The best representative value for a set of observations

What is the best representative value of a set of discordant results The arithmetical mean naturally suggests itself, and some mathematicians start from the axiom the arithmetical mean is the best representative value of a series of discrepant observations . Various attempts, based upon the law of errors, have been made to show that the arithmetical mean is the best representative value of a number of observations made under the same conditions and all equally trustworthy. The roof rests upon the fact that the positive and negative deviations, being equally probable, will ultimately balance each other as shown in Ex. (1), below.1 

Examples.—(1) If alf a, ... an are a series of observations, a their arithmetical mean, show that the algebraic sum of the residual errors is 

En passant, notice that in calculating the mean of a number of observations which agree to a certain number of digits, it is not necessary to perform the whole of the addition. For example, the mean of the above nine measurements is written.  

The best representative value of a constant interval. When 

---