counted and the number in the total sample is 

 estimated on the basis of the weights of the sub- 

 sample and total sample. Since the estimate is 

 based on weight rather than volume, the three 

 vanes need not divide the subsampler into exactly 

 0.2, 0.2, and 0.6 segments. 



If a length-frequency estimate is desired, the 

 subsample can be further subsampled. Since the 

 number in the first subsample is now knovra, any 

 desired number for the length-frequency subsam- 

 ple can be closely approximated by selecting the 

 proper sequence of subsamples. For instance, sup- 

 pose the first subsample contains 3,371 anchovies 

 and a length frequency is desired from approxi- 

 mately 100 fish; 3,371 x 0.2 x 0.4 x 0.4 = 108. 

 Therefore, subsamples taken in this sequence 

 should produce the desired number for measuring. 



The consistency with which the desired number 

 is obtained may be judged (Table 1) by comparing 

 the "theoretical" and "actual" numbers obtained 

 in 20 successive trials. The two subsamplers used 

 in these trials had a tendency to slightly exceed 

 the desired number; one or both of the smaller 

 compartments in each subsampler probably con- 

 tained a bit more than 0.2 of the whole. However, 

 the increased subsample size actually improves 

 the probability of obtaining an accurate length- 

 frequency estimate. Also, with use, one soon 



learns whether the tendency is to obtain more or 

 fewer than the theoretical number and can select 

 the subsampling sequence accordingly. 



How well the length-frequency estimates de- 

 rived from subsampling groups of anchovies and 

 menhaden represented the true length frequen- 

 cies of the groups was examined by using the 

 Kolmogorov-Smirnov one-sample, two-tailed test, 

 which is a test of goodness of fit. The test involves 

 comparing the observed cumulative frequency 

 distribution from a subsample with the cumula- 

 tive frequency distribution of the total sample. It 

 is sensitive to any kind of difference between the 

 two distributions — differences in location (central 

 tendency), in dispersion, in skewness, etc. Accord- 

 ing to Siegel ( 1956) the Kolmogorov-Smirnov test 

 is definitely more powerful than the chi-square 

 test when samples are small, and may be more 

 powerful in all cases. 



The cumulative length-frequency distribution 

 for only one subsample was significantly different 

 (oc = 0.05) from its corresponding total sample 

 (Table 1 ). In the other 19 tests, the probability was 

 greater than 0.20 that a divergence of the observed 

 magnitude would occur if the observations were 

 really a random subsample from the total sample 

 (0.20 is the highest probability listed in Siegel's 

 table). 



Table L— Results of 20 tests to determine the correspondence between: 1) the theoretical and actual number of bay 

 anchovies or gulf menhaden in the subsample, and 2) the cumulative length frequency distribution of fish in the 

 subsample and in the corresponding total sample. Subsamples were returned to the total sample after each trial. The 

 cumulative distribution shown in italics (in the same row with the number in the total sample) was the true 

 distribution obtained by measuring every fish in the sample. 



'Measured in 5-mm increments: i.e.. 15 = 15.0-19.9, 20 = 20.0-24.9, etc. 



^The probability of a divergence this large in a random subsample from the total sample was between 0.05 and 0.01 The probability for the 1 9 

 other subsamples was >0.20 



493 



