Sample

It is very rare that all units in a population can be measured. More commonly, a selection of units from a population is chosen, and only these selected units are measured. A population may be all the people who live in a city, all the children currently enrolled in a school, all the trees of a particular species in a forest, or all the sales transactions of a shop this month. The selection of elements from a population is called a sample. In these examples, the samples would be samples of people, of children, of trees, or of transactions.

Samples are selected according to a sampling scheme. The size of the sample to be selected should be defined in the scheme. Summary statistics are calculated from the sample and are used to estimate population parameters. Choosing the best sample scheme and selecting the most appropriate method to estimate population parameters are the main interests in research in sample surveys.

The most elementary sample scheme is simple random sampling. With a finite population of N units, the number of ways the population can be arranged into subsets of n distinct units can be calculated as follows:

The subsets represent all the possible samples of size n distinct units that could be taken from a finite population of size N. In simple random sampling, all these possible samples have an equal chance of being selected. For example, consider a population in which N = 5. For convenience, the units are labeled A, B, C, D, and E. Ten possible samples of size 3 could be taken:

Samples can be chosen so that each unit appears in the sample only once, or they can be chosen so that each unit can in the sample more than once. When units can appear only once in the sample (as in the example above), the process is called sampling without replacement, and when units can appear more than once, the process is sampling with replacement. Drawing names or numbers out of a hat is an example[Page 856] of sampling without replacement, provided none of the names or numbers is returned to the hat. Selecting units from the population by means of a random table of numbers that correspond to the unit labels is an example of sampling with replacement because the same random number can appear more than once, and hence the same population unit can be selected more than once. Sampling without replacement may be more informative than sampling with replacement because every unit in the sample appears only once and brings new information. If the same unit appeared twice, the second occurrence of the unit would not bring new information.

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