246 BEroRT— 1881. 



On the other hand, the Median, Decile, and Qaai'tile values possess a 

 trustworthiness of the same order as that of the Average or Ai'ithmetic 

 Mean values. They are not sensibly affected by a solitary accident, and 

 a moderately large series of observations is sufficient to determine them 

 with as much precision as is needful for ordinary statistical purposes. 



A small error in the position of the medians, quartiles, &c., causes an 

 error in their values proportional to the tangent of the circumscribing curve 

 at the corresponding points. On protracting the curves for height, weight, 

 and strength from their tabular values, it appears that the tangents at 

 the quartiles are but little greater than those at the medians, but that the 

 tangents at the deciles are about twice as great. Again, the tangents at 

 corresponding points in two of these curves, drawn from different num- 

 bers of observations (the ordinates relating to the successive values being 

 supposed in all cases to stand at the same distances apart), must vary 

 inversely as the number of observations. Consequently, in order to as- 

 certain decile values in the series with which we are now dealing, with 

 the same accuracy as medians and quartiles, we require to have about 

 twice the number of observations. 



It appears to be well worth while to print, not only summary tables of 

 results, as Table VIII. for the height, and Table IX. for the weight, but 

 to supplement these by other tables going more into detail and referring 

 to the classes separately. So much has been written on the applicability 

 of the Exponential Law of Error to statistical results, that it is important 

 to publish material for the more complete discussion of the subject. Into 

 the discussion itself, this is hardly the place to enter, further than by 

 saying that the median values will be found to conform very closely in- 

 deed with the arithmetical means, that the distribution of variations on 

 either side of the median value is so symmetrical that the difference be- 

 tween either quartile or decile and the median is almost exactly one-half 

 of the difference between the two quartiles or the two deciles, and, lastly, 

 that the range between the two deciles is very commonly a trifle short 

 of double the range between the two quartiles. According to the Ex- 

 ponential Law of Error, the results in every case would have been nearly 

 the same as these. 



I would refer those who desire to pursue the subject on a theoretical 

 basis to a paper of my own on the ' Geometric Mean in Vital and Social 

 Statistics,' in the Proceedings of the Royal Society, October, 1879, and 

 more especially to the subsequent one by Dr. Donald McAlister on the 

 ' Law of the Geometric Mean,' in which the equation is given to the cir- 

 cumscribing curve, both on the assumption of the arithmetical mean of 

 two fallible observations of the same fact being the most probable inference 

 from them, and on that of the Geometric Mean being accepted, as I have 

 argued that it ought to be, as the more probable inference in all physiolo- 

 gical phenomena. 



On the Calculation of Deciles, Quartiles, and Medians. 



The deciles, quartiles, and medians are ordinates to an ideal curve, 

 supposed to be constructed as follows : — An infinite number of measure- 

 ments, belonging to the same statistical group, are arranged in the order 

 of their magnitudes, and ordinates of lengths corresponding I'espectively 

 to each of them are erected side by side, at equal, but infinitesimally 

 small, distances apart, along a given line AB ; then the curve passing 

 through their tops is the curve in question. The median is the ordinate 

 corresponding to the abscissa of i'AB ; the lower and upper quai'tiles 



