The objective of this chapter is to describe a practical sampling system for length composition of the annual total landings of one fishery resource along the coast of a certain area. The reason why only one resource is being used is just to simplify the explanations and calculations, but the sampling system could be developed for several resources. The sampling methods starts with a stratification of the population of landings of sardine, followed by multi-stage sampling applied to each stratum. Two of the sampling designs dealt with in previous chapters, stratified sampling and simple random sampling, are the basis for the whole plan.
The sampling considers a region with several landing ports, different components of the fleet and the months of the annual landings. The specific example used in this chapter is an example of how to calculate the annual length composition of the Sardinella aurita fishery in a coastal area with several landing ports, where small purse seiners and trawlers fish this species.
The annual length composition of the landings is important in itself, and equally as a basis for estimating other biological characteristics of the landings. Combining the information on the length composition of the landings with other information, such as:
The characteristics of the landings as a whole (e.g. length composition of the overall landings) can be estimated.
In order to achieve these objectives, landing statistics (total annual catch) are needed. In this case it is assumed that all the fish caught in one year, in the area and by all fleet components come from a single population of the resource.
If only the annual length composition of the landings is needed, one single sampling scheme will be set up. If on the other hand, other biological characteristics like those mentioned above (percentage of adults, age-length keys by length group, etc.) are intended, separate sampling systems should be established.
A crucial issue in all these sampling designs is to ensure that the observation techniques used to determine the characteristics (length, weight, maturation, sex, age, etc.) are well defined. Although this may seem obvious, often sampling systems are invalidated due to inconsistencies during sampling activities. Therefore if, for example, total length is to be measured, every effort should be made to ensure that all the samplers are measuring the total length and not the fork length. Attention should also be paid as to whether the fish are measured to the cm below, to the cm above or to the nearest cm.
Definition of the population and sampling scheme
As mentioned above, the population of interest is composed of all fishes caught by the entire fleet fishing the resource that are landed in all the ports of the area being considered, throughout one year.
Total annual landings, which is the population to be sampled, will be divided into strata following a separation by month, fleet component and landing port.
The number of strata is the product of the number of months (m) by the number of fleet components (fc) by the number of landing ports (p):
Number of strata = m × fc × p
Number of strata = 12 months × 2 fleet
components × 3 ports = 72 strata
Let us take the example of the Sardinella aurita fishery mentioned above and assume that the region has three landing ports. Remember that the Sardinella aurita was being fished by two fleet components (small seiners and trawlers) and that the objective is to estimate the annual (twelve months) length composition.
According to sampling theory (see Chapter 4 on stratified sampling) all strata must be sampled. This means that a minimum of 72 samples must be taken. But only one sample from each stratum (month, port and fleet component) is insufficient for estimating the precision or the error of the annual length composition (at least 2 or preferably 3 samples from each stratum should be taken). Let us consider only one stratum, represented by the trawlers landing Sardinella aurita in Port A during the month of January. Figure 7.1 illustrates the sampling scheme adopted in this situation.
The multistage sampling scheme represented in Figure 7.1 could be explained as follows:
in a stratum (defined by one fleet category, one landing port and one month) several days (selected by simple random sampling) will be sampled (sampling unit 1 -SU1);
for each day selected for sampling, out of the total number of trawler landings, several trawler landings are selected (by simple random sampling) (sampling unit 2 -SU2);
for each trawler landing selected in the previous step, several fish boxes are selected (by simple random sampling) (sampling unit 3 -SU3);
finally, from each of these boxes a number of fishes is selected (using simple random sampling without replacement) and measured (sampling unit 4 -SU4).

In the last stage the length composition is calculated from the sampling unit 4. Then, by successive back-calculations, the sampled length composition is raised to the length composition of the stratum.
Table 7.1 summarizes our sampling scheme. The notation is the same as that used in previous chapters. In order to simplify the explanation, only one stratum (one month, one port and one fleet component) has been considered. The procedure is repeated independently, for each stratum.
Table 7.1
Summary of a sampling scheme for
estimating the length composition of landings
![]() | h is the index of a stratum: one fleet component, one port and one month h=1…K | ![]() |
| (Nh) total number of days (stratum size) in stratum h | i, index for days i=1…Nh | (nh) number of days sampled (sample size) in stratum h (simple random sampling) |
| (Nhi) total number of landings in day i in stratum h | j, index for landings j=1…Nhi | (nhi) total number of landings sampled in day i in stratum h (simple random sampling) |
| (Nhij) total number of boxes in landing j of day i in stratum h | k, index for boxes k=1…Nhij | (nhij) total number of boxes sampled from landing j of day i in stratum h (simple random sampling) |
| (Nhijk) total number of fishes in box k of landing j from day i in stratum h | l, index for fishes l=1…Nhijk | (nhijk) total number of fishes sampled from box k of landing j from day i in stratum h (simple random sampling) |
Sampling
The aim of this example was to illustrate how to raise successive estimates of each stage up to the stratum level. However to simplify the presentation we have reduced the number of some sampling units to one (as mentioned in the text this is not advisable in practical cases). Another simplification was to eliminate the first sub-index (days).
Let us consider once again the Sardinella aurita example in our particular area. The area under consideration has three landing ports (Port A, Port B and Port C). Each port has two fleet components fishing for Sardinella aurita (trawlers and small seiners). The objective is to estimate the annual (12 months) length composition of the landings. Remember that according to sampling theory, it is mandatory that all strata be sampled. In this particular case study there are hence 72 strata.
Only the sampling of one stratum (Trawlers, Port A and January) is worked in detail, with the aim of simplifying the calculations. Also for simplicity the sub-index for stratum is not indicated in the text and tables. The procedure is carried out for all the 72 strata in order to obtain the annual length composition.
One day (“Day 1”) is selected (by simple random sampling) from the 31 days of the month of January in Port A. During the selected day, 30 trawlers landed Sardinella aurita at Port A. Out of the 30 landings, three landings (“Landing 1”, “Landing 2” and “Landing 3”) were selected (by simple random sampling).
“Landing 1” was composed of 20 boxes in total, “Landing 2” of 10 boxes in total and “Landing 3” of 8 boxes in total. From “Landing 1” two boxes (“Box 1.1” and “Box 1.2”) out of the 20 are selected; from “Landing 2” one box (“Box 2.1”) is selected out of 10 and from “Landing 3” one box is also selected (“Box 3.1”) out of 8.
All boxes were selected using a simple random sampling method. For simplicity more complicated sampling methods were avoided, e.g. selection with probabilities proportional to sizes of landings measured by the number of boxes.
From each of the boxes selected one bucket (“Bucket 1.1.1”, “Bucket 1.2.1”, “Bucket 2.1.1”, and “Bucket 3.1.1”) of fishes is sampled (simple random sampling without replacement). The number of fishes in each bucket is:
Previous calculations estimated that the weight of a box of Sardinella aurita was 30kg and the weight of a bucket of Sardinella aurita was 10 kg.
The length of each fish is measured with the same observation technique, and the fishes are grouped into length classes. In the example 10 length classes are considered.
The example could be represented by the following figure (Figure 7.2):

Calculations
The measurements are organized in frequency tables (one table for each bucket).
Table 7.2
Frequency table for fish length -fish from buckets
| Bucket 1.1.1 | Bucket 1.2.1 | Bucket 2.1.1 | Bucket 3.1.1 | ||||
| Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency |
| l1 | 4 | l1 | 0 | l1 | 8 | l1 | 8 |
| l2 | 8 | l2 | 0 | l2 | 10 | l2 | 14 |
| l3 | 30 | l3 | 5 | l3 | 23 | l3 | 24 |
| l4 | 35 | l4 | 7 | l4 | 30 | l4 | 30 |
| l5 | 37 | l5 | 17 | l5 | 30 | l5 | 37 |
| l6 | 30 | l6 | 23 | l6 | 27 | l6 | 38 |
| l7 | 27 | l7 | 20 | l7 | 20 | l7 | 27 |
| l8 | 10 | l8 | 15 | l8 | 15 | l8 | 10 |
| l9 | 8 | l9 | 10 | l9 | 12 | l9 | 8 |
| l10 | 1 | l10 | 5 | l10 | 7 | l10 | 4 |
| Total | 190 | Total | 102 | Total | 182 | Total | 200 |
The sample frequencies are raised to the total of each box. In order to do this, a factor, RF, that is the quotient of the box weight to the sample weight (in this particular case the sample weight is the weight of the full bucket of Sardinella aurita), is needed. It is known that the weight of each box is 30 kg and the weight of each bucket is 10 kg, therefore the factor would be:
RF = box weight / sample weight = 30/10 = 3
Each frequency is multiplied by the respective RF, in this case equal to 3. Table 7.3 shows the results.
Table 7.3
Length composition by box after raising frequencies
| Box 1.1 | Box 1.2 | Box 2.1 | Box 3.1 | ||||
| Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency |
| l1 | 12 | l1 | 0 | l1 | 24 | l1 | 24 |
| l2 | 24 | l2 | 0 | l2 | 30 | l2 | 42 |
| l3 | 90 | l3 | 15 | l3 | 69 | l3 | 72 |
| l4 | 105 | l4 | 21 | l4 | 90 | l4 | 90 |
| l5 | 111 | l5 | 51 | l5 | 90 | l5 | 111 |
| l6 | 90 | l6 | 69 | l6 | 81 | l6 | 114 |
| l7 | 81 | l7 | 60 | l7 | 60 | l7 | 81 |
| l8 | 30 | l8 | 45 | l8 | 45 | l8 | 30 |
| l9 | 24 | l9 | 30 | l9 | 36 | l9 | 24 |
| l10 | 3 | l10 | 15 | l10 | 21 | l10 | 12 |
Now the length composition of each landing is calculated. In the case of Landing 1, since two different boxes were sampled, the length compositions of Box 1.1 and Box 1.2 are summed (Table 7.4).
Table 7.4
Length composition by landing (boxes 1.1 and 1.2 are
summed)
| Box 1.1 + Box 1.2 | Box 2.1 | Box 3.1 | |||
| Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency |
| l1 | 12 | l1 | 24 | l1 | 24 |
| l2 | 24 | l2 | 30 | l2 | 42 |
| l3 | 105 | l3 | 69 | l3 | 72 |
| l4 | 126 | l4 | 90 | l4 | 90 |
| l5 | 162 | l5 | 90 | l5 | 111 |
| l6 | 159 | l6 | 81 | l6 | 114 |
| l7 | 141 | l7 | 60 | l7 | 81 |
| l8 | 75 | l8 | 45 | l8 | 30 |
| l9 | 54 | l9 | 36 | l9 | 24 |
| l10 | 18 | l10 | 21 | l10 | 12 |
The frequencies of the boxes are raised to the total of each landing. The raising factor, RF, is the quotient of the total number of boxes landed by the total number of boxes sampled. Therefore the factors are:



Table 7.5 shows the results of this raising.
Table 7.5
Length composition of the landings sampled
| Landing 1 | Landing 2 | Landing 3 | |||
| Classes (cm) | Frequency | Classes (cm) | Frequency | Classes (cm) | Frequency |
| l1 | 120 | l1 | 240 | l1 | 192 |
| l2 | 240 | l2 | 300 | l2 | 336 |
| l3 | 1050 | l3 | 690 | l3 | 576 |
| l4 | 1260 | l4 | 900 | l4 | 720 |
| l5 | 1620 | l5 | 900 | l5 | 888 |
| l6 | 1590 | l6 | 810 | l6 | 912 |
| l7 | 1410 | l7 | 600 | l7 | 648 |
| l8 | 750 | l8 | 450 | l8 | 240 |
| l9 | 540 | l9 | 360 | l9 | 192 |
| l10 | 180 | l10 | 210 | l10 | 96 |
The length compositions of the sampled landings are now summed as shown in Table 7.6.
Table 7.6
Sum of the length composition of landings
| Landing 1 + Landing 2 + Landing 3 | |
| Classes (cm) | Frequency |
| l1 | 552 |
| l2 | 876 |
| l3 | 2316 |
| l4 | 2880 |
| l5 | 3408 |
| l6 | 3312 |
| l7 | 2658 |
| l8 | 1440 |
| l9 | 1092 |
| l10 | 486 |
In order to raise the length composition to the total landing of the day, a factor, RF, that is, the quotient between the total number of landings during the day by the total number of landings sampled, is needed. Therefore the raising factor would be:

Table 7.7 gives the final result.
Table 7.7
Length composition for the
total landings of trawlers in Port A
on January 1
| Total landings of day | |
| Classes (cm) | Frequency |
| l1 | 5520 |
| l2 | 8760 |
| l3 | 23160 |
| l4 | 28800 |
| l5 | 34080 |
| l6 | 33120 |
| l7 | 26580 |
| l8 | 14400 |
| l9 | 10920 |
| l10 | 4860 |
The length composition of the total landings from the stratum (Trawlers, Port A and January) is obtained applying procedures similar to the ones just described to all the days sampled, summing them, and then raising the total length composition to the whole month. In this last raising, the raising factor is the ratio between the total number of days with landings in the month and the number of days sampled,

The total length composition of Sardinella aurita landings during a month, and during a year are obtained adding together the raised length compositions for the month or for the year.