106 BELL SYSTEM TECHNICAL JOURNAL 



Substituting log X for x, a new equation may be obtained, in which 

 y is the frequency per unit of x (or log X). An expression for the 

 frequency per unit of X is desired, which may be called F. It may be 

 shown that the desired equation is 



1 (IogX)2 



Y = —= e - -^^ . (2) 



This is the equation of what we have called above the logarithmic 

 skew curve, which is really not a curve of error in the same sense as 

 equation (1) is. 



In the course of this discussion it will be convenient to refer to 

 certain features of the frequency curves by the accustomed terminol- 

 ogy of statistics. The median item of a group is such that one-half 

 of all the items are larger, and one-half are smaller, and is the central 

 item when they are arrayed in order of size. The quartiles, upper and 

 lower, together with the median, divide the array into four parts, 

 each containing one-fourth of the items. The percentiles divide the 

 array into 100 equal parts. The mode is that value of the variable 

 which is of most frequent occurrence. 



In the normal curve of error a, the standard deviation, is tech- 

 nically defined as the square root of the mean of the squares of the 

 deviations of the items from their mean. For present purposes it 

 may be regarded as a measure of dispersion approximately equal to 

 the difference between the values of x at the 84th percentile and the 

 median. In the logarithmic skew curve a is the difference between 

 the corresponding logarithms. 



The origin of x in the normal curve of error is the arithmetic mean, 

 median, or mode, which are coincident. When a logarithmic scale 

 of abscissas is introduced, the median value of x (or log X) corre- 

 sponds to the median value of X, which is smaller than the mean 

 value of X, and larger than the mode. In a logarithmic skew curve 

 the median may be considered the origin, and at this point x (or 

 log X) is equal to zero, and X is equal to unity. When this curve 

 is applied to house rents the median rent occurs at this point. The 

 relation between rents and values of X is a simple one. If rents be 

 denoted by R, and if M be the median rent, then 



R = MX. (3) 



The relationship of the various scales is presented in Fig. 9. The 

 scales for X and log X may be considered fixed, and the scale for 



