THE GEOMETRY OF THE TRANSLATED NORMAL 



CURVE. 



Carl J. West, Ph. D. 



Inixcduct.icn. In curve tracing the graphic representation 

 is constructed from the equation. Pue largely to the require- 

 ments of statistics the converse, namely, to find the equation of 

 the curve when the distribution of points is given, has become 

 of interest. This problem is very different from the exercises of 

 analytical geometry in which a given law of distribution of 

 points is to be translated into algebraic language. For the 

 presence in the statistical data of accidental irregularities makes 

 it undesirable as well as practically impossible to obtain a curve 

 passing through the points. Instead, a curve is "fitted" to the 

 points, that is, a curve is passed among the points in accordance 

 with some generally accepted principal such as that of least 

 squares or the agreement of moments. 



Aside from the straight line and the parabolas, the curves 

 proposed by Pearson"^' have found acceptance. In order to 

 derive curves which can be fitted to widely varying distributions 

 of points. Professor F. Y. Edgeworthj has proposed to modify, 

 to translate, the normal probability curve with unit standard 



deviation, 



t= 

 1 ~^r 

 y = / — e 



\/27r 



In this article we shall discuss the geometry of the curves 

 which Edgeworth obtains by this transformation and derive a 

 method for an approximate solution of the two equations, one 

 of the fourth and the other of the sixth degree, which arise in 

 the fitting of a curve of this class. 



* Pearson, Karl:^"Skew Variation in Homogeneous Material;" Phil. Trans. 

 1895, Vol. CLXXXVI, A, pp. 253 et seq. 



"On the Systematic Fitting of Curves to Observations and Measurements," 

 Biometrika, I, pp. 265 et seq. and Biometrika II, pp. 1 et seq. 



Elderton: — "Frequency Curves and Correlation," pp. 1-105; C. & E. Layton, 

 1906. 



t Edgerton, F. Y.': — "On the Representation of .Statistics by Means of Analyti- 

 cal Geometry," Jour. Roy. Stat. Soc, 1914, Feb., Mar., May, June and July. 



6o 



