Miscellanea 
437 
II. The Frequency Constants of a Variable z=f{xi, x^)*. 
By RAYMOND PEARL, Ph.D. 
It often happens in practical biometric work that one desires to find the frequency constants 
of a compound character, from a previous knowledge of the constants of the separate components. 
Thus, for example, one measures the length, the breadth, and the height of each of a series of 
skulls. He wishes to know at least the mean and the standard deviation of the diametral product 
{LxBx H). There are two ways open to find the values of these constants. On the one hand 
the length, breadth, and height may be multiplied together for each individual skull, a frequency 
distribution of the products made, and the constants calculated in the ordinary way ; or, on the 
other hand, by the use of the appropriate formulae one can deduce straight ofi' the constants for 
the product knowing those for the components which enter into the product. The latter procedure 
will obviously eflfect a great saving of labour. 
The formulae for determining the mean and standard deviation of a character 5=/ (.Tj, .^2) 
when the same constants and the coefficient of correlation for and x., are known, are well known 
to mathematicians. The writer ventures to think, however, that they are not so familiar to 
many of those who have approached the field of biometry along the biological pathway. For 
the convenience of such it seems desirable to publish all together those formulae of the sort under 
discussion which are most likely to be used in practical work. 
The general method of deducing these formulae will be clear to anyone who will carefully 
study Pearson's paper "On a Form of Spurious Correlation which may arise when Indices are 
used in the Measurement of Organs t," wherein the formulae for 2=— are discussed. The general 
formulae for z=f{x, y) will also be found discussed in the Phil. Trans. Vol. 187 A, p. 278, 1896. 
In the formulae given in Table I the various letters have the following meanings : 
Xy , X2 and x^ the separate characters involved in the compound character z. 
wii , m<i and the means of the characters x\ , X2 and .^3 . 
o"! , 0-2 and 0-3 the standard deviations of .t\ , x<i and .r^ . 
V, = v.i = — , = (The y's are the ordinary coefficients of variation divided 
by 100.) 
r denotes the coefficient of correlation between the two characters designated by the 
subscripts. 
The table gives the formulae for the mean and standard deviation of 
(a) the sum of two and of three variables, 
{b) the difference of two variables, 
(c) the product of two and of three variables, 
{d) the quotient of two variables (index). 
In certain of the cases the formulae are approximations, but very close ones. The nature of 
the appi'oximations made is indicated in the table. 
* Papers from the Biological Laboratory of the Maine Agricultural Experiment Station, No. 4. 
t Pioc. Roy. Soc. Vol. 60, pp. 489—498, 1897. 
