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This note provides the theoretical background for a series of manuals, studies and computer models that have been prepared with the view of providing some practical guidance in the design and implementation of sample-based fishery surveys. It is a methodological supplement to a family of statistical approaches and computer software, developed by the Fishery Information, Data and Statistics Unit (FIDI) of the FAO Fisheries Department, for assisting member countries in improving and upgrading their national data collection programmes. The presented approach addresses the question of sampling efficiency when data collection is performed under operational constraints, a frequent concern of fishery administrations with limited budget and human resources.

During the design phase of sample-based fishery surveys the question often arises as to what should be the appropriate sample size guaranteeing an acceptable level of reliability for the estimated population parameters. In several occasions, and particularly at the early stages of implementing a fishery statistical monitoring programme, little is known about the distribution and variability of the target population. Consequently, statistical developers tend to initially operate on a large-sample basis, with the intention however of scaling down data collection operations as soon as some guiding statistical indicators, used for improving the cost-effectiveness of the sampling schemes, become available. Usually such indicators can only be formulated and verified after a complete operational cycle (i.e. a year) of a fishery statistical programme, which means that for long periods data collection is performed at high operational capacity. Generally, lack of any a priori guidance on sample size requirements tends to increase the size and complexity of field operations and this, in turn, has a direct impact on the logistical aspects of data collection and data management procedures.

The presented study attempts to provide some answers to the above problem and proposes a supplementary practical tool for use in the design of sample-based fishery surveys. It is clearly not an alternative to conventional statistical techniques and methods. Its main feature is the limited use of probabilistic approaches and the adoption of geometrical criteria and techniques that would be described as "reasonably pessimistic".

The paper makes use of three major assumptions. The first two concern population size and method of sampling. It is assumed that all populations under study are finite and of known size and that samples are always taken at random and without replacement. No other assumptions are made concerning the distribution of the target population, although some knowledge about the general shape of its distribution should result in more realistic and less pessimistic indicators.

The third assumption concerns a geometrical property of sampling efficiency when it is expressed as an exponential function of the sample size.

The method is based on a simple definition of a relative index of proximity of a sample mean to the population mean. This index, referred to as accuracy A throughout the paper, has several geometrical properties that are only a function of the population size. Using these properties it is possible to formulate geometrical constraints that can be used for predicting a lower limit for accuracy at any sample size between 1 and the population size. Construction of these geometrical constraints is fairly simple and can be quickly achieved through the use of standard computing tools (such as worksheets) that are available in most personal computers. The method can also be generalized for infinite populations and a description of the related analytical model is given in the last section.

Formulation of geometrical limits for sampling accuracy is based on the following stepwise process:

(a) Definition of sampling accuracy;

(b) Finding a relationship between accuracy and sample size;

(c) Determining the global domain and its boundary for all possible patterns of accuracy growth;

(d) Transforming the found relationship of accuracy-sample size into a new system of coordinates;

(e) Setting-up new geometrical limits, this time only for the most probable patterns of accuracy growth.

The various observations and conclusions of the above process are based on mathematically demonstrated propositions.

Some technical reports and notes concerning methodological and operational aspects of sample-based fishery surveys are given in the list of references. Admittedly the list is very limited for despite his efforts the author has been unable to identify in the available literature any paper or study on the subject of geometrical approaches and a priori indicators of sampling efficiency, although he recognizes that such literature must certainly exist.

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