388 THE PEINCIPLES OF SCIENCE. [chap. 



observations, and we get the mean error of the mean 

 result. 



8. Lastly, multiply by the natural constant 0*6745 (or 

 approximately by O'Gy^, or even by f ), and we arrive at 

 the probable error of the mean result. 



Suppose, for instance, that five measurements of the 

 height of a hill, by the barometer or otherwise, have given 

 the numbers of feet as 293, 301, 306, 307, 313 ; we want 

 to know tlie probable error of the mean, namely 304. Now 

 the differences between this mean and the above numbers, 

 paying no regard to dirnction, are II, 3, 2, 3, 9 ; their 

 squares are 121, 9, 4, 9, 81, and the sum of the squares 

 of the errors consequently 224. The number of observa- 

 tions being 5, we divide by i less, or 4, getting 56. This 

 is the square of the mean error, and taking its square root 

 we have 748 (say 7^), the mean error of a single obser- 

 vation. Dividing by 2*236, the square root of 5, the 

 number of observations, we find the mean error of the mean 

 result to be 3*35, or say 3 J, and lastly, multiplying by 

 •6745, we arrive at the probable error of the 7nean result, 

 which is found to be 2*259, or say 2^. The meaning of 

 this is that the probability is one half, or the odds are 

 even that the true height of the mountain lies between 

 30 if and 306 J feet. We have thus an exact measure of 

 the degree of credibility of our mean result, which mean 

 indicates the most likely point for the truth to fall 

 upon. 



The reader should observe that as the object in these 

 calculations is only to gain a notion of the degree of con- 

 fidence with which we view the mean, there is no real use 

 in carrying the calculations to any great degree of pre- 

 cision ; and whenever the neglect of decimal fractions, or 

 even the slight alteration of a number, will much abbreviate 

 the computations, it may be fearlessly done, except in 

 cases of high importance and precision. Brodie has shown 

 how the law of error may be usefully applied in chemical 

 investigations, and some illu.strations of its employment 

 may be found in his paper.^ 



The experiments of Benzenberg to detect the revolution 

 of the eaith, by tlie deviation of a ball from the perpen- 



^ Philosophical Transactions, 1S73, p. 83. 



