286 



REPORT 1887. 



the mmimuin shows the lopsided character of the price-curves. The median has 

 been iised instead of the arithmetic mean only for convenience of calculation. 

 Much the same conclusions would evidently have followed from the use of the 

 arithmetic mean, as the writer has verified for the years 1801, 1821, 1831, 1851. 

 The figures in each row overlined and underlined respectively are the pene- 

 maximum and penemiuimum. If we compare the distances between each of 

 these and the median the series of signs is found to become 0+ + + — h + + +. 



The exceptional year is 1821. If we examine the arithmetic mean for that year 

 the exception still exists, but in a less marked degree. 



Such a curve is well represented by the equation t/= —- e~''"' "°s ■'''', where h is 



a constant corresponding to the dispersion, or ecai-t, of the curve (see Dr. Macalister's 

 paper On the Lcnv of the Geometric Mean (' Proe. Roy. Soc' 1879), and compare the 

 present writer's Observations and Statistics ('Cam. Phil. Trans. ' p. 149). Hence, 

 given a number of observations de'sdating from the mean about which they are 

 grouped, each according to a law of the general form above stated, the most prob- 

 able value of the mean deduced from these observations wiU be the toeighted 

 geometric mean given by the equation 



° " Aj + /*„ + &c. + h 



where x is the sought mean, x\, x„, &c., are the given observations, and h^, h.-,, &c., 

 are the weights, of which more hereafter. 



It must be remembered, however, that there maj^ be other means adapted to 

 represent the bias which has been observed, in particular what may be called the 

 unsymmetrical probability-curve, elsewhere described by the present writer {Land. 

 Phil. Mag. April 1886). Nor, again, is it to be supposed that all statistics of prices 

 are grouped unsymmetrically. Where the entries are average prices based on a 

 great number of items it is agreeable both to ' theory and the writer's observations 

 that the normal symmetrical ' probability '-curve will set in. It will be found 



' See in Methods of Statistics the statement of the ioro2Dosition that the average of 

 a large number of returns obeying individually any law of grouping tends to conform 

 to the Probability-curve. 



