Simple random sample
Encyclopedia
In statistics
, a simple random sample is a subset of individuals (a sample
) chosen from a larger set (a population
). Each individual is chosen randomly
and entirely by chance, such that each individual has the same probability
of being chosen at any stage during the sampling process, and each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals. This process and technique is known as simple random sampling, and should not be confused with Random Sampling
.
In small populations and often in large ones, such sampling is typically done "without replacement" ('SRSWOR'), i.e., one deliberately avoids choosing any member of the population more than once. 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 ('SRSWR').
Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the odds of choosing the same individual twice is low.
An unbiased random selection of individuals 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.
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 other than the frame. Its simplicity also makes it relatively easy to interpret data collected via SRS. For these reasons, simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity. If these conditions are not true, stratified sampling
or cluster sampling
may be a better choice.
Let us assume you had a school with 1000 students, divided equally into boys and girls, and you wanted to select 100 of them for further study. You might put all their names in a bucket and then pull 100 names out. Not only does each person have an equal chance of being selected, we can also easily calculate the probability of a given person being chosen, since we know the sample size (n) and the population (N) and it becomes a simple matter of division:
n/N or 100/1000 = 0.10 (10%)
This means that every student in the school has a 10% or 1 in 10 chance of being selected using this method. Further, all combinations of 100 students have the same probability of selection.
If a systematic pattern is introduced into random sampling, it is referred to as "systematic (random) sampling". For instance, if the students in our school had numbers attached to their names ranging from 0001 to 1000, and we chose a random starting point, e.g. 0533, and then pick every 10th name thereafter to give us our sample of 100 (starting over with 0003 after reaching 0993). In this sense, this technique is similar to cluster sampling, since the choice of the first unit will determine the remainder. This is no longer simple random sampling, because some combinations of 100 students have a larger selection probability than others - for instance, {3, 13, 23, ..., 993} has a 1/10 chance of selection, while {1, 2, 3, ..., 100} cannot be selected under this method.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a simple random sample is a subset of individuals (a sample
Sample (statistics)
In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size...
) chosen from a larger set (a population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...
). Each individual is chosen randomly
Randomization
Randomization is the process of making something random; this means:* Generating a random permutation of a sequence .* Selecting a random sample of a population ....
and entirely by chance, such that each individual has the same probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
of being chosen at any stage during the sampling process, and each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals. This process and technique is known as simple random sampling, and should not be confused with Random Sampling
Random sample
In statistics, a sample is a subject chosen from a population for investigation; a random sample is one chosen by a method involving an unpredictable component...
.
In small populations and often in large ones, such sampling is typically done "without replacement" ('SRSWOR'), i.e., one deliberately avoids choosing any member of the population more than once. 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 ('SRSWR').
Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the odds of choosing the same individual twice is low.
An unbiased random selection of individuals 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.
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 other than the frame. Its simplicity also makes it relatively easy to interpret data collected via SRS. For these reasons, simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity. If these conditions are not true, stratified sampling
Stratified sampling
In statistics, stratified sampling is a method of sampling from a population.In statistical surveys, when subpopulations within an overall population vary, it is advantageous to sample each subpopulation independently. Stratification is the process of dividing members of the population into...
or cluster sampling
Cluster sampling
Cluster Sampling is a sampling technique used when "natural" groupings are evident in a statistical population. It is often used in marketing research. In this technique, the total population is divided into these groups and a sample of the groups is selected. Then the required information is...
may be a better choice.
Distinction between a systematic random sample and a simple random sample
In a simple random sample, one person must take a random sample from a population, and not have any order in which one chooses the specific individual.Let us assume you had a school with 1000 students, divided equally into boys and girls, and you wanted to select 100 of them for further study. You might put all their names in a bucket and then pull 100 names out. Not only does each person have an equal chance of being selected, we can also easily calculate the probability of a given person being chosen, since we know the sample size (n) and the population (N) and it becomes a simple matter of division:
n/N or 100/1000 = 0.10 (10%)
This means that every student in the school has a 10% or 1 in 10 chance of being selected using this method. Further, all combinations of 100 students have the same probability of selection.
If a systematic pattern is introduced into random sampling, it is referred to as "systematic (random) sampling". For instance, if the students in our school had numbers attached to their names ranging from 0001 to 1000, and we chose a random starting point, e.g. 0533, and then pick every 10th name thereafter to give us our sample of 100 (starting over with 0003 after reaching 0993). In this sense, this technique is similar to cluster sampling, since the choice of the first unit will determine the remainder. This is no longer simple random sampling, because some combinations of 100 students have a larger selection probability than others - for instance, {3, 13, 23, ..., 993} has a 1/10 chance of selection, while {1, 2, 3, ..., 100} cannot be selected under this method.
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.See also
- Sampling (statistics)Sampling (statistics)In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....
- Probability Proportional to Size Sampling
- Marketing researchMarketing researchMarketing research is "the function that links the consumer, customer, and public to the marketer through information — information used to identify and define marketing opportunities and problems; generate, refine, and evaluate marketing actions; monitor marketing performance; and improve...
- Multistage samplingMultistage samplingMultistage sampling is a complex form of cluster sampling.Advantages * cost and speed that the survey can be done in* convenience of finding the survey sample* normally more accurate than cluster sampling for the same size sampleDisadvantages...
- Nonprobability samplingNonprobability samplingSampling is the use of a subset of the population to represent the whole population. Probability sampling, or random sampling, is a sampling technique in which the probability of getting any particular sample may be calculated. Nonprobability sampling does not meet this criterion and should be...
- Opinion pollOpinion pollAn opinion poll, sometimes simply referred to as a poll is a survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions of a population by conducting a series of questions and then extrapolating generalities in ratio or within confidence...
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- Random sampleRandom sampleIn statistics, a sample is a subject chosen from a population for investigation; a random sample is one chosen by a method involving an unpredictable component...
- Systematic samplingSystematic samplingSystematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equal-probability method, in which every kth element in the frame is selected, where k, the sampling interval , is calculated as:k =...