442 



TRANSMISSION 



If a circle be drawn about the center with a radius equal to 

 the average deviation of A, this circle will fairly well represent 

 his marksmanship ; that is to say, the average distance of all his 

 shots from the center is the same as if they all lay on this circle. 

 If the marksmanship of B is not so good as that of A, his average 

 is greater and the circle correspondingly larger. 1 



Now neither of these circles is an absolute index of the marks- 

 manship of either A or B, unless an infinite number of shots 

 has been made ; that is to say, if only a few shots have been 

 fired, the probability of error is great if we assume these circles 

 to be fully representative of the marksmanship, because there 

 is practical certainty that succeeding shots will be either better 

 or worse ; indeed, there is always a chance that the next may be 

 a lucky shot and lower the deviation, or a wild one and raise it. 



We should therefore conceive of two other circles lying 

 neighbor to each of those representing the calculated deviations. 

 These are represented in the cut by the light-line circles and 

 give a graphic meaning to the probable error. They are drawn 

 so that if the heavy circles do not accurately represent A's and 

 B's marksmanship then the chances are even that the true 

 position of the circle representing A's marksmanship, for exam- 

 ple, lies somewhere between the light lines representing the 

 probable error of the computation. The chances are of course 

 also even that it lies beyond these limits, either within or outside. 

 Obviously, the smaller the probable error the greater the confi- 

 dence to be placed in the calculated deviation. The student is 

 cautioned here that in this illustration the " probable error" 

 refers not to A's or B's failure to hit the target, but to our 



1 The question may be raised as to whether there is not a better measure of 

 marksmanship than the average departure of the shots from the bull's-eye. For 

 instance, with the bull's-eye as a center, we may describe circles through each of 

 the shots of A, and construct a circle with the average area of these circles for 

 its "area. This circle may then be selected as a measure of A's marksmanship 

 instead of the circle above discussed. The radius of this circle can be obtained 

 by taking the square root of the mean square of the deviations from the bull's- 

 eye. It is not important for us to discuss here the relative merits of these two 

 methods of measuring marksmanship, but it is important that we recognize that 

 the method explained in the text is based upon what we may well call " average 

 deviation from an ideal," while that suggested in this footnote may well 1;e called 

 " standard deviation from an ideal." 



