TYPE AND VARIABILITY 



439 



narrow margin, giving us much confidence in the determination, 

 with only a small chance of being wide of the truth. 



Of course the error in a determination has also an even 

 chance of lying outside the limits set by the probable error (E), 

 but the following table will show that it is very unlikely that 

 the error is many times as great as E. Thus the chances that 

 the true value lies within the range set by E, 2 E, etc., are 

 as follows : 1 



E the chances are even 

 2 E the chances are 4.5 to i 

 3 "the chances are 21 to i 

 4 E the chances are 142 to i 

 5 E the chances are 1310 to i 

 6 E the chances are 19,200 to i 

 7 E the chances are 420,000 to i 

 8 -" the chances are 1 7,000,000 to i 

 9 E the chances are about 1,000,000,000 to i 



It is extremely improbable, therefore, that an error will be 

 many times as large as the probable error. For instance, it is 

 practically certain that the error is not as large as 9 E, since 

 the table shows that the chances are about a billion to one in 

 favor of its being smaller than 9 E. 



Thus by giving, along with any result, the calculated probable 

 error, the reader may know what degree of confidence is to be 

 placed in the results. 



For a graphic illustration of the meaning of E, suppose in 

 the following figure the line AB represents our determination, 

 and the lines ab and a'b' the location of -f- E and E. 



* B ' 



Now this means that if AB is not the true location of the value 

 in question, the chances are even that this location is not outside 

 the limits set by the lines ab and a'b' representing E. 



1 C. B. Davenport, Statistical Methods, p. 14. 



