Sampling Proceedures
Statisticians employ different procedures in choosing the observations that will constitute their random samples of the population. The objective of these procedures is to select samples that will be representative of the population from where they originate. These samples, also known as random samples, will have the property that each sample has the same probability of being drawn from the population as another sample.
Simple random sampling is the process of selecting a random sample from a finite or infinite population. There are a total of 
different samples having size n that can be obtained from a finite population of size N. If the n observation are selected randomly, then each of the samples are random samples that have an equal probability 
, of being selected. For an infinite population, the sample is random if each of the n observations correspond to n independent random variables, i.e., each observations is selected independently of the others. Oftentimes, the size of a population under study is large enough so that the population can be considered infinite; if the sample size is small relative to the population size, the population can usually be considered infinite.
EX.
different samples of 5 letters that can be obtained from the 26 letters of the alphabet. If a procedure for selecting a sample of 5 letters was devised such that each of these 65780 samples had an equal probability (equal to 1/65780) of being selected, then the sample selected would be a random sample.
Systematic sampling is the sampling procedure wherein the kth element of the population under study is selected for the sample, with the starting point randomly determined from the first k elements. The value of k is often dependent on the structure and objectives of the sampling experiment, as well as the population under study. In systematic sampling, the sample values are spread more evenly across the population; thus, many systematic samples are highly representative of the population from which they were selected. Yet, one must be careful that the value of k does not result in a sampling interval whose periodicity would compromise the randomness of the observations.
EX. In inspecting a batch of 1000 pipes for defects, we can choose to inspect every 10th item in the batch. The items inspected are the 10th, 20th, 30th, and so on until the 1000th item. In doing so, we must ensure that each 10th item is not specially produced by a special process or machine; otherwise, the proportion of defects in the sample consisting of every 10th item will be fairly homogenous within the sample, and the sample will not be representative of the entire batch of 1000 pipes.
Stratified random sampling is the sampling procedure that divides the population under study into mutually exclusive sub populations, and then selects random samples from each of these sub populations. The sub populations are determined in such a way that the parameter of interest is fairly homogenous within a sub population. By doing so, the variability of the population parameter within each sub population should be considerably less than its variability for the entire population. Oftentimes, there is a relationship between the characteristics of a certain population and the population parameter.
EX. In determining the distribution of incomes among engineers in the Bay Area, we can divide the population of engineers into sub populations corresponding to each major engineering specialty (electrical, chemical, mechanical, civil, industrial, etc.). Random samples can then be selected from each of these sub populations of engineers. The logic behind this sampling structure is the reasonable assumption that the income of an engineer depends, to a large extent, on his particular specialty.
Cluster sampling is the sampling procedure that randomly selects clusters of observations from the population under study, and then chooses all, or a random selection, of the elements of these clusters, as the observations of the sample. Often, cluster sampling is a cost efficient procedure for selecting a sample representative of the population; this is especially true for a widely scattered population.
EX. In conducting a poll of voter preferences for a statewide election, we can randomly select congressional districts (or some other applicable grouping of voters), and then conduct the poll among the people in the chosen congressional district. Many voter polls that utilize cluster sampling would carefully choose their clusters so that they best represent the voter preferences for the whole state.
