g. g- "It follows that the arithmetic mean 



1. V N 



7 ± N 



^ = N 2 C (9) 



1 V 



is asymptotically normal "with a mean m and a standard deviation, 



°^1 ^ V^N." 



This central linnit theorem works remarkably well in many cases 

 for small values of N. For example, the faltiing (or convolution) of three 

 samples from a rectangular distribution even then approximates a normal 

 distribution, and for the target distribution, it may well be that the 

 theorem is applicable for values of N as low as 9 or 16 for practical 

 purposes. 



It is necessary to calculate the second moment about the mean of the 

 distribution given by equation (1). This is found by means of equation 



= Jo ^ ^ h ^ 7o E (10) 



= E (I--J) 



From the above theorem, the computed mean of a sample of N observed 

 wave amplitudes will be normally distributed with a mean given by 

 equation (2) and a second moment about the mean given by equation 

 (10) divided by N. 



The variable z defined below is therefore distributed according to 

 a normal distribution with a zero mean and a unit standard deviation. 



" ^ ^ ^ - — (11) 



yiTJTETN 



It is now possible to compute the probability that z will lie between 

 any two values under the \init normal curve. The probability that z will 



14 



