clxxxviif Tables for Statisticians and Biometricians [XXXVIII XLI 



curves. Here q may be given any value from slightly over zero to slightly under 



dii 

 unity. For with this condition -^- will become infinite when x = 0, but the area Z 



of the auxiliary curve remain finite. Pairman and Pearson, seeing the difficulty of 

 determining q, took its value equal to the mean of its possible range, i.e. = |, but 

 they realised that q was actually a measure of the "intensity of asymptoting," and 

 would vary from one frequency curve to a second. 



The reader must be warned straightway of the difficulty of the subject. In 

 practical statistics we are given the frequencies on certain subranges, say %, r? 2 , 

 w 3 , n&, n& and n^ on the first six subranges supposed equal. We cannot, however, 

 assert, if these quantities be in order of magnitude, n : being the greatest, that the 

 curve is asymptotic at the terminal. They may be "exponential," or they may rise 

 with extreme abruptness from the base, so that % is the greatest frequency. Or 

 again they may be asymptotic although n is not the greatest frequency, because 

 the asymptote may spring from the base at a point within the first subrange. There 

 are not a few cases in which we do not know where the range of the frequency curve 

 starts, and this must be determined from the data themselves. But it is just in 

 such cases that the abruptness-coefficients make the most substantial changes in 

 the crude moment-coefficients, and until we know the true moments we are unable 

 to determine whether the curve be asymptotic or non-asymptotic. We thus move 

 in a vicious circle, and may have to proceed by trial -and -error processes. No method 

 of calculating abruptness-coefficients has yet been devised, which leaves the start of 

 the frequency curve, and therefore the start of the auxiliary curve, one of the factors 

 to be determined. We assume a knowledge of the start of the curve be it asym- 

 ptotic or abrupt, but this never flows from the subrange frequencies themselves. For 

 example, when we start a curve of house rentals with the first group 10, it 

 is clear that no cottage is really rented as ; if it were it would be because it 

 would be given as part wages. The least rental value before the War might be 2s. Qd. 

 weekly or 6 a year. Hence, if the distribution be asymptotic, the asymptote does 

 not lie at the start of the first subrange 10, but inside it, at a point not con- 

 veyed by the statistical data, but to be obtained from other considerations. Thus 

 in dealing with infantile mortality the first year's frequency may be greater than 

 that of any later year, but to obtain a reasonable fit we may have to take prenatal 

 deaths into consideration, and put the start of the curve even in the early days of 

 pregnancy. 



Other considerations also arise, when we wish, for instance, to fit continuous 

 curves to really discrete data, or to data given in discrete form. For example, if 

 we are told that a buttercup may have 5 to 11 petals, there is no great forcing of 

 the data if we suppose the subrange of 5 to be a block of frequency on the base 4'5 

 to 5'5, and place the asymptote at 4'5. On the other hand, if we are given data 

 for "cloudiness" in the 11 groups 0, 1, 2, ... 9, 10, which signify that none of the 

 heavens y 1 ^, T 2 <y, ... T 9 <y, and the whole sky was covered by clouds at a certain hour 



