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THE AMERICAN NATURALIST [Vol. XL VIII 



Pearson's theory to cover the class of curves, formerly not 

 amenable to such test. The result forms an extremely valuable 

 extension of biometric theory. 



Briefly Slutsky's essential result may be put as follows. He 

 finds (the complete proof is not given in this paper) that 



where x 2 is the quantity denoted by the same letter m Pearson's 

 original work, and is the argument in Elderton's table; n, p is 

 the frequency in the x p array, i. e., the number of observations 

 on which each observed ordinate is based; e p is the difference 

 between the observed and the calculated mean y for each x p 

 array; and a, lr is the standard deviation of each x p array; i. e., 

 the standard deviation of the group of observations from which 

 each particular y was calculated. 8, as usual, denotes summa- 

 tion. Knowing x 2 , P is read directly from Elderton's tables. 



Slutsky gives a couple of examples of the application of the 

 method in his paper. For illustration here I have preferred to 

 take an example from my own unpublished data. The observa- 

 tions (y. Xp ) in this case are the mean butter productions of 

 American Jersey cattle, based on seven-day tests. 5 



The theoretical points Y. rp are calculated from the equation, 

 y = 14.21098 + .0250a; — .0038a; 2 + 3.0104 log x, 

 the constants of which were determined from the observations 

 by the method of least squares. 



The test for goodness of fit is carried out in Table I. It should 

 be said that, following the suggestion given by Slutsky in his 

 paper, I have used in the <r nx column the graduated rather than 

 the observed values. In the present case the scedastic curve is 

 hopelessly far from a straight line. It is, in point of fact, 

 logarithmic. 



From this table we have x 2 = 32.115. This is beyond the range 

 of Elderton's table. By a rough, but sufficientlv accurate, graph- 

 ical extrapolation, I find for present values of n' and x 2 > 

 P = .417 about. 



In other words, if the butter production of Jersey cows changes 

 with age according to the curve given, we should expect to 



6 For data see "Jersey Sires and Their Tested Daughters," published by 

 American Jersey Cattle Club, New York, 1909. 



