha test 



4 is the truncated variable, so we have for s t = 4.0109 and for s n = 4.5261; these two standard 



deviations are calculated in the usual straightforward way. The calculation of the product 



13 5394 

 moment is also straightforward, and is in this case 13.5394. Thus p is ,. „,.-, " x/ ,, r ^, x = 0.7458 



(4.0109)(4.5261) 



J is calculated in the following way: In this case we are dealing with 62.46 per cent of the 



cases, i. e., -*> is 0.6246. If we turn now to Sheppard's tables ' and look up 0.6246 in the column 



headed "Permille," we find the entry in the table corresponding to 0.624 to be 0.3160. Then 

 by making the proper interpolation for our fourth decimal place we get 0.3175. This quantity 

 is negative and is "h." To find "zu," we turn over to Table II (same volume) and look up 

 0.3175 in the column headed "x," and the corresponding entry in the column headed "z" is 

 the required quantity. After making the proper interpolation for the third and fourth decimal 

 place we have "z h " equal to 0.3793327. From this point on the work is simply a matter of 

 substitution in the various formulae given above. 



By substituting -t>, h, and z h in the formula for J we have: 



. / - (0.3175) (0.3793327^ / 0.3793327 V n - MIM *o 

 J = \ 0^246 )~\ 0.6246 ) -°- 5616632 



Then by substituting J and p and p 2 in the proper formula we get r: 



°' 7458 =0.8608 



Vl + (1 - 0.5562) ( - 0.5616632) 



We therefore consider 0.8608 to be the correlation between variable alpha 4 and variable alpha 8, 

 as exhibited in our sample. It should be remembered that in applying this method we have 

 not cut off 393 cases and thrown them away, but have redistributed them in accordance with 

 the normal correlation surface. Our calculations have been made from all the cases. For the 

 relative size of p and r depends on the size of J, which in turn is large or small as we cut off 

 large or small numbers. In applying this method to the contingency tables (Tables 10 to 

 154) an effort was made in each case to remove sufficient cases to relieve the jamming (i. e., the 

 piling up of cases in the extreme arrays), yet at the same time to leave as many cases in the 

 truncated portion as possible. A large number of the tables are jammed on one side only. 

 These are simple to deal with, for all that is necessary is to cut off the array or arrays in which 

 the cases are piled up. Other tables show jamming on two sides; still others are jammed on 

 three sides. In cutting tables which exhibit jamming on two adjacent sides an effort was made 

 to remove enough cases to relieve the jam on each of the variables. In tables like the one for 

 alpha 3 and beta 7 (Table 48), which show jamming on three sides, the J method was abandoned 

 and the method of tetrachoric functions employed. This method is fully described in Pearson's 



1 Table No. I In "Tables for Statisticians and Biometrlcians," by Karl Pearson, 1914. 



