42G 
Miscellanea 
Then, taken between appropriate limits, 
^ydx= J?/(,e-'-. . ff« = jyad-^HBi + 2B,t + SBst- + ...)dt (4). 
This method is most useful where the range of t is either - oo to + oo or 0 to + oo . The first case 
gives the "whole" area N ; the second gives N^, the area from the mode to the upper limit of x. 
Therefore 
N = f^_y»^~" (^1+ 3^3'' + •••) = Uo^'^ (B, + iB^ + ...) (5). 
N,„ = IN + \ yi,e-'- (2B4 + -iB^fi + ...) di = JA' + yo (^j + 2 ! £4 + 3 ! So + ...) (6). 
Hence the area from the mode to the median is 
y^{B^ + 21 B, + 31 Bs+ ...) (7)- 
The evaluation of B^, B.,, ... we shall perform by means of the method of indeterminate coefficients, 
using (3) and , ; . 
i-t'''^=' 
which is obtained by logarithmic differentiation of (2) with respect to i. De Morgan* gives the 
values of the first five coefficients in terms of the differential coefficients at the mode of log (/> {x). 
It may be shown that when <^ {x) contains high exponents or a large number of factors the 
coefficients Sj, jB,, Bg, ... are of decreasing magnitude and the series (5) and (7) are convergent 
enough for use. This is generally the case with frequency curves. When the skewness is small 
the coefficients B<^, B^, ... must be small; in this case, then, if d be the abscissa of the median we 
have the area from the mode to the median approximately equal to ygd, and usually we may 
neglect B^, Bg, ... in comparison with B.,. Hence approximately 
d = B^ (9). 
If we let D be the distance of the mean from the mode we have 
i) = ^ ["'^ xydx = ^ y^e-" 1- {2C\t + 3C,fi + ...) dt (10), 
where C,. is the coefficient of f in the expansion of x- in powers of i. Then from (5) we have 
' D = ^,fn(tC,i-...)=i^^±-^ (11). 
Therefore to a first approximation D = ^B2 = '.Id (12). 
This is the relation that has been noticed in practical statistical work and it holds under the 
conditions stated. 
§ 3. The relation just found is quite general provided that the frequency curves are moderately 
asymmetrical. It would, of course, be possible to give general formulae for the more accurate 
representation of the relation between the distances of the mean and median from the mode in 
the very general type of curve already used. But in practice nearly all the types of frequency 
curves are included in Professor Pearson's system and it is for these curves that we shall now 
obtain a closer approximation to the true value of d/D. In the memoir already quoted Professor 
Pearson takes the differential equation of frequency curves in the form 
^ ., (13), 
y dx «(, + a^x + a.^x^ 
the origin of x being at the mode. 
If we write y = e""'- we obtain, as in (8), 
dx- 
+ ^ta.^x + U (Oo + a^x) = 0 (14), 
and writing x- = C^^ + Cy^ + (15), 
* Differential und Integral Calculus, p. 602. 
