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statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
,
quality assurance Quality assurance (QA) is the term used in both manufacturing and service industries to describe the systematic efforts taken to ensure that the product(s) delivered to customer(s) meet with the contractual and other agreed upon performance, design ...
, and
survey methodology Survey methodology is "the study of survey methods". As a field of applied statistics concentrating on human-research surveys, survey methodology studies the sampling of individual units from a population and associated techniques of survey da ...
, sampling is the selection of a subset (a statistical sample) of individuals from within a
statistical population In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypoth ...
to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population in question. Sampling has lower costs and faster data collection than measuring the entire population and can provide insights in cases where it is infeasible to measure an entire population. Each
observation Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...
measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In
survey sampling 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 ...
, weights can be applied to the data to adjust for the sample design, particularly in
stratified sampling In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations. In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each s ...
. Results from
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical ...
are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population.
Acceptance sampling Acceptance sampling uses statistical sampling to determine whether to accept or reject a production lot of material. It has been a common quality control technique used in industry. It is usually done as products leave the factory, or in some c ...
is used to determine if a production lot of material meets the governing
specification A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard. There are different types of technical or engineering specificati ...
s.


Population definition

Successful statistical practice is based on focused problem definition. In sampling, this includes defining the "
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
" from which our sample is drawn. A population can be defined as including all people or items with the characteristic one wishes to understand. Because there is very rarely enough time or money to gather information from everyone or everything in a population, the goal becomes finding a representative sample (or subset) of that population. Sometimes what defines a population is obvious. For example, a manufacturer needs to decide whether a batch of material from
production Production may refer to: Economics and business * Production (economics) * Production, the act of manufacturing goods * Production, in the outline of industrial organization, the act of making products (goods and services) * Production as a stati ...
is of high enough quality to be released to the customer, or should be sentenced for scrap or rework due to poor quality. In this case, the batch is the population. Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions. In other cases, the examined 'population' may be even less tangible. For example,
Joseph Jagger Joseph Hobson Jagger (2 September 1830 – 25 April 1892) was an English textile industry businessman from Yorkshire, who in around 1881 is said to have "broken the bank at Monte Carlo" by identifying and exploiting biases in the wheels of the ...
studied the behaviour of
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
wheels at a casino in
Monte Carlo Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
, and used this to identify a biased wheel. In this case, the 'population' Jagger wanted to investigate was the overall behaviour of the wheel (i.e. the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of its results over infinitely many trials), while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of some physical characteristic such as the
electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allow ...
of
copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...
. This situation often arises when seeking knowledge about the
cause system Causality (also referred to as causation, or cause and effect) is influence by which one Event (relativity), event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''eff ...
of which the ''observed'' population is an outcome. In such cases, sampling theory may treat the observed population as a sample from a larger 'superpopulation'. For example, a researcher might study the success rate of a new 'quit smoking' program on a test group of 100 patients, in order to predict the effects of the program if it were made available nationwide. Here the superpopulation is "everybody in the country, given access to this treatment" – a group which does not yet exist, since the program isn't yet available to all. The population from which the sample is drawn may not be the same as the population about which information is desired. Often there is large but not complete overlap between these two groups due to frame issues etc. (see below). Sometimes they may be entirely separate – for instance, one might study rats in order to get a better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009. Time spent in making the sampled population and population of concern precise is often well spent, because it raises many issues, ambiguities, and questions that would otherwise have been overlooked at this stage.


Sampling frame

In the most straightforward case, such as the sampling of a batch of material from production (acceptance sampling by lots), it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at a forthcoming election (in advance of the election). These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical theory. As a remedy, we seek a
sampling frame In statistics, a sampling frame is the source material or device from which a sample is drawn. It is a list of all those within a population who can be sampled, and may include individuals, households or institutions. Importance of the sampling fra ...
which has the property that we can identify every single element and include any in our sample. The most straightforward type of frame is a list of elements of the population (preferably the entire population) with appropriate contact information. For example, in an
opinion poll An opinion poll, often simply referred to as a survey or a poll (although strictly a poll is an actual election) is a human research survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions ...
, possible sampling frames include an
electoral register An electoral roll (variously called an electoral register, voters roll, poll book or other description) is a compilation that lists persons who are entitled to vote for particular elections in a particular jurisdiction. The list is usually broke ...
and a
telephone directory A telephone directory, commonly called a telephone book, telephone address book, phonebook, or the white and yellow pages, is a listing of telephone subscribers in a geographical area or subscribers to services provided by the organization tha ...
. A probability sample is a sample in which every unit in the population has a chance (greater than zero) of being selected in the sample, and this probability can be accurately determined. The combination of these traits makes it possible to produce unbiased estimates of population totals, by weighting sampled units according to their probability of selection.
''Example: We want to estimate the total income of adults living in a given street. We visit each household in that street, identify all adults living there, and randomly select one adult from each household. (For example, we can allocate each person a random number, generated from a uniform distribution between 0 and 1, and select the person with the highest number in each household). We then interview the selected person and find their income.'' ''People living on their own are certain to be selected, so we simply add their income to our estimate of the total. But a person living in a household of two adults has only a one-in-two chance of selection. To reflect this, when we come to such a household, we would count the selected person's income twice towards the total. (The person who ''is'' selected from that household can be loosely viewed as also representing the person who ''isn't'' selected.)''
In the above example, not everybody has the same probability of selection; what makes it a probability sample is the fact that each person's probability is known. When every element in the population ''does'' have the same probability of selection, this is known as an 'equal probability of selection' (EPS) design. Such designs are also referred to as 'self-weighting' because all sampled units are given the same weight. Probability sampling includes:
Simple Random Sampling In statistics, a simple random sample (or SRS) is a subset of individuals (a sample (statistics), sample) chosen from a larger Set (mathematics), set (a statistical population, population) in which a subset of individuals are chosen randomization, ...
,
Systematic Sampling In survey methodology, 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. In this approach, progression through ...
,
Stratified Sampling In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations. In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each s ...
, Probability Proportional to Size Sampling, and
Cluster may refer to: Science and technology Astronomy * Cluster (spacecraft), constellation of four European Space Agency spacecraft * Asteroid cluster, a small asteroid family * Cluster II (spacecraft), a European Space Agency mission to study t ...
or
Multistage Sampling In statistics, multistage sampling is the taking of samples in stages using smaller and smaller sampling units at each stage. Multistage sampling can be a complex form of cluster sampling because it is a type of sampling which involves dividing ...
. These various ways of probability sampling have two things in common: # Every element has a known nonzero probability of being sampled and # involves random selection at some point.


Nonprobability sampling

Nonprobability sampling is any sampling method where some elements of the population have ''no'' chance of selection (these are sometimes referred to as 'out of coverage'/'undercovered'), or where the probability of selection can't be accurately determined. It involves the selection of elements based on assumptions regarding the population of interest, which forms the criteria for selection. Hence, because the selection of elements is nonrandom, nonprobability sampling does not allow the estimation of sampling errors. These conditions give rise to exclusion bias, placing limits on how much information a sample can provide about the population. Information about the relationship between sample and population is limited, making it difficult to extrapolate from the sample to the population.
''Example: We visit every household in a given street, and interview the first person to answer the door. In any household with more than one occupant, this is a nonprobability sample, because some people are more likely to answer the door (e.g. an unemployed person who spends most of their time at home is more likely to answer than an employed housemate who might be at work when the interviewer calls) and it's not practical to calculate these probabilities.''
Nonprobability sampling methods include
convenience sampling Convenience sampling (also known as grab sampling, accidental sampling, or opportunity sampling) is a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand. This type of samplin ...
,
quota sampling Quota sampling is a method for selecting survey participants that is a non-probabilistic version of stratified sampling. Process In quota sampling, a population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. ...
, and
purposive sampling Sampling is the use of a subset of the population to represent the whole population or to inform about (social) processes that are meaningful beyond the particular cases, individuals or sites studied. Probability sampling, or random sampling, is a ...
. In addition, nonresponse effects may turn ''any'' probability design into a nonprobability design if the characteristics of nonresponse are not well understood, since nonresponse effectively modifies each element's probability of being sampled.


Sampling methods

Within any of the types of frames identified above, a variety of sampling methods can be employed individually or in combination. Factors commonly influencing the choice between these designs include: * Nature and quality of the frame * Availability of auxiliary information about units on the frame * Accuracy requirements, and the need to measure accuracy * Whether detailed analysis of the sample is expected * Cost/operational concerns


Simple random sampling

In a simple random sample (SRS) of a given size, all subsets of a sampling frame have an equal probability of being selected. Each element of the frame thus has an equal probability of selection: the frame is not subdivided or partitioned. Furthermore, any given ''pair'' of elements has the same chance of selection as any other such pair (and similarly for triples, and so on). This minimizes bias and simplifies analysis of results. In particular, the variance between individual results within the sample is a good indicator of variance in the overall population, which makes it relatively easy to estimate the accuracy of results. Simple random sampling can be vulnerable to sampling error because the randomness of the selection may result in a sample that doesn't reflect the makeup of the population. For instance, a simple random sample of ten people from a given country will ''on average'' produce five men and five women, but any given trial is likely to over represent one sex and underrepresent the other. Systematic and stratified techniques attempt to overcome this problem by "using information about the population" to choose a more "representative" sample. Also, simple random sampling can be cumbersome and tedious when sampling from a large target population. In some cases, investigators are interested in research questions specific to subgroups of the population. For example, researchers might be interested in examining whether cognitive ability as a predictor of job performance is equally applicable across racial groups. Simple random sampling cannot accommodate the needs of researchers in this situation, because it does not provide subsamples of the population, and other sampling strategies, such as stratified sampling, can be used instead.


Systematic sampling

Systematic sampling (also known as interval sampling) relies on arranging the study population according to some ordering scheme and then selecting elements at regular intervals through that ordered list. Systematic sampling involves a random start and then proceeds with the selection of every ''k''th element from then onwards. In this case, ''k''=(population size/sample size). It is important that the starting point is not automatically the first in the list, but is instead randomly chosen from within the first to the ''k''th element in the list. A simple example would be to select every 10th name from the telephone directory (an 'every 10th' sample, also referred to as 'sampling with a skip of 10'). As long as the starting point is
randomized In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
, systematic sampling is a type of
probability sampling In statistics, quality assurance, and Statistical survey, survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a population (statistics), statistical population to estimate characteristics o ...
. It is easy to implement and the
stratification Stratification may refer to: Mathematics * Stratification (mathematics), any consistent assignment of numbers to predicate symbols * Data stratification in statistics Earth sciences * Stable and unstable stratification * Stratification, or str ...
induced can make it efficient, ''if'' the variable by which the list is ordered is correlated with the variable of interest. 'Every 10th' sampling is especially useful for efficient sampling from
databases In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases spa ...
. For example, suppose we wish to sample people from a long street that starts in a poor area (house No. 1) and ends in an expensive district (house No. 1000). A simple random selection of addresses from this street could easily end up with too many from the high end and too few from the low end (or vice versa), leading to an unrepresentative sample. Selecting (e.g.) every 10th street number along the street ensures that the sample is spread evenly along the length of the street, representing all of these districts. (Note that if we always start at house #1 and end at #991, the sample is slightly biased towards the low end; by randomly selecting the start between #1 and #10, this bias is eliminated.) However, systematic sampling is especially vulnerable to periodicities in the list. If periodicity is present and the period is a multiple or factor of the interval used, the sample is especially likely to be ''un''representative of the overall population, making the scheme less accurate than simple random sampling. For example, consider a street where the odd-numbered houses are all on the north (expensive) side of the road, and the even-numbered houses are all on the south (cheap) side. Under the sampling scheme given above, it is impossible to get a representative sample; either the houses sampled will ''all'' be from the odd-numbered, expensive side, or they will ''all'' be from the even-numbered, cheap side, unless the researcher has previous knowledge of this bias and avoids it by a using a skip which ensures jumping between the two sides (any odd-numbered skip). Another drawback of systematic sampling is that even in scenarios where it is more accurate than SRS, its theoretical properties make it difficult to ''quantify'' that accuracy. (In the two examples of systematic sampling that are given above, much of the potential sampling error is due to variation between neighbouring houses – but because this method never selects two neighbouring houses, the sample will not give us any information on that variation.) As described above, systematic sampling is an EPS method, because all elements have the same probability of selection (in the example given, one in ten). It is ''not'' 'simple random sampling' because different subsets of the same size have different selection probabilities – e.g. the set has a one-in-ten probability of selection, but the set has zero probability of selection. Systematic sampling can also be adapted to a non-EPS approach; for an example, see discussion of PPS samples below.


Stratified sampling

When the population embraces a number of distinct categories, the frame can be organized by these categories into separate "strata." Each stratum is then sampled as an independent sub-population, out of which individual elements can be randomly selected. The ratio of the size of this random selection (or sample) to the size of the population is called a
sampling fraction In sampling theory, the sampling fraction is the ratio of sample size to population size or, in the context of stratified sampling, the ratio of the sample size to the size of the stratum. The formula for the sampling fraction is :f=\frac, where ...
. There are several potential benefits to stratified sampling. First, dividing the population into distinct, independent strata can enable researchers to draw inferences about specific subgroups that may be lost in a more generalized random sample. Second, utilizing a stratified sampling method can lead to more efficient statistical estimates (provided that strata are selected based upon relevance to the criterion in question, instead of availability of the samples). Even if a stratified sampling approach does not lead to increased statistical efficiency, such a tactic will not result in less efficiency than would simple random sampling, provided that each stratum is proportional to the group's size in the population. Third, it is sometimes the case that data are more readily available for individual, pre-existing strata within a population than for the overall population; in such cases, using a stratified sampling approach may be more convenient than aggregating data across groups (though this may potentially be at odds with the previously noted importance of utilizing criterion-relevant strata). Finally, since each stratum is treated as an independent population, different sampling approaches can be applied to different strata, potentially enabling researchers to use the approach best suited (or most cost-effective) for each identified subgroup within the population. There are, however, some potential drawbacks to using stratified sampling. First, identifying strata and implementing such an approach can increase the cost and complexity of sample selection, as well as leading to increased complexity of population estimates. Second, when examining multiple criteria, stratifying variables may be related to some, but not to others, further complicating the design, and potentially reducing the utility of the strata. Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling). ; A stratified sampling approach is most effective when three conditions are met: # Variability within strata are minimized # Variability between strata are maximized # The variables upon which the population is stratified are strongly correlated with the desired dependent variable. ; Advantages over other sampling methods # Focuses on important subpopulations and ignores irrelevant ones. # Allows use of different sampling techniques for different subpopulations. # Improves the accuracy/efficiency of estimation. # Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size. ; Disadvantages # Requires selection of relevant stratification variables which can be difficult. # Is not useful when there are no homogeneous subgroups. # Can be expensive to implement. ; Poststratification Stratification is sometimes introduced after the sampling phase in a process called "poststratification". This approach is typically implemented due to a lack of prior knowledge of an appropriate stratifying variable or when the experimenter lacks the necessary information to create a stratifying variable during the sampling phase. Although the method is susceptible to the pitfalls of post hoc approaches, it can provide several benefits in the right situation. Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to implement weighting, which can improve the precision of a sample's estimates. ; Oversampling Choice-based sampling is one of the stratified sampling strategies. In choice-based sampling, the data are stratified on the target and a sample is taken from each stratum so that the rare target class will be more represented in the sample. The model is then built on this
biased sample 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 (or non-human f ...
. The effects of the input variables on the target are often estimated with more precision with the choice-based sample even when a smaller overall sample size is taken, compared to a random sample. The results usually must be adjusted to correct for the oversampling.


Probability-proportional-to-size sampling

In some cases the sample designer has access to an "auxiliary variable" or "size measure", believed to be correlated to the variable of interest, for each element in the population. These data can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, as discussed above. Another option is probability proportional to size ('PPS') sampling, in which the selection probability for each element is set to be proportional to its size measure, up to a maximum of 1. In a simple PPS design, these selection probabilities can then be used as the basis for
Poisson sampling In survey methodology, Poisson sampling (sometimes denoted as ''PO 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 sampl ...
. However, this has the drawback of variable sample size, and different portions of the population may still be over- or under-represented due to chance variation in selections. Systematic sampling theory can be used to create a probability proportionate to size sample. This is done by treating each count within the size variable as a single sampling unit. Samples are then identified by selecting at even intervals among these counts within the size variable. This method is sometimes called PPS-sequential or monetary unit sampling in the case of audits or forensic sampling.
''Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and 490 students respectively (total 1500 students), and we want to use student population as the basis for a PPS sample of size three. To do this, we could allocate the first school numbers 1 to 150, the second school 151 to 330 (= 150 + 180), the third school 331 to 530, and so on to the last school (1011 to 1500). We then generate a random start between 1 and 500 (equal to 1500/3) and count through the school populations by multiples of 500. If our random start was 137, we would select the schools which have been allocated numbers 137, 637, and 1137, i.e. the first, fourth, and sixth schools.''
The PPS approach can improve accuracy for a given sample size by concentrating sample on large elements that have the greatest impact on population estimates. PPS sampling is commonly used for surveys of businesses, where element size varies greatly and auxiliary information is often available – for instance, a survey attempting to measure the number of guest-nights spent in hotels might use each hotel's number of rooms as an auxiliary variable. In some cases, an older measurement of the variable of interest can be used as an auxiliary variable when attempting to produce more current estimates. * *


Cluster sampling

Sometimes it is more cost-effective to select respondents in groups ('clusters'). Sampling is often clustered by geography, or by time periods. (Nearly all samples are in some sense 'clustered' in time – although this is rarely taken into account in the analysis.) For instance, if surveying households within a city, we might choose to select 100 city blocks and then interview every household within the selected blocks. Clustering can reduce travel and administrative costs. In the example above, an interviewer can make a single trip to visit several households in one block, rather than having to drive to a different block for each household. It also means that one does not need a
sampling frame In statistics, a sampling frame is the source material or device from which a sample is drawn. It is a list of all those within a population who can be sampled, and may include individuals, households or institutions. Importance of the sampling fra ...
listing all elements in the target population. Instead, clusters can be chosen from a cluster-level frame, with an element-level frame created only for the selected clusters. In the example above, the sample only requires a block-level city map for initial selections, and then a household-level map of the 100 selected blocks, rather than a household-level map of the whole city. Cluster sampling (also known as clustered sampling) generally increases the variability of sample estimates above that of simple random sampling, depending on how the clusters differ between one another as compared to the within-cluster variation. For this reason, cluster sampling requires a larger sample than SRS to achieve the same level of accuracy – but cost savings from clustering might still make this a cheaper option.
Cluster sampling 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 this sampling plan, the total populat ...
is commonly implemented as
multistage sampling In statistics, multistage sampling is the taking of samples in stages using smaller and smaller sampling units at each stage. Multistage sampling can be a complex form of cluster sampling because it is a type of sampling which involves dividing ...
. This is a complex form of cluster sampling in which two or more levels of units are embedded one in the other. The first stage consists of constructing the clusters that will be used to sample from. In the second stage, a sample of primary units is randomly selected from each cluster (rather than using all units contained in all selected clusters). In following stages, in each of those selected clusters, additional samples of units are selected, and so on. All ultimate units (individuals, for instance) selected at the last step of this procedure are then surveyed. This technique, thus, is essentially the process of taking random subsamples of preceding random samples. Multistage sampling can substantially reduce sampling costs, where the complete population list would need to be constructed (before other sampling methods could be applied). By eliminating the work involved in describing clusters that are not selected, multistage sampling can reduce the large costs associated with traditional cluster sampling. However, each sample may not be a full representative of the whole population.


Quota sampling

In quota sampling, the population is first segmented into
mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
sub-groups, just as in
stratified sampling In statistics, stratified sampling is a method of sampling from a population which can be partitioned into subpopulations. In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each s ...
. Then judgement is used to select the subjects or units from each segment based on a specified proportion. For example, an interviewer may be told to sample 200 females and 300 males between the age of 45 and 60. It is this second step which makes the technique one of non-probability sampling. In quota sampling the selection of the sample is non-
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
. For example, interviewers might be tempted to interview those who look most helpful. The problem is that these samples may be biased because not everyone gets a chance of selection. This random element is its greatest weakness and quota versus probability has been a matter of controversy for several years.


Minimax sampling

In imbalanced datasets, where the sampling ratio does not follow the population statistics, one can resample the dataset in a conservative manner called minimax sampling. The minimax sampling has its origin in
Anderson Anderson or Andersson may refer to: Companies * Anderson (Carriage), a company that manufactured automobiles from 1907 to 1910 * Anderson Electric, an early 20th-century electric car * Anderson Greenwood, an industrial manufacturer * Anderson Ra ...
minimax ratio whose value is proved to be 0.5: in a binary classification, the class-sample sizes should be chosen equally. This ratio can be proved to be minimax ratio only under the assumption of
LDA LDA may refer to: Aviation *Localizer type directional aid, an instrument approach to an airport *Landing distance available, the length of runway that is available for the ground run of an airplane landing Law *Legal document assistant, a non-la ...
classifier with Gaussian distributions. The notion of minimax sampling is recently developed for a general class of classification rules, called class-wise smart classifiers. In this case, the sampling ratio of classes is selected so that the worst case classifier error over all the possible population statistics for class prior probabilities, would be the best.


Accidental sampling

Accidental sampling Convenience sampling (also known as grab sampling, accidental sampling, or opportunity sampling) is a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand. This type of sampl ...
(sometimes known as grab, convenience or opportunity sampling) is a type of nonprobability sampling which involves the sample being drawn from that part of the population which is close to hand. That is, a population is selected because it is readily available and convenient. It may be through meeting the person or including a person in the sample when one meets them or chosen by finding them through technological means such as the internet or through phone. The researcher using such a sample cannot scientifically make generalizations about the total population from this sample because it would not be representative enough. For example, if the interviewer were to conduct such a survey at a shopping center early in the morning on a given day, the people that they could interview would be limited to those given there at that given time, which would not represent the views of other members of society in such an area, if the survey were to be conducted at different times of day and several times per week. This type of sampling is most useful for pilot testing. Several important considerations for researchers using convenience samples include: # Are there controls within the research design or experiment which can serve to lessen the impact of a non-random convenience sample, thereby ensuring the results will be more representative of the population? # Is there good reason to believe that a particular convenience sample would or should respond or behave differently than a random sample from the same population? # Is the question being asked by the research one that can adequately be answered using a convenience sample? In social science research,
snowball sampling In sociology and statistics research, snowball sampling (or chain sampling, chain-referral sampling, referral sampling) is a nonprobability sampling technique where existing study subjects recruit future subjects from among their acquaintances. T ...
is a similar technique, where existing study subjects are used to recruit more subjects into the sample. Some variants of snowball sampling, such as respondent driven sampling, allow calculation of selection probabilities and are probability sampling methods under certain conditions.


Voluntary Sampling

The voluntary sampling method is a type of non-probability sampling. Volunteers choose to complete a survey. Volunteers may be invited through advertisements in social media. The target population for advertisements can be selected by characteristics like location, age, sex, income, occupation, education, or interests using tools provided by the social medium. The advertisement may include a message about the research and link to a survey. After following the link and completing the survey, the volunteer submits the data to be included in the sample population. This method can reach a global population but is limited by the campaign budget. Volunteers outside the invited population may also be included in the sample. It is difficult to make generalizations from this sample because it may not represent the total population. Often, volunteers have a strong interest in the main topic of the survey.


Line-intercept sampling

Line-intercept sampling In statistics, more specifically in biostatistics, line-intercept sampling (LIS) is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a “transect”, intersects the element.Kaiser, L, 1983. ...
is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a "transect", intersects the element.


Panel sampling

Panel sampling is the method of first selecting a group of participants through a random sampling method and then asking that group for (potentially the same) information several times over a period of time. Therefore, each participant is interviewed at two or more time points; each period of data collection is called a "wave". The method was developed by sociologist
Paul Lazarsfeld Paul Felix Lazarsfeld (February 13, 1901August 30, 1976) was an Austrian-American sociologist. The founder of Columbia University's Bureau of Applied Social Research, he exerted influence over the techniques and the organization of social resea ...
in 1938 as a means of studying
political campaign A political campaign is an organized effort which seeks to influence the decision making progress within a specific group. In democracies, political campaigns often refer to electoral campaigns, by which representatives are chosen or referend ...
s. This
longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...
sampling-method allows estimates of changes in the population, for example with regard to chronic illness to job stress to weekly food expenditures. Panel sampling can also be used to inform researchers about within-person health changes due to age or to help explain changes in continuous dependent variables such as spousal interaction. Groves, et alia. ''Survey Methodology'' There have been several proposed methods of analyzing
panel data In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data set, data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time s ...
, including
MANOVA In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests i ...
,
growth curves Growth curve can refer to: *Growth curve (statistics), an empirical model of the evolution of a quantity over time. *Growth curve (biology), a statistical growth curve used to model a biological quantity. * Curve of growth (astronomy), the relation ...
, and
structural equation modeling Structural equation modeling (SEM) is a label for a diverse set of methods used by scientists in both experimental and observational research across the sciences, business, and other fields. It is used most in the social and behavioral scienc ...
with lagged effects.


Snowball sampling

Snowball sampling In sociology and statistics research, snowball sampling (or chain sampling, chain-referral sampling, referral sampling) is a nonprobability sampling technique where existing study subjects recruit future subjects from among their acquaintances. T ...
involves finding a small group of initial respondents and using them to recruit more respondents. It is particularly useful in cases where the population is hidden or difficult to enumerate.


Theoretical sampling

Theoretical sampling occurs when samples are selected on the basis of the results of the data collected so far with a goal of developing a deeper understanding of the area or develop theories. Extreme or very specific cases might be selected in order to maximize the likelihood a phenomenon will actually be observable.


Replacement of selected units

Sampling schemes may be ''without replacement'' ('WOR' – no element can be selected more than once in the same sample) or ''with replacement'' ('WR' – an element may appear multiple times in the one sample). For example, if we catch fish, measure them, and immediately return them to the water before continuing with the sample, this is a WR design, because we might end up catching and measuring the same fish more than once. However, if we do not return the fish to the water or
tag and release Tag and release is a form of catch and release fishing in which the angler attaches a tag to the fish, records data such as date, time, place, and type of fish on a standardized postcard, and submits this card to a fisheries agency or conservatio ...
each fish after catching it, this becomes a WOR design.


Sample size determination

Formulas, tables, and power function charts are well known approaches to determine sample size. Steps for using sample size tables: # Postulate the effect size of interest, α, and β. # Check sample size table ## Select the table corresponding to the selected α ## Locate the row corresponding to the desired power ## Locate the column corresponding to the estimated effect size. ## The intersection of the column and row is the minimum sample size required.


Sampling and data collection

Good data collection involves: * Following the defined sampling process * Keeping the data in time order * Noting comments and other contextual events * Recording non-responses


Applications of sampling

Sampling enables the selection of right data points from within the larger data set to estimate the characteristics of the whole population. For example, there are about 600 million tweets produced every day. It is not necessary to look at all of them to determine the topics that are discussed during the day, nor is it necessary to look at all the tweets to determine the sentiment on each of the topics. A theoretical formulation for sampling Twitter data has been developed. In manufacturing different types of sensory data such as acoustics, vibration, pressure, current, voltage, and controller data are available at short time intervals. To predict down-time it may not be necessary to look at all the data but a sample may be sufficient.


Errors in sample surveys

Survey results are typically subject to some error. Total errors can be classified into sampling errors and non-sampling errors. The term "error" here includes systematic biases as well as random errors.


Sampling errors and biases

Sampling errors and biases are induced by the sample design. They include: #
Selection bias Selection bias is the bias introduced by the selection of individuals, groups, or data for analysis in such a way that proper randomization is not achieved, thereby failing to ensure that the sample obtained is representative of the population int ...
: When the true selection probabilities differ from those assumed in calculating the results. # Random sampling error: Random variation in the results due to the elements in the sample being selected at random.


Non-sampling error

Non-sampling errors are other errors which can impact final survey estimates, caused by problems in data collection, processing, or sample design. Such errors may include: # Over-coverage: inclusion of data from outside of the population # Under-coverage: sampling frame does not include elements in the population. # Measurement error: e.g. when respondents misunderstand a question, or find it difficult to answer # Processing error: mistakes in data coding # Non-response or Participation bias: failure to obtain complete data from all selected individuals After sampling, a review should be held of the exact process followed in sampling, rather than that intended, in order to study any effects that any divergences might have on subsequent analysis. A particular problem involves ''non-response''. Two major types of non-response exist:Dillman, D. A., Eltinge, J. L., Groves, R. M., & Little, R. J. A. (2002). "Survey nonresponse in design, data collection, and analysis". In: R. M. Groves, D. A. Dillman, J. L. Eltinge, & R. J. A. Little (Eds.), ''Survey nonresponse'' (pp. 3–26). New York: John Wiley & Sons. * unit nonresponse (lack of completion of any part of the survey) * item non-response (submission or participation in survey but failing to complete one or more components/questions of the survey) In
survey sampling 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 ...
, many of the individuals identified as part of the sample may be unwilling to participate, not have the time to participate (opportunity cost), or survey administrators may not have been able to contact them. In this case, there is a risk of differences between respondents and nonrespondents, leading to biased estimates of population parameters. This is often addressed by improving survey design, offering incentives, and conducting follow-up studies which make a repeated attempt to contact the unresponsive and to characterize their similarities and differences with the rest of the frame. The effects can also be mitigated by weighting the data (when population benchmarks are available) or by imputing data based on answers to other questions. Nonresponse is particularly a problem in internet sampling. Reasons for this problem may include improperly designed surveys, over-surveying (or survey fatigue), and the fact that potential participants may have multiple e-mail addresses, which they don't use anymore or don't check regularly.


Survey weights

In many situations the sample fraction may be varied by stratum and data will have to be weighted to correctly represent the population. Thus for example, a simple random sample of individuals in the United Kingdom might not include some in remote Scottish islands who would be inordinately expensive to sample. A cheaper method would be to use a stratified sample with urban and rural strata. The rural sample could be under-represented in the sample, but weighted up appropriately in the analysis to compensate. More generally, data should usually be weighted if the sample design does not give each individual an equal chance of being selected. For instance, when households have equal selection probabilities but one person is interviewed from within each household, this gives people from large households a smaller chance of being interviewed. This can be accounted for using survey weights. Similarly, households with more than one telephone line have a greater chance of being selected in a random digit dialing sample, and weights can adjust for this. Weights can also serve other purposes, such as helping to correct for non-response.


Methods of producing random samples

*
Random number table Random number tables have been used in statistics for tasks such as selected random samples. This was much more effective than manually selecting the random samples (with dice, cards, etc.). Nowadays, tables of random numbers have been replaced b ...
* Mathematical algorithms for
pseudo-random number generator A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
s * Physical randomization devices such as coins, playing cards or sophisticated devices such as
ERNIE Ernie is a masculine given name, frequently a short form (hypocorism) of Ernest, Ernald, Ernesto, or Verner. It may refer to: People * Ernie Accorsi (born 1941), American football executive * Ernie Adams (disambiguation) * Ernie Afaganis (born c ...


History

Random sampling by using lots is an old idea, mentioned several times in the Bible. In 1786 Pierre Simon
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
estimated the population of France by using a sample, along with
ratio estimator The ratio estimator is a statistical parameter and is defined to be the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymme ...
. He also computed probabilistic estimates of the error. These were not expressed as modern
confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
s but as the sample size that would be needed to achieve a particular upper bound on the sampling error with probability 1000/1001. His estimates used
Bayes' theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
with a uniform
prior probability In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
and assumed that his sample was random.
Alexander Ivanovich Chuprov Alexander Ivanovich Chuprov (Александр Иванович Чупров; 1841–1908) was a professor of political economy and statistics at Moscow University whose lectures provided the standard introduction to economics for late 19th-centu ...
introduced sample surveys to
Imperial Russia The Russian Empire was an empire and the final period of the List of Russian monarchs, Russian monarchy from 1721 to 1917, ruling across large parts of Eurasia. It succeeded the Tsardom of Russia following the Treaty of Nystad, which ended th ...
in the 1870s. In the US the 1936 ''
Literary Digest ''The Literary Digest'' was an influential American general interest weekly magazine published by Funk & Wagnalls. Founded by Isaac Kaufmann Funk in 1890, it eventually merged with two similar weekly magazines, ''Public Opinion'' and '' Current ...
'' prediction of a Republican win in the
presidential election A presidential election is the election of any head of state whose official title is President. Elections by country Albania The president of Albania is elected by the Assembly of Albania who are elected by the Albanian public. Chile The pre ...
went badly awry, due to severe
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
br>
More than two million people responded to the study with their names obtained through magazine subscription lists and telephone directories. It was not appreciated that these lists were heavily biased towards Republicans and the resulting sample, though very large, was deeply flawed.


See also

*
Data collection Data collection or data gathering is the process of gathering and measuring information on targeted variables in an established system, which then enables one to answer relevant questions and evaluate outcomes. Data collection is a research com ...
*
Estimation theory Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...
*
Gy's sampling theory Gy's sampling theory is a theory about the sampling of materials, developed by Pierre Gy from the 1950s to beginning 2000sGy, P (2004), Chemometrics and Intelligent Laboratory Systems, 74, 61-70. in articles and books including: *(1960) Sampling no ...
*
German tank problem In the statistical theory of estimation theory, estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number o ...
*
Horvitz–Thompson estimator In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample. Inverse probability weighting is applied to acco ...
*
Official statistics Official statistics are statistics published by government agencies or other public bodies such as international organizations as a public good. They provide quantitative or qualitative information on all major areas of citizens' lives, such as e ...
*
Ratio estimator The ratio estimator is a statistical parameter and is defined to be the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymme ...
*
Replication (statistics) In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "... the repetition of the ...
*
Random-sampling mechanism A random-sampling mechanism (RSM) is a truthful mechanism that uses sampling in order to achieve approximately-optimal gain in prior-free mechanisms and prior-independent mechanisms. Suppose we want to sell some items in an auction and achieve m ...
*
Resampling (statistics) In statistics, resampling is the creation of new samples based on one observed sample. Resampling methods are: # Permutation tests (also re-randomization tests) # Bootstrapping # Cross validation Permutation tests Permutation tests rely on r ...
*
Pseudo-random number sampling Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution. Methods are typically based on the availability of a unifor ...
*
Sample size determination Sample size determination 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 populatio ...
*
Sampling (case studies) A case study is an in-depth, detailed examination of a particular case (or cases) within a real-world context. For example, case studies in medicine may focus on an individual patient or ailment; case studies in business might cover a particular fi ...
*
Sampling bias 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 (or non-human fa ...
*
Sampling distribution In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were s ...
*
Sampling error In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample ( ...
*
Sortition In governance, sortition (also known as selection by lottery, selection by lot, allotment, demarchy, stochocracy, aleatoric democracy, democratic lottery, and lottocracy) is the selection of political officials as a random sample from a larger ...
*
Survey sampling 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 ...
*
Design effect In survey methodology, the design effect (generally denoted as D_ or D_^2) is a measure of the expected impact of a sampling design on the variance of an estimator for some parameter. It is calculated as the ratio of the variance of an estimator b ...


Notes

The textbook by Groves et alia provides an overview of survey methodology, including recent literature on questionnaire development (informed by
cognitive psychology Cognitive psychology is the scientific study of mental processes such as attention, language use, memory, perception, problem solving, creativity, and reasoning. Cognitive psychology originated in the 1960s in a break from behaviorism, which ...
) : *
Robert Groves Robert Martin Groves (born September 27, 1948) is an American sociologist and expert in survey methodology who has served as the Provost of Georgetown University in Washington, D.C. since August 2012. He also served as the Director of the Unit ...
, et alia. ''Survey methodology'' (2010 2nd ed.
004 004, 0O4, O04, OO4 may refer to: * 004, fictional British 00 Agent * 0O4, Corning Municipal Airport (California) * O04, the Oversea-Chinese Banking Corporation * Abdul Haq Wasiq, Guantanamo detainee 004 * Junkers Jumo 004 turbojet engine * Lauda ...
. The other books focus on the
statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical ...
of survey sampling and require some knowledge of basic statistics, as discussed in the following textbooks: *
David S. Moore David Sheldon Moore is an American statistician, who is known for his leadership of statistics education for many decades. Biography David S. Moore received his A.B. from Princeton University and the Ph.D. from Cornell University in mathematics ...
and George P. McCabe (February 2005). "''Introduction to the practice of statistics''" (5th edition). W.H. Freeman & Company. . * The elementary book by Scheaffer et alia uses quadratic equations from high-school algebra: * Scheaffer, Richard L., William Mendenhal and R. Lyman Ott. ''Elementary survey sampling'', Fifth Edition. Belmont: Duxbury Press, 1996. More mathematical statistics is required for Lohr, for Särndal et alia, and for Cochran (classic): * * * The historically important books by Deming and Kish remain valuable for insights for social scientists (particularly about the U.S. census and the
Institute for Social Research The Institute for Social Research (german: Institut für Sozialforschung, IfS) is a research organization for sociology and continental philosophy, best known as the institutional home of the Frankfurt School and critical theory. Currently a part ...
at the
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
): * * Kish, Leslie (1995) ''Survey Sampling'', Wiley,


References


Further reading

* Singh, G N, Jaiswal, A. K., and Pandey A. K. (2021), Improved Imputation Methods for Missing Data in Two-Occasion Successive Sampling, Communications in Statistics: Theory and Methods. DOI:10.1080/03610926.2021.1944211 *Chambers, R L, and Skinner, C J (editors) (2003), ''Analysis of Survey Data'', Wiley, * Deming, W. Edwards (1975) On probability as a basis for action, ''The American Statistician'', 29(4), pp. 146–152. * Gy, P (2012) ''Sampling of Heterogeneous and Dynamic Material Systems: Theories of Heterogeneity, Sampling and Homogenizing'', Elsevier Science, * Korn, E.L., and Graubard, B.I. (1999) ''Analysis of Health Surveys'', Wiley, * Lucas, Samuel R. (2012). "Beyond the Existence Proof: Ontological Conditions, Epistemological Implications, and In-Depth Interview Research."], ''Quality & Quantity'', . * Stuart, Alan (1962) ''Basic Ideas of Scientific Sampling'', Hafner Publishing Company, New York * * (Portrait of T. M. F. Smith on page 144) * * *


Standards


ISO

* ISO 2859 series * ISO 3951 series


ASTM

* ASTM E105 Standard Practice for Probability Sampling Of Materials * ASTM E122 Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process * ASTM E141 Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling * ASTM E1402 Standard Terminology Relating to Sampling * ASTM E1994 Standard Practice for Use of Process Oriented AOQL and LTPD Sampling Plans * ASTM E2234 Standard Practice for Sampling a Stream of Product by Attributes Indexed by AQL


ANSI, ASQ

* ANSI/ASQ Z1.4


U.S. federal and military standards

*
MIL-STD-105 MIL-STD-105 was a United States defense standard that provided procedures and tables for sampling by attributes based on Walter A. Shewhart, Harry Romig, and Harold F. Dodge sampling inspection theories and mathematical formulas. Widely adopted ...
* MIL-STD-1916


External links

* {{DEFAULTSORT:Sampling (Statistics) Survey methodology Scientific method