452 THE PRINCIPLES OF SCIENCE. 



Rules for finding the probable error of a mean result : 



1. Draw the mean of all the observed results. 



2. Find the excess or defect, that is, the error of each 

 result from the mean. 



3. Square each of these reputed errors. 



4. Add together all these squares of the errors. 



5. Take the square root of this sum. 



6. Divide the square root by the number of results. 



7. Multiply the quotient by 0*67449 (or approxi- 

 mately by 0*674, or even 0*67), a natural constant 

 number derived from the Law of Error in a manner 

 which is described in mathematical works upon the 

 subject. 



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 the probable error of the mean, namely 304. Now 

 the differences between this mean and the above numbers, 

 paying no regard to direction, are n, 3, 2, 3, 9; their 

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

 consequently 224. Taking the square root of this sum by 

 the common arithmetic process, or by logarithms, we ob- 

 tain 14-966, and dividing by five, the number of observa- 

 tions, we have 2-99, which has only to be multiplied by 

 67 to yield us 2-019. This number is so close to 2, that 

 we may call the probable error equal to two. Thus the 

 probability is one-half, or the odds are even, that the true 

 height of the mountain lies between 302 and 306 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 



