ON ERRORS OP OBSERVATION. 137 



equally likely, each is one half the average error, con- 

 sidered without reference to sign. 



By the probable error I mean that amount of error 

 wjiich is such, that there is an even chance for exceed- 

 ing or falling short of it. Thus if it be 1 to 1 that 

 the error shall lie between and 10, and of course 

 the same that it shall exceed 10, then 10 is called the 

 probable error. For any number greater than 10, the 

 chances are (no matter how little) in favour of the 

 error being within that number ; for any thing less 

 than 10, the chances are against the error falling 

 within that amount. 



PROBLEM. The number of observations being n, 

 and positive and negative errors being equally likely, 

 required the probability that the average of the n 

 observations lies within a given quantity k of the truth ; 

 or, M being the average, that the truth lies within 

 M + k and M k. 



RULE. (By Table I.) Take the average of the 

 observations, find all the errors upon the supposition 

 that the average is the true result, add together the 

 squares of the errors, and divide the square of the 

 number of observations by twice the sum of the squares 

 of the errors. Call the result * the weight of the 

 average. Multiply k by the square root of the weight ; 

 let the result be t ; then the H answering to t in table 

 I. is the probability required. 



RULE. (By Table II.) Find the weight as in the 

 last rule, and divide 62 by 130 times the square root 

 of the weight. The result is the probable error of the 

 average. Divide k by the probable error, and let the 

 quotient be t ; then the K answering to t in table II. is 

 the probability required. 



EXAMPLE. In the preceding instance, what is the 

 probability that the average 1051 lies within 50 of the 

 truth. The squares of the errors are, 108241, 13924, 

 324, 17956, 67600, 1444, 6241, 59049, 7056, 



* The reason of the appellation will be afterwards explained. 



