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261 
probable error of the difference, or the odds are only about 3 to 1 against the 84 group actually 
coming out larger than the 93 group. In other words in every four random samples of this 
material we should find on the average the mode in one case (if judged by inspection) to fall in 
the 84 and not the 93 element ! Thus large frequencies about the modal value are subject 
to large absolute probable errors, and unless we have investigated the probable error of the 
differences of such frequencies, we shall have no security for even having found the elementary 
range within which the mode really lies. In fact, if we want to find the mode satisfactorily we 
must take into account the frequency in elements lying outside the groups 52 and 84. Five 
groups will be better than three, seven than five, and so on. But where are we to stop ? 
Clearly it will be best to take all the component frequencies, or the mode can only be found 
with the maxivium of accuracy when we deduce it like the mean from the whole series of obser- 
vations. It will rarely if ever lie at the mid-point of the group of apparent maximum frequency 
of observation, and very often will lie outside the range of this group altogether. It is quite 
fallacious to supjiose it a constant of the distribution easily determined by inspection. To 
discover it involves some theory of the nature of the distribution of the frecpieucy or some 
interpolation hypothesis ; it cannot be found until the errors of random sampling have been 
smoothed by some such process. 
For practical purjioses the median is one of the easiest qxiantities to determine, and this can 
be found in a very few minutes from iuspection of the measurements, i.e. count half the observa- 
tions from either end of the frequency distribution, and this will land the counter part-way, say 
n individuals, into some elementary fi-equency groujx Look out the individuals in the obser- 
vation-book falling into this group and arrange them in order of size, the ?ith individual from the 
proper end of the group either gives the median value of the organ (total number of individuals 
odd) or we place the median value (total number of individuals even) mid-way between the n and 
Ti-l-lth individuals. If the median has been found as well as the mean, then a quite good value 
of the mode may be deduced by remembering that the median lies between the mode and the 
mean and that the distance from the median to the mode is double the distance from the mean 
to the median ; this is close enough for practical purposes in the majority of frequency distri- 
butions*. Unless the mode be determined in this manner or from a complete treatment of the 
frequency distribution the mere tabulation of modes by inspection seems of small value, and the 
reasoning upon modes so determined liable to lead to fallacious conclusions. 
K. PEARSON. 
V. On the Change in Expectation of Life in Man during a period 
of circa 2000 years. 
It is well known that the expectation of life at each age has changed in England very 
sensibly during the last 50 years — Farr's table difi'ers very considerably from Ogle's table. The 
same remark applies, if we compare the Registrar-General's life table for 1881-90 with that of 
J. P., F.R.S., based on the London bills of mortality for 1728 to 1757 1. But an opportunity has 
occurred for comparing the expectation of life in man at an interval of nearly 2000 years. The 
change that has taken place in this period cannot fail to be one of the greatest interest from the 
standpoint of evolution. 
Professor Flinders Petrie has drawn my attention to the fact that the ages at death of a 
certain number of Egyptian mummies in the Roman period have been recorded and are published 
* Phil. Trans., Vol. 186, A. p. 375. Formulae for determining the mode and the probable error 
of its determination from the moments directly are given iu a paper : "On the Mathematical Theory 
of Errors of Judgment, with Special Keference to the Personal Equation," in type for the Phil. Trans. 
See also R. S. Proc. Vol. 68, p. 369. 
t See jR. S. Proc. Vol. 67, p. 169. 
