38 Mr. F. Galton on Statistics by Inter comparison, 



nitudes differing from the mean value by such and such mul- 

 tiples of the probable error, will occur with such and such 

 degrees of frequency." My proposal is to reverse the process, 

 and to say, u since such and such magnitudes occur with such 

 and sueh degrees of frequency, therefore the differences between 

 them and the mean value are so and so, as expressed in units 

 of probable error." According to this process, the positions 

 of the first divisions of the scale of divergence, which are those 

 of the mean value plus or minus one unit of probable error, 

 are of course p and q, lying at the ^ and f points of the 

 ogive, or, if the base consist of 1000 units, at the 250th point 

 from the appropriate end. The second divisions being those 

 of mean value plus or minus two units of probable error, will, 

 according to the usual Tables, be found at the 82nd point 

 from the appropriate end, the third divisions will be at the 

 17th, and the fourth at the 3rd. If we wished to pursue the 

 scale further, we should require a base long enough to include 

 very many more than 1000 units. 



Remarks on the Law of Frequency of Error. 



Considering the importance of the results which admit of 

 being derived whenever the law of frequency of error can be 

 shown to apply, I will give some reasons why its applicability is 

 more general than might have been expected from the highly 

 artificial hypotheses upon which the law is based. It will be 

 remembered that these are to the effect that individual errors of 

 observation, or individual differences in objects belonging to the 

 same generic group, are entirely due to the aggregate action of 

 variable influences in different combinations, and that these in- 

 fluences must be (1) all independent in their effects, (2) all 

 equal, (3) all admitting of being treated as simple alternatives 

 " above average " or " below average •/* and (4) the usual Tables 

 are calculated on the further supposition that the variable influ- 

 ences are infinitely numerous. 



As I shall lay much stress on matters connected with the 

 last condition, it will save reiteration if I be permitted the use 

 of a phrase to distinguish between calculations based on the sup- 

 position of a moderate number (r) of elements (in which case the 

 frequency of error or the divergence is expressed by the coefficients 

 of the expansion of the binomial [a + b) r ) and one based on the 

 supposition of the number being infinite (which is expressed by 



the exponential e~&), by calling the one the binomial and the 

 other the exponential process, the latter being the process to 

 be understood whenever the " law of frequency of error " is 

 spoken of without further qualification. When the results of 



