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BELL SYSTEM TECHNICAL JOURNAL 



carries us, see where it breaks down, sec why it breaks down, and then 

 avail ourselves of the new theory^^a powerful tool of great value, be- 

 cause it makes possible for the first time the solution of many practical 

 problems. Here is the experiment. Take 998 small circular chips, 

 499 green and 499 white. Mark 20 white ones with 0, 40 white ones 

 with 0.1, 39 white ones with 0.2, etc., in accordance with the normal 

 law. Do the same for the green chips except that all numbers on the 

 chips are minus. Put the 998 chips in a bowl, mix thoroughly, draw 

 out one and record it. Replace the chip, again mix thoroughly, and 

 repeat the process until 4000 values are observed. A little reflection 

 shows that this experiment is equivalent to making 4000 measure- 

 ments of a quantity by a method subject to a normal law of error with 

 a root mean square error of approximately unity. 



Let us group these 4000 values into 1000 groups of 4, and determine 

 the average for each group, taking the first four observations as the 

 first group, the second four as the second group and so on. This gives 



Fig. 2 — Curve showing customary error theory' to be satisfactory on one condition 

 not often met in practice; i.e., a is known 



• Distribution of 1000 averages of 4 



a 

 — Normal law with root mean square error ^7^ 



V4 



us 1000 averages. Suppose we subtract the true value m (in this case 

 zero) from each average and divide this result by the root mean square 

 error of the frequency distribution of values within the bowl. This 

 gives us 1000 observations of the error of the average of 4 observations 

 measured in terms of a. Customary error theory shows that these 

 averages should be distributed normally as indicated by the smooth 



