438 TRANSMISSION 



The real nature of probable error and the methods for 

 deducing the formulas for its calculation will be covered in 

 the Appendix, which is devoted especially to mathematical 

 methods and conceptions, but enough should here be said to 

 give the student an intelligent idea of what is meant by probable 

 error and to acquaint him with the bare formulas for its calcu- 

 lation as to the values here under discussion. 



The probable error (denoted by E) is a pair of divergencies 

 lying one above and the other below the value determined, and 

 of which we can say with confidence that there is an even chance 

 that the true value lies between these limits. 1 These numbers 

 are numerically equal, but one is regarded as plus, the other as 

 minus (^), and the two define a range within which, out of a 

 very large number of determinations, at least half the true values 

 would be found. This being the case, we may say of any single 

 determination that the chances are even that any error involved 

 will not fall outside the limits set by E. It is obvious, there- 

 fore, that the smaller the probable error the narrower this range, 

 the greater confidence we should place in our determination, and 

 the smaller are the chances of a large error having been made. 



The expression " probable error" may be misleading. It is 

 not, as might be supposed from the words, the most probable 

 error. The most probable value is our determination and the 

 most probable error is zero. Neither does the probable error 

 fix the limits of error, but it is an extremely good measure of 

 accuracy in that it fixes a range above and below the determined 

 value such that the chances are even that the true value lies 

 within this range. 



Thus, if a series of calculations results in a final number 27.4, 

 with a probable error of .12, it means that out of a great 

 number of cases the true value of one half will lie between 

 27.52 (27.4 + . 1 2) and 27.28 (27.4 .12). 



If another calculation involving larger numbers or more ac- 

 curate methods should result in the same value, 27.4, but a 

 probable error of only .04, then the true value has an even 

 chance of lying between 27.44 and 27.36, which is a very 



1 There is, of course, also an even chance that the true value lies outside the 

 same limits. 



