VARIABILITY EST TRAWL CATCHES 



159 



The total number of species expected is 



gjy=-«io gi (i- 



-x) 



(3) 



from which it is seen that the distribution is 

 related to the algebraic expansion of the loga- 

 rithm, hence its name. 



The total number of individuals expected is 



(4) 



y, ca T = otx/(l — x) . 



r = l 



Letting S represent the total number of species 

 observed and N the total number of individuals, 

 we have from equations 3 and 4 



<S= — a log, (l—x) 

 N=ax/(l—x) 



(5) 

 (6) 



so that both a and x may be determined if S and 

 N are known. Fisher (1943) and Williams (1947) 

 give tables of log 10 N/a in terms of logi N/S for 

 solving equations 5 and 6, given S and N. The 

 computations are straightforward and the param- 

 eters x and a, as well as the terms of the expected 

 series (equation 2), may be readily determined 

 for a given set of data. 



Fisher (1943) gives the following formula for 

 the variance of 



'{<#+«>' log. ^±f-«Ar} 



(SN + Sa-Naf 



...(7) 



The formula enables one to calculate the standard 

 error of a but Anscombe (1950) points out that 

 equation 7 is appropriate only under certain 

 conditions of sampling, these conditions not 

 being completely met in the census sampling. 

 Anscombe gives the formula 



F„~ 



■(8) 



ktt 



which appears to apply to the conditions met in 

 the census sampling, that is, comparisons of 

 observations on different sorts of biological 

 associations and comparisons of a from observa- 

 tions obtained in different years and with probable 

 relative differences in abundance between areas 

 and years. The variances of a use in this paper 

 are estimated from equation 8. 



APPENDIX D 



A negative binomial distribution fitted to numbers of 

 plaice eggs caught by a plankton net 



The published data on sampling with plankton 

 nets do not often give the total catches in suc- 

 cessive tows, or sufficient data on number of 

 tows with zero observations, to study the nature of 

 the distributions encountered. Table 17 is a 

 summary of the catches of plaice eggs in all stages 

 of development in the southern part of the North 

 Sea (Buchanan- Wollaston, 1923, table 3). The 

 data were collected in 1914. Table 3 (Buchanan- 

 Wollaston) indicates that 27 of the 50 stations 

 yielded no plaice eggs in the plankton tows. 

 The mean catch is 1.5.1 eggs with a standard 

 deviation of 30.3. 



Estimating k from formula 2, appendix B, we 

 have 



27/50= (1+15.1/*)-* 



from which £=0.1292. 



The expected variance from equation 1, 

 pendix B, is 



15.1 = 15.170.1292 



ap- 



or 1779.9, as compared to the observed variance, 

 915.5. 



The observed frequency of plaice eggs per tow 

 and the frequency expected on the hypothesis of 

 a negative binomial distribution are presented in 

 table 18. The probability of obtaining a worse 

 fit by chance for 2x 2 with three degrees of freedom 

 is somewhat greater than 0.60, so that the hypoth- 

 esis of a negative binomial distribution of the 

 data may not be rejected. 



APPENDIX E 



Proof, smaller sampling unit more efficient with hetero- 

 geneity present 



When the variance is a function of the mean of 

 a distribution, the effect of the size of sampling 

 unit on the efficiency of sampling and on the 

 precision of estimates cannot be expected to be 

 obvious to the biologist. Experimental data con- 

 firming the validity of the efficiency of the smaller 

 sampling unit, under conditions of heterogeneity, 

 have been published by Fleming and Baker (1936), 

 Beall (1939), and Finney (1946). 



The precision of an estimate is defined as the 

 reciprocal of its variance. Efficiency is defined in 

 terms of the relative amounts of sampling required 



