Simple random sample
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In statistics, a simple random sample is a group of subjects (a sample) chosen from a larger group (a population). Each subject from the population is chosen randomly and entirely by chance, such that each subject has the same probability of being chosen at any stage during the sampling process. This process and technique is known as Simple Random Sampling, and should not be confused with Random Sampling.
In small populations such sampling is typically done "without replacement", i.e., one deliberately avoids choosing any member of the population more than once. An unbiased random selection of subjects is important so that in the long run, the sample represents the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement.
Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete sampling frame, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population.
Advantages are that it is free of classification error, and it requires minimum advance knowledge of the population. It best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items. If these conditions are not true, stratified sampling or cluster sampling may be a better choice.
[edit] Distinction between a random sample and a simple random sample
In a simple random sample, not only does each element have the same probability of being selected for the sample, but furthermore every possible sample has the same probability. The latter, stronger property distinguishes the notion of "simple random sample" from the notion of just "random sample".
Consider a sample of 4 integers from the population of integers 1 through 8. One way to draw this sample is to flip a coin. If the coin comes up heads, then select the four even integers. If the coin comes up tails, then select the four odd integers. This is a random sample because the probability that any one element appears in the sample is exactly 50%. But it is not a simple random sample because one possible sample (1,3,5,7) has 50% probability while another possible sample (1,2,3,4) has 0% probability. Since there are 8!/(4!4!)=70 different samples of size 4 from a population of 8, for a sample to be a simple random sample, each sample has to have probability 1/70.
[edit] Sampling a dichotomous population
If the members of the population come in two kinds, say "red" and "black", one can consider the distribution of the number of red elements in a sample of a given size. That distribution depends on the numbers of red and black elements in the full population. For a simple random sample with replacement, the distribution is a binomial distribution. For a simple random sample without replacement, one obtains a hypergeometric distribution.
