MULLIN and CONVERSE: EUPHAL'SIID AND ZOOPLANKTON BIOMASSES 



APPENDIX A 



Caution must be used when performing a logarithmic (In) transformation [;/, = ln(.r,)] 

 on data (.r,) belonging to a nonnormal distribution. The resulting mean of the log-trans- 

 formed data 



n 



E In(Xi) 



y = ^ 



n 

 is equal to the logarithm of the geometric mean of the untransformed data 



y = in("VA 



1^1 



Since the geometric mean is always less than the arithmetic mean (Zar 1984), the 

 antilog of the mean of the log-transformed data must be corrected before it can be used 

 as an unbiased estimation of the arithmetic mean of the untransformed data. 



Bagenal (1955) showed a relationship between the means of transformed (^7^ and 

 untransformed [x) data, which is valid when the transformed data belong to a normal 

 distribution, with mean = y and variance = ct|. 



^^^-i (1) 



and 



y = ln(.f) - — c4 (2) 



We applied this relation to our calculations of the critical differences between the 

 means of log-transformed biomasses after and before the beginning of the fishery. The 

 data had to be log-transformed in order to perform the two-sample f-test, since this 

 assumes that the distributions underlying the two samples are normal. 



The critical differences were so calculated, from the two-sample f-test formula: 



E%-E% = t!s,^l^^^^^^ (3) 



where E%, E% are, respectively, the means of the log-transformed biomasses after and 

 before the beginning of the fishery, equal to // in Equation (2), 



t! is the critical f-value at a = 0.05, 



Sp is the pooled variance for the two samples, and 



?Za,j?h are, respectively, the numbers of samples after and before the beginning 

 of the fishery. 



These differences were then related to the means of untransformed data by 

 applying Equation (2): 



E% - E% = ME J - 0.5 s; - \n{Eb) + 0.5 s^ = ln( ^ ) (4) 



E^ 

 E,, 



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