462 



SCIENCE 



[N. S. Vol. L. No. 1298 



way of interpreting these weights is to give 

 each observation a number of " votes," so to 

 speak, equal to its weight; that is, to count 

 its result that number of times in making up 

 the average. An observation with weight 5, 

 for example, is considered to be worth five 

 times as much as one with weight 1, what- 

 ever that signifies. It merits as much confi- 

 dence as the average of five observations with 

 weight 1. 



If each result to be weighted is actually 

 the mean of a number of similar observations, 

 the weighting is comparatively easy. But we 

 refer to the weighting of the original observa- 

 tions; and how is one to decide, without in- 

 dulging in mere guesswork, what this factor, 

 to be assigned to each of such a series of 

 results, should be? 



Probably many scientific observers do not 

 weight their measurements because they do 

 not know how. They know that some results 

 are much more trustworthy than others, but 

 they are at a loss when it comes to expressing 

 how much more. It is the purpose of this 

 paper to suggest a simple means whereby any 

 scientific worker may arrive at a consistent 

 practise in this matter. 



There are very few people engaged in work 

 involving precise measurement who have not 

 reached, through experience either as teachers 

 or as students, a prety well defined interpre- 

 tation of the ordinary percentage grades as- 

 signed to pupils in school or college. When a 

 boy comes home with a grade of only 72 in 

 grammar, the occasion is not one for con- 

 gratulation. The whole family knows that 72 

 stands for poor quality of scholarship, for the 

 reason that the vast majority of pupils are 

 assigned a higher grade than this. 



Now, it is not difficult for an observer to 

 assign percentage grades to his experimental 

 results, passing judgment upon them very 

 much as he would upon work done by a stu- 

 dent in laboratory or classroom. He may 

 even take separate account of the various 

 factors which may aiiect the observation, such 

 as weather conditions, visibility, constancy of 

 temperature, etc., and combine all these in 

 estimating the final grade of the result, just 

 as a teacher combines recitations, notebooks. 



tests and examination in grading a student. 



And if, when the several observations of a 

 set have been thus graded, a means is at hand 

 to translate the grades into relative weights, 

 our problem is solved. Such a means is pro- 

 vided by the following considerations. 



The variable conditions attending the ma- 

 king- of measurements, which alone afiect 

 their relative weight, are of course fortuitous. 

 The experimenter tries to have everything 

 constant and to maintain a uniformly high 

 standard of precision, just as a marksman 

 tries repeatedly to hit the same target. But 

 he can not control all the conditions, and these 

 fluctuate in accordance with the well-known 

 law of departures, based upon the theory of 

 probabilities. 



The theory puts no limit to these fluctua- 

 tions, and one observation might, theoretically, 

 be a thousand times more reliable than an- 

 other; but practically no such range need be 

 considered. Probably for most purposes, pri- 

 marily assigned-"^ weights need not go outside 

 the simple scale of integers from to 10; 

 the weight denoting absolute worthlessness 

 (observation to be discarded) and the weight 

 10 denoting practical certainty. Either of 

 these cases would be extraordinary and of very- 

 rare occurrence. The general run of results 

 will vary from a little doubtful to a little 

 more than usually reliable, being situated not 

 far either way from weight 5, the middle of 

 the scale. 



Wlaen the unit of weight has been chosen 

 with such significance that the distribution is 

 as described, it is possible at once to predict 

 what proportion of observations should, in the 

 long run, have weight 2, what proportion 

 should have weight 3, etc. This is done by 

 means of the prohahilHy integral, tables of 

 Wihich are given in most books on the theory of 

 errors. These proportions are here given as 

 percentages, accurate to the second decimal 

 place : 



1 This does not apply to weights -computed for 

 adjusted values based on long series of observa- 

 tions or upon indirect measurements of connected 

 quantities. In such cases, large and even mixed 

 fractional numbers may be consistently assigned 

 as weights. 



