130 
On Polycho7'ic Coefficients of Correlation 
(2) The developments we require involve the use of the tetrachoric functions. 
The tetrachoric function of the order t is given by* 
Tt = - J- -7= e ^■'^ (m). 
The tetrachoric functions t, to T,j are tabled for positive values of x in Tables 
for Statisticians and Biometricians'f to five decimal places. For negative values of x 
tetrachoric functions of an odd order remain unchanged, but those of an even order 
must have their sign as given in the tables reversed. 
It will frequently be needful to take the difference of the tetrachoric functions 
at the boundaries of a marginal category. Thus if Tt {h) denotes the value of the 
tetrachoric function for x = h, we shall need for the sth marginal total 
rt (hs) - Tt 
This difference we shall write, for brevity, 
and in obtaining its numerical value from tables of the tetrachoric functions it is 
essential to remember that s (or s') is supposed to increase in the positive direction 
of the axis of x (or y), and that when h (or k) is negative attention must be paid 
to changing the sign of the tabled value of Tj, if i be even. 
The formula for determining the successive tetrachoric functions for a given 
value of X is 
Tt - xpt Tt-i - qt Tt_2 (iv), 
where pt and qt are given hy the following table : 
t 
Pt 
It 
* 
Pt 
Qt 
2 
•707,1068 
•000,0000 
14 
•267,2612 
-889,4990 
3 
■577,.3503 
•408,2483 
15 
•258,1989 
-897,0851 
4 
-.500,0000 
•577,3503 
16 
•250,0000 
-903,6962 
5 
•447,21,36 
•670,8204 
17 
•242,5356 
-.909,5085 + 
6 
•408,2483 
•730,2968 
18 
•235,7023 
•914,6592 
7 
•377,9645 
•771,5168 
19 
•229,4157 
•919,2547 
8 
•353,5534 
•801,7838 
20 
•223,6068 
■923,3804 
9 
•333,3333 
•824,9578 
21 
•218,2179 
•927,1051 
10 
•316,2278 
-843,2740 
22 
•213,2007 
•930,4842 
11 
-.301,5113 
•858,1163 
23 
•208,5144 
•933,5637 
12 
•288,6751 
•870,3880 
24 
•204,1241 
•936,3819 
13 
-277,3.501 
•880,7047 
25 
•200,0000 
•938,9709 
Since Ti = e it can be found at once from the tables for the ordinates 
\/27r 
of the normal curve, and will indeed have been computed at each division in order 
* The reasons why the tetrachoric functions are tabled with the factor l/Vf ! are: (a) because this 
factor greatly simplifies our formulae and (/;) because a factor of some such order is essential, if we are 
to have manageable tabulated values. As a matter of fact the factor chosen reduces all tetrachoric 
functions to numerical values lying between 0 and 1. 
+ Cambridge University Press, pp. 42 — 51. 
