The unrestricted, simple random sample is the simplest form of probability sampling. Since all probability samples must provide a known nonzero chance of selection for each population element, the simple random sample is considered a special case in which each population element has a known and equal chance of selection. In this section, we use the simple random sample to build a foundation for understanding sampling procedures and choosing probability samples.

1. Simple Random Sampling

In simple random sampling, all study objects have an equal chance of being included in the sample. Researchers begin with a complete list of all members of a population and then choose sample items at random. It should be noted that in simple random sampling, each study object is selected completely independently of other objects.

The sampling process involves assigning a unique identification number to each study object in the sampling frame. After this, the researcher must design a method of selecting study objects in a manner that allows all equal chance of being selected. One way of doing this is writing these identification numbers on small pieces of paper, mixing them thoroughly in a box, and then picking the papers without looking. The numbers on the pieces of paper picked identify the study objects to be included in the sample. In some cases, however, this procedure (lottery method) may be impractical or tedious.

Another procedure used in selecting study objects in simple random sampling involves the use of tables of random numbers. The researcher begins picking randomly objects from any preselected place in the table of random numbers. Then s/he systematically chooses numbers by either moving vertically or horizontally. The sample will therefore consist of the study objects whose numbers are chosen.

2. Complex probability Sampling

Simple random sampling is often impractical. It requires a population list that is often not available. The design may also be wasteful because it fails to use all the information about a population. In addition, the carrying out of a simple random design may be expensive in time and money. These problems have led to the development of alternative designs that are superior to the simple random design in statistical and/or economic efficiency. A more efficient sample in a statistical sense is one that provides a given precision (standard error of the mean) with a smaller sample size. A sample that is economically more efficient is one that provides a desired precision at a lower dollar cost. We achieve this with designs that enable us to lower the costs of data collecting, usually through reduced travel expense and interviewer time. In the discussion that follows, four alternative probability sampling approaches are considered:

systematic, stratified, cluster and multi-stage.

3. Systematic Sampling

This method is frequently used in production and quality control sampling. In this approach, every n‘th element in the population is sampled, beginning with a random start of an element in the range of 1 to n. After a randomly selected start point(s) a sample item would be selected every n‘th item. Assume that in an assembly line it was decided to sample every 100th item and a start point of 67 was chosen randomly, the sample would be the following items: 67th; 167th; 267th; 367th; and so on The gap between selections is known as the sampling interval and is itself often randomly selected.

A concern with this technique is the possible periodicity in the population that may coincide with the sampling interval and cause bias.

4. Stratified Sampling

Most populations can be segregated into several mutually exclusive sub-populations, or strata. Thus, the process by which the sample is constrained to include elements from each of the segments is called stratified random sampling. There are three reasons why a researcher chooses a stratified sample:

- To increase a sample‘s statistical efficiency;
- To provide adequate data for analysing the various subpopulations, and
- To enable different research methods and procedures to be used in different strata.

With the ideal stratification, each stratum is homogeneous internally and heterogeneous with other strata.

The size of the strata samples is calculated with two pieces of information:

- How large the total sample should be and
- How the total sample should be allocated among strata.