4 On the Probability of Error [January, 



theory of probabilities dictated a variation of this rule, em- 

 bodied in the formula previously stated. This formula may 

 be understood in words as follows : — Square the number of 

 observations, and divide this product by twice the sum of 

 the squares of the errors. 



Having now defined the term " weight," we have to trace 

 the meaning of the terms mean risk and probable error. 



If we consider positive and negative errors as equally 

 probable, they will balance each other, so that the average 

 of positive or of negative errors will be equal. We thus 

 arrive at the meaning of the term average error, and can 

 proceed easily to the determination of the mean risk ; and as 

 the mean risk of positive error is the average positive error, 

 the mean risk of negative error the average negative error, 

 the mean risk may be taken as half the average error. Re- 

 presenting the mean risk by m, and the weight by w, we have — 



Mean risk = 200 ,_ , 

 709 v^ 



, 0*28200=53 

 or more nearly, = _r J . 



yqjD 



Or, mean risk = prob. error x 0*591473. 

 The probable error is that error for which there are equal 

 chances of exceeding or of not attaining. For instance, 

 suppose the chances are equal that the error is included be- 

 tween o and 2, or that it should exceed 2, assigning this as 

 the limit of error, then, of course, for any number greater 

 than (say) 10 the chances are in favour of the error being in- 

 cluded within the number. It has been calculated that — 



62 



The probable error = — , 



130 V w 



or more nearly, = 47 93 # 

 V w 

 Or, the probable error = m 1*690694. 

 Calling the mean risk m, the probable error p, and the 

 weight w, we have from the preceding reasoning the fol- 

 lowing formulae : — 



w = ^- . 



1420W 2 



w = J_ _ 0-227468 



22^ 2 p 2 



From the foregoing formulae, when either the weight, 

 probable error, or mean risk is given, the other two can be 

 determined. As we are capable, in most scientific observa- 

 tions, of so adjusting our instrumental means that the errors 



