4. Reliability of the Average Height 



The expected value of the average height equals the expected 

 value of a random variable from the population. Therefore the average 

 value of a sample of N observations is an unbiased estimate of the mean 

 of the theoretical population. There will be no systematic error in the 

 estimate of the average. 



If enough assumptions are made about the nature of the observations, 

 the confidence limits of a particular estimate of an average wave 

 height can be found. These assumptions are not too realistic. They 

 will have to be modified qualitatively after the derivation is complete, 

 but at least they permit the statement of some practical rules applicable 

 to height observations. The following assumptions are made: 



1. Each trough is correlated with the value of unity to the 

 preceding crest. 



2. The crests are completely uncorrelated (i.e., the height of a 

 wave is independent of the height of preceding and following 

 wave s ) . 



3. The amplitudes, and therefore the heights, by reason of the 

 first assumption are distributed according to the target 

 distribution. 



A total of N wave heights then corresponds to a total of N independent 

 amplitude observations, and a wave record with these properties (it 

 does not exist) would look like the record sketched in figure 3. (It is 

 interesting to note that the amplitude values could be completely 

 uncorrelated and that the autocorrelation function of such a record 

 would still show a well-developed oscillation through plus and minus 

 values.) 



A theorem in statistics can now be used to study the average value 

 of these N wave amplitudes. The central limit theorem of statistics 



states (Cramer, 1946, page 215, for example) that: "If ^ ^ ^ 



are independent random variables all having the sanie probability 

 distribution, and if m and a denote the mean and the standard 

 deviation of every ^ , then the sum 





(8) 



is asymptotically normal" withameanNm and the standard deviation, 



13 



