Systematic sampling

Last updated May 14, 2024

In survey methodology, one-dimensional systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equiprobability method.^[1] This applies in particular when the sampled units are individuals, households or corporations. When a geographic area is sampled for a spatial analysis, bi-dimensional systematic sampling on an area sampling frame can be applied.^[2]

In one-dimensional systematic sampling, progression through the list is treated circularly, with a return to the top once the list ends. The sampling starts by selecting an element from the list at random and then every k^th element in the frame is selected, where k, is the sampling interval (sometimes known as the skip): this is calculated as:^[3]

k={\frac {N}{n}}

where n is the sample size, and N is the population size.

Using this procedure each element in the population has a known and equal probability of selection (also known as epsem). This makes systematic sampling functionally similar to simple random sampling (SRS). However, it is not the same as SRS because not every possible sample of a certain size has an equal chance of being chosen (e.g. samples with at least two elements adjacent to each other will never be chosen by systematic sampling). It is, however, much more efficient (if the variance within a systematic sample is more than the variance of the population).^{[ citation needed ]}

Systematic sampling is to be applied only if the given population is logically homogeneous, because systematic sample units are uniformly distributed over the population. The researcher must ensure that the chosen sampling interval does not hide a pattern. Any pattern would threaten randomness.

Example: Suppose a supermarket wants to study buying habits of their customers, then using systematic sampling they can choose every 10th or 15th customer entering the supermarket and conduct the study on this sample.

This is random sampling with a system. From the sampling frame, a starting point is chosen at random, and choices thereafter are at regular intervals. For example, suppose you want to sample 8 houses from a street of 120 houses. 120/8=15, so every 15th house is chosen after a random starting point between 1 and 15. If the random starting point is 11, then the houses selected are 11, 26, 41, 56, 71, 86, 101, and 116. As an aside, if every 15th house was a "corner house" then this corner pattern could destroy the randomness of the sample.

If, more frequently, the population is not evenly divisible (suppose you want to sample 8 houses out of 125, where 125/8=15.625), should you take every 15th house or every 16th house? If you take every 16th house, 8*16=128, there is a risk that the last house chosen does not exist. On the other hand, if you take every 15th house, 8*15=120, so the last five houses will never be selected. The random starting point should instead be selected as a non-integer between 0 and 15.625 (inclusive on one endpoint only) to ensure that every house has an equal chance of being selected; the interval should now be non-integral (15.625); and each non-integer selected should be rounded up to the next integer. If the random starting point is 3.6, then the houses selected are 4, 20, 35, 50, 66, 82, 98, and 113, where there are 3 cyclic intervals of 15 and 4 intervals of 16.

To illustrate the danger of systematic skip concealing a pattern, suppose we were to sample a planned neighborhood where each street has ten houses on each block. This places houses No. 1, 10, 11, 20, 21, 30... on block corners; corner blocks may be less valuable, since more of their area is taken up by street front etc. that is unavailable for building purposes. If we then sample every 10th household, our sample will either be made up only of corner houses (if we start at 1 or 10) or have no corner houses (any other start); either way, it will not be representative.

Systematic sampling may also be used with non-equal selection probabilities. In this case, rather than simply counting through elements of the population and selecting every k^th unit, we allocate each element a space along a number line according to its selection probability. We then generate a random start from a uniform distribution between 0 and 1, and move along the number line in steps of 1.

Example: We have a population of 5 units (A to E). We want to give unit A a 20% probability of selection, unit B a 40% probability, and so on up to unit E (100%). Assuming we maintain alphabetical order, we allocate each unit to the following interval:

A: 0 to 0.2 B: 0.2 to 0.6 (= 0.2 + 0.4) C: 0.6 to 1.2 (= 0.6 + 0.6) D: 1.2 to 2.0 (= 1.2 + 0.8) E: 2.0 to 3.0 (= 2.0 + 1.0)

If our random start was 0.156, we would first select the unit whose interval contains this number (i.e. A). Next, we would select the interval containing 1.156 (element C), then 2.156 (element E). If instead our random start was 0.350, we would select from points 0.350 (B), 1.350 (D), and 2.350 (E).

Related Research Articles

In statistics, cluster sampling is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research.

In statistics, sampling bias is a bias in which a sample is collected in such a way that some members of the intended population have a lower or higher sampling probability than others. It results in a biased sample of a population in which all individuals, or instances, were not equally likely to have been selected. If this is not accounted for, results can be erroneously attributed to the phenomenon under study rather than to the method of sampling.

A random variable is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which

Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

In statistics, survey sampling describes the process of selecting a sample of elements from a target population to conduct a survey. The term "survey" may refer to many different types or techniques of observation. In survey sampling it most often involves a questionnaire used to measure the characteristics and/or attitudes of people. Different ways of contacting members of a sample once they have been selected is the subject of survey data collection. The purpose of sampling is to reduce the cost and/or the amount of work that it would take to survey the entire target population. A survey that measures the entire target population is called a census. A sample refers to a group or section of a population from which information is to be obtained.

In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample of individuals from within a statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.

Informally, in frequentist statistics, a confidence interval (CI) is an interval which is expected to typically contain the parameter being estimated. More specifically, given a confidence level $, a CI is a random interval which contains the parameter being estimated % of the time. The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.$

In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value". The error of an observation is the deviation of the observed value from the true value of a quantity of interest. The residual is the difference between the observed value and the estimated value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances.

The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

In population genetics, Ewens's sampling formula describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.

A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards.

Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complex studies, different sample sizes may be allocated, such as in stratified surveys or experimental designs with multiple treatment groups. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.

The Fisher–Yates shuffle is an algorithm for shuffling a finite sequence. The algorithm takes a list of all the elements of the sequence, and continually determines the next element in the shuffled sequence by randomly drawing an element from the list until no elements remain. The algorithm produces an unbiased permutation: every permutation is equally likely. The modern version of the algorithm takes time proportional to the number of items being shuffled and shuffles them in place.

In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample. An essential property of Bernoulli sampling is that all elements of the population have equal probability of being included in the sample.

In statistics, a simple random sample is a subset of individuals chosen from a larger set in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way. In SRS, each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals. Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods.

Reservoir sampling is a family of randomized algorithms for choosing a simple random sample, without replacement, of $k$ items from a population of unknown size $n$ in a single pass over the items. The size of the population $n$ is not known to the algorithm and is typically too large for all $n$ items to fit into main memory. The population is revealed to the algorithm over time, and the algorithm cannot look back at previous items. At any point, the current state of the algorithm must permit extraction of a simple random sample without replacement of size $k$ over the part of the population seen so far.

In survey research, the design effect is a number that shows how well a sample of people may represent a larger group of people for a specific measure of interest. This is important when the sample comes from a sampling method that is different than just picking people using a simple random sample.

References

↑ Levy, Paul (2003). Sampling of Populaions. Methods and applications (3rd ed.). Wiley. pp. 68–98. ISBN 978-0471455066.
↑ Wang, J.F. (2012). "A review of spatial sampling". Spatial Statistics (2): 1–14 – via Elsevier Science Direct.
↑ Ken Black (2004). Business Statistics for Contemporary Decision Making (Fourth (Wiley Student Edition for India) ed.). Wiley-India. ISBN 978-81-265-0809-9.

External links

TRSL – Template Range Sampling Library is a free-software and open-source C++ library that implements systematic sampling behind an (STL-like) iterator interface.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Levy, Paul (2003). Sampling of Populaions. Methods and applications (3rd ed.). Wiley. pp. 68–98. ISBN 978-0471455066.

[2] Wang, J.F. (2012). "A review of spatial sampling". Spatial Statistics (2): 1–14 – via Elsevier Science Direct.

[ken_black_india-3] Ken Black (2004). Business Statistics for Contemporary Decision Making (Fourth (Wiley Student Edition for India) ed.). Wiley-India. ISBN 978-81-265-0809-9.

[1]

[2]

[3]

Systematic sampling

Contents

See also

Related Research Articles

References

External links