DISTRIBtTTION OF EGGS AND LARVAE OF JACK MACKEREL 



259 



spaced stations is highly desirable, such samphng 

 cannot be effected, since it would raise the current 

 cost of sampling prohibitively. 



The monthly samphng of the Cahfornia Coop- 

 erative Oceanic Fisheries Investigations grid of 

 stations has all but precluded the simple assess- 

 ment of the error associated with these estimates 

 of abundance. Although the construction of the 

 proper statistical model was not mthin the scope 

 of this investigation, I was able to make an 

 estimate of the error arising from the practice 

 of hnear interpolation of egg numbers in time and 

 space. This calculation was possible because in 

 1953 and 1954 stations which were only 20 miles 

 apart were occupied, and the samples contained 

 jack mackerel eggs. In 1952, a few stations 

 containing jack mackerel eggs were occupied in 

 late March. 



The errors arising from stratification in space 

 (i.e., spacing sampling stations 40 miles apart) and 

 time (i.e., spacing sampling cruises 1 month apart) 

 were considered. Standard numbers of eggs for 

 stations 20 miles apart- — obtained by linear inter- 

 polation of values obtained from stations 40 miles 

 apart^ — ^were compared with values actually ob- 

 served. In hke manner, standard numbers of 

 eggs for stations sampled at intervals of one-half 

 month^ — obtained by linear interpolation of ob- 

 served values from stations sampled at monthly 

 intervals — were compared with the values actually 

 observed. The differences should be zero if no 

 error arises from linear interpolation through space 

 and time. Since the average difference cannot be 

 expected actually to equal zero, owing to sampling 

 variability, the 95-percent confidence limits for 

 both estimates were computed. These limits 

 should include zero. 



The error arising from spacing the sampling 

 stations 40 miles apart was estimated using data 

 from selected stations on selected lines. (Stations 

 having no eggs were not used.) These stations 

 were placed 20 miles apart. An estimated value 

 was given for every other station by a process of 

 linear interpolation of values between the remain- 

 ing stations which were 40 miles apart. These 

 interpolated values were then compared with the 

 actual number of eggs found at the stations and 

 the difference (Aj) calculated. (A(=obscrved 

 standard number of jack mackerel eggs minus 

 estimated number of jack mackerel eggs.) The 

 deltas were averaged to give A, the average differ- 



5S0553 O— 61 3 



ence between the observed standard number of 

 jack mackerel eggs and the number calculated by 

 linear interpolation. 



The frequency distribution of A< was plotted 

 and appeared to be normally distributed (fig. 6). 



-600 -400 -200 



200 400 600 800 



A, 



Figure 6. — Frequency distribution of A, (the diflFerence 

 between observed and estimated numbers of jack 

 mackerel eggs). 



Therefore A( was considered to be a normally dis- 

 tributed random variable with mean A and 

 variance 5^. 



-137.2$ AS 152.8 



The average difference is close to zero. The 

 wide confidence limits indicate the high variability 

 associated with any one observation. 



The A( were compared with the size of each ith 

 haul to determine whether the differences were 

 related to population size (fig. 7). No relation 

 was evident and it was concluded that the A< were 

 not related to population size. 



The error arising from stratification in time was 

 estimated by considering the standard numbers 

 of jack mackerel eggs taken at selected stations 

 during cruises spaced about 2 weeks apart (March, 

 late March, and April, 1952). By linear inter- 

 polation of the number of eggs taken at a station 

 between the March and April occupancies, an 

 estimate of the number of eggs that should occur 

 at the station during the late March occupancy 

 was obtained. This estimated number was then 



