— 8 - 



i) The number of individuals in the catches varies greatly and we cannot 

 draw general conclusions except on the basis of a large material. Our method 

 therefore when dealing with such material as we possess must be, like in other 

 statistics, to adopt the principle of averages. lyct us say that we wish to find 

 in the portion of the North Sea examined the average number of haddock ' per 

 unit of space for a given time. As unit of space we take the stretch swept by 

 a trawl in an hour, which we shall designate a trawl-hour. The problem is much 

 simplified if the observations are equally distributed over the whole area within 

 the time stated; for we have then only to calculate the proportion which the 

 total number of individuals caught bears to the total number of trawl-hours. This 

 proportion we may term the total average. We can also calculate the average 

 of several parts of the whole area and take the mean of their averages. This 

 we may designate the mean average. 



Now let us assume that the observations have been evenly distributed; that 

 is to say that fishing has taken place for an equal number of trawl-hours within 

 the different parts of the area, and that these different parts are of equal size. 

 We will now get the same value by either mode of procedure 2. Unless however 

 these assumptions hold good we shall get a false value for our average by blindly 

 employing either the one or the other method in our calculations. 



In our material the trawl-hours are very unevenly distributed. Fishing has 

 been carried on largely — that is to say over large portions of the bottom — in 

 some areas and to a limited extent in others, while the various areas somewhat 

 arbitrarily included differ not a little in size. The following would appear to be 

 the best manner of obtaining exact mathematical values: find the proportion 

 between the sizes of the different areas and thence obtain a factor of proportion 

 for each area. The averages of the different areas must each be multiplied by 

 its factor of proportion, and the sum of all the different values thus obtained be 

 divided by the sum of the factors of proportions. 



1 We refer here to trawled haddock only. 



2 If a, h denote the number of individuals within n different areas and t denotes the 



number of trawl-hours for each of these areas, then: 



the mean average the total average 



a b 

 J) y are naturally the total averages for the single areas. 



3 Let us assume that we wish to find the average number of individuals per trawl-hour within 

 an area-group consisting of three different areas, whose sizes respectively are m:n\o\ and let us 

 further suppose that within these different areas a, h and c individuals have been caught during the 



course of a, ß and ;- trawl-hours. The average-number for each area is then — , -0 and — . Granting 



that the average of these different areas is correct, (i. e. that the material has been really representa- 

 tive of these areas) the true average for the whole group af areas will be: — 



m .a n ,h , o . c 



m ■\- n-\- o 



