# Population

## Population and sampling frame

A sampling study starts with a number of questions that refer to a certain domain of interest. That domain is called the population and is defined as the totality of all elements. It is important to distinguish this definition from the definition of a biological population that is the collection of inter-breeding organisms of a particular species (Zöhrer 1980 ).

The number of elements from which a sample should be drawn is called the sampling frame which is a list of all elements that can be selected during statistical sampling (all elements that have a inclusion probability larger than 0). It is important to note that a sampling frame has the property that we can identify every single element and include any in our sample.

In the ideal case the sampling frame contains all elements of a population, however one can imagine reasons for differences of both. It is good practice that both, population and sampling frame, should be clearly defined for any sampling study. Reasons for a sample frame that is smaller as the population is for example, that parts of the population can not be sampled, because they are not accessible. In forest inventories we can imagine, that areas with extreme steep slopes can not be sampled. In those cases one should consider to re-define the population. Note:
By means of a sampling study one is able to derive statistical sound estimations for the part of the population that is in the sampling frame. We can derive estimates for those sampling units that have a probability to be selected only! If the population is larger than the sampling frame, we can not justify any estimations assigned to the whole population.

### The sampling frame in forest inventories

In forestry we are typically interested in estimating variables of forests or trees. Nevertheless the sampling frame is rarely the set of all trees in a forest area, but the area itself. This area consists of an infinite number of dimensionless points from which a certain number is selected as sample points (Kleinn 2007). This definition is also called an areal sampling frame. Around such a sample point we define a certain area that is the sample plot where the observation one makes on this area is assigned to the respective point. Important!
The elements that are sampled and of which the sample frame consists are typically the sample plots and not single trees! In other words: one selects areas in the forest (and observe the trees on these plots) and not single trees. This fact has far reaching consequences for the statistical issues, for example for the definition of the population.

In contrast to the infinite size of the sample frame one obviously can only observe a discrete number of trees (and combinations of trees) in the area of interest. The infinite sample frame can be decomposed in areas related to this discrete number of possible clusters or samples of trees. More information about the decomposition of the sample frame can be found in an article about the so called jigsaw puzzle view.