maximum . For the degree of skewness usually 

 encountered, the distribution of sample means 

 from such frequency distributions can be pre- 

 sumed normal . The variance of the mean will 

 be the variance of the characters' frequency dis- 

 tribution, divided by the sample size . 



Similarly, the distribution of the differ- 

 ence between such means will be normal with 

 variance equal to the sum of the variances of 

 the two means whose difference is considered. 



The distribution— of the sample estimate 

 of "P", say T, will generally not by symmetri- 

 cal. If 



/\ m - m 



P = b2 bl = y_ 



m al " m bl x 



where the m's are the sample means, the^ % 

 confidence limits for Y are 



A 

 P 



where 



J ~ n.gtj^ " y n'^Cx y l 

 L 1 - tlozC'i J 



d = o^y) = 4 b2 ) + ff 2 ( m bi) 



( m b2 " m bl> 



and 



G 2 = Q^x) = <T 2 (m al ) + ^(m. ) 



~=2 

 x 



( m „ 



TT 



•al m bl>' 



and tc?C is the "a^- ' level of the normal or 

 "Student" distribution. These roots may exceed 

 unity or may be negative, and may even be imag- 

 inary, but only when the coefficients of variation 

 are large. For coefficients of variation of y 

 and x greater than . 7 the roots will often be 

 imaginary . Although one might be tempted to 

 conclude in such cases that one knows less than 

 nothing about the reality of P, they will usually 

 be interpreted as simply inadequate sample size . 



AN EXAMPLE FROM THE PACIFIC 

 SARDINE (SARDINOPS CAERULEA) 

 FISHERY 



1/ Cochran, W. G. "Sampling Techniques, 

 New York, John Wiley and Sons (1953), see 

 p. 121. 



There are a number of different kinds of 

 data available from the sardine fishery which 

 could serve to illustrate the method. Among 

 these are calculated lengths of fish at earlier 

 ages, based on the proportionality of scale growth 

 and fish growth. Such characters should be con- 

 stant throughout life for a given individual, 

 satisfying the conditions stated above. 



We have selected the calculated length at 



the end of the first year of life of fish of the 



1945 -class taken in the 1947-48 and 1948-49 



seasons in the San Pedro and Monterey fisheries. 



The frequency distributions of these calculated 



"1 ' are given in table 1 . The means, their 



1 s 

 estimated variances and other statistics are 



given in table 2. The 1945 year class is being 



used as an example. Each year class could be 



similarly treated. 



As may be noted from table 2, the mean 

 1 of the 1945 -class in San Pedro in the 1947-48 

 season was 138.1mm., some 37.5 mm. greater 

 than the comparable mean at Monterey. In the 

 following (1948-49) season, the mean 1 of the 

 1945-class at San Pedro had changed to 129.9 mm., 

 or 8.2 mm. less than it had been in the previous 

 season. 



If an influx of fish from Monterey were 

 responsible for the decreased San Pedro average, 

 one can estimate what portion of the San Pedro 

 1948-49 landings must have been fish from 

 Monterey of the past year . This portion can be 

 estimated as the ratio of the change in mean at 

 San Pedro to the difference between San Pedro and 

 Monterey means in the "first" year, 1947-48. 

 This would be the ratio of 8.2 to 37.5, or 0.2187, 

 approximately 22 percent . In other words, if 

 22 percent of the San Pedro samples in 1948-49 

 were fish from Monterey in 1947-48, the San 

 Pedro mean would have changed from 138 . 1 mm . 

 to 129.9 mm. 



Referring now not to the samples, but to 

 the population from which they were taken, some 

 allowance for sampling variation must be made . 

 A 95 percent confidence interval for the ratio .2187 

 in this example would be .328 and .1121, or ap- 

 proximately 11 to 33 percent. The computation of 

 these limits is outlined in table 2 . 



31 



