2 On the Probability of Error [January, 



include perception and correct estimation of the most minute 

 deviation from the absolute standard. Between even this 

 deviation and the absolute standard there are an infinite 

 number of results that may be obtained. Approximate 

 correctness is all, then, that the experimentalist has the 

 probability of attaining. So that, given a series of experi- 

 mental results, there remains the decision of two points — 

 (i), the determination of the standard ; and (2), the probable 

 error of the assumed standard. 



There are many engaged in experimental research, even 

 in research of a relatively high order, to whom the methods 

 of determining these two points are unknown. To these I 

 shall endeavour to explain the laws of probability as they 

 have been laid down by eminent mathematicians. To those 

 (and I am afraid their number is not legion) acquainted 

 with these laws, I can offer illustrations only of the appli- 

 cation of these laws. Without a due consideration of 

 these principles astronomy could not claim its character of 

 exactness ; and there appears no reason why the physical 

 and chemical sciences should not, as means of observa- 

 tion increase in delicacy, attain to the rank and estimation 

 of exact sciences. Our chronographs measure easily to the 

 i-ioo,oooth of a second of time ; our balances turn with a 

 fragment of a hair weighing i-io,oooth of a grain ; the re- 

 sults of electrical experiments have been obtained varying 

 only 5 in the 1000. With this exactness, surely we may 

 think it carelessness that does not ascertain the closest ap- 

 proximation to accuracy, as well as the limit of error to be 

 allowed this approximation. 



The subject is a most subtle one. It may be defined as 

 the best mode of combining observations so as to yield the 

 most trustworthy mean ; and in this light I am unable to 

 mention any work affording so popular and so profound a 

 discussion as the little volume, by Prof. De Morgan, entitled 

 "An Essay on Probabilities." The author shows how the 

 observer may measure the degree of confidence to which the 

 average of any series of observations is entitled. Thus, 

 taking his own example, — that of fifteen observations 

 giving the following results : 722, 933, 1033, 917, 1311, 1089, 

 972, 1294, 967, 1344, 1250, 744> I 309> 8 5S, 1029,— if the 

 average, or the arithmetic mean, of all the observations be 

 1051, the theory does not assist the observer in estimating 

 the probability of this average being true. For true, in the 

 absolute sense of the word, it, in all probability, is not, and 

 therefore no theory is needed to assist in drawing that con- 

 clusion. But let any definite departure from truth be 



