108 BELL SYSTEM TECHNICAL JOURNAL 



Equation 6 above defines the peak of the curve, or the mode of the 

 variable. 



The arithmetic mean for a distribution agreeing with the logarith- 

 mic skew curve probably can not be defined by any mathematical 

 expression sufficiently simple for practical use. It is a function of 

 <r, but its exact position on the curve has not been determined. As 

 applied to rent data, the mean may be computed direct from a house 

 count summary with an error of two or three per cent. Thus found, 

 its position on the curve is in the neighborhood of the 65th to 70th 

 percentile. 



The geometric mean coincides with the median for a logarithmic 

 skew distribution. This follows from the fact that the median value of 

 X corresponds to the median, which is also the mean, value of log X. 



The measure of dispersion for a logarithmic skew curve is also a 

 measure of skewness. Up to this point a has been used as the measure 

 of dispersion, in agreement with conventional usage. For practical 

 purposes another measure may be substituted, which has a more 

 readily understood meaning. This is the quartile deviation, known 

 also by the misleading term probable error. The quartile deviation 

 for a logarithmic skew curve is that deviation either above or below 

 the median which includes one-fourth of all the items in the array. 

 It may, like o-, be measured in logarithms, and 



Quartile Deviation = 0.6745 a. 



Perhaps the easiest mathematical conception of a measure of skew- 

 ness and dispersion is that of the ratio of the upper quartile to the 

 median. This is identical with the ratio of the median to the lower 

 quartile, and is the number whose logarithm is the quartile deviation 

 as defined above. We shall let this ratio be denoted by Q. 



Either a or the quartile deviation for a given set of data may be 

 best determined from a straight line graph on logarithmic probability 

 paper. The following table gives the positions in the array for certain 

 convenient multiples of a and the quartile deviation, when measured 

 in logarithms. 



