Sampling (statistics)

# Sampling (statistics)

Discussion

Encyclopedia
In statistics
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....

and survey methodology
Statistical survey
Survey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....

, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population.

Researchers rarely survey the entire population because the cost of a census
Census
A census is the procedure of systematically acquiring and recording information about the members of a given population. It is a regularly occurring and official count of a particular population. The term is used mostly in connection with national population and housing censuses; other common...

is too high. The three main advantages of sampling are that the cost is lower, data collection is faster, and since the data set is smaller it is possible to ensure homogeneity and to improve the accuracy and quality of the data.

Each observation
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...

measures one or more properties (such as weight, location, color) of observable bodies distinguished as independent objects or individuals. In survey sampling
Survey sampling
In statistics, survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey.A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to...

, weights can be applied to the data to adjust for the 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...

design, particularly 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...

(blocking
Blocking
Blocking may refer to:- Telecommunications and computing :*Block , a sequence of bytes or bits, having a nominal length*Block , technical measures to restrict users' access to certain internet resources...

). Results from probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistical theory
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 inference, and the actions and deductions that...

are employed to guide practice. In business and medical research, sampling is widely used for gathering information about a population.

## Process

The sampling process comprises several stages:
• Defining the population of concern
• Specifying a sampling frame, a set of items or events possible to measure
• Specifying a sampling method for selecting items or events from the frame
• Determining the sample size
• Implementing the sampling plan
• Sampling and data collecting

## Population definition

Successful statistical practice is based on focused problem definition. In sampling, this includes defining the 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...

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 that which defines a population is obvious. For example, a manufacturer needs to decide whether a batch
Batch
Batch may refer to:Food and drink*Batch , an alcoholic fruit beverage*Batch loaf, a type of bread popular in Ireland*A dialect term for a bread roll used in Nuneaton and Coventry, England*Small batch, bourbon whiskey blended from selected barrels...

of material from production
Batch production
Batch production is a technique used in manufacturing, in which the object in question is created stage by stage over a series of workstations. Batch production is common in bakeries and in the manufacture of sports shoes, pharmaceutical ingredients , inks, paints and adhesives. In the manufacture...

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 we need 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, our 'population' may be even less tangible. For example, Joseph Jagger
Joseph Jagger
Joseph Hobson Jagger was a British engineer, widely known as The Man Who Broke the Bank at Monte Carlo, though he is not the only person to have done so. His name is sometimes reported as Jaggers, but the International Genealogical Index indicates that Jagger is more likely...

studied the behaviour of roulette
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

wheels at a casino in Monte Carlo
Monte Carlo
Monte Carlo is an administrative area of the Principality of Monaco....

, 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
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

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 of copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

.

This situation often arises when we seek knowledge about the cause system 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.

Note also that the population from which the sample is drawn may not be the same as the population about which we actually want information. 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, we might study rats in order to get a better understanding of human health, or we 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 sentencing of a batch of material from production (acceptance sampling by lots), it is possible 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 possible. 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 actually 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
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....

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
Opinion poll
An 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...

, possible sampling frames include an electoral register
Electoral register
The electoral roll is a listing of all those registered to vote in a particular area. The register facilitates the process of voting, helps to prevent fraud and may also be used to select people for jury duty...

and a telephone directory
Telephone directory
A telephone directory is a listing of telephone subscribers in a geographical area or subscribers to services provided by the organization that publishes the directory...

.

## Probability and nonprobability sampling

A probability sampling scheme is one 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
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

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
Simple random sample
In statistics, a simple random sample is a subset of individuals chosen from a larger set . 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...

, Systematic Sampling
Systematic sampling
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 equal-probability method, in which every kth element in the frame is selected, where k, the sampling interval , is calculated as:k =...

, 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...

, Probability Proportional to Size Sampling, and Cluster
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...

or Multistage Sampling
Multistage sampling
Multistage 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...

. These various ways of probability sampling have two things in common:
1. Every element has a known nonzero probability of being sampled and
2. involves random selection at some point.

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
Selection bias
Selection bias is a statistical bias in which there is an error in choosing the individuals or groups to take part in a scientific study. It is sometimes referred to as the selection effect. The term "selection bias" most often refers to the distortion of a statistical analysis, resulting from the...

, 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 ccidental sampling, quota sampling
Quota sampling
Quota sampling is a method for selecting survey participants. In quota sampling, a population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgment is used to select the subjects or units from each segment based on a specified proportion. For example,...

and purposive sampling. 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 frame 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
Simple random sample
In statistics, a simple random sample is a subset of individuals chosen from a larger set . 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...

('SRS') of a given size, all such subsets of the frame are given an equal probability. 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 minimises 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.

However, SRS 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 overrepresent one sex and underrepresent the other. Systematic and stratified techniques, discussed below, attempt to overcome this problem by using information about the population to choose a more representative sample.

SRS may also be cumbersome and tedious when sampling from an unusually 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. SRS cannot accommodate the needs of researchers in this situation because it does not provide subsamples of the population. Stratified sampling, which is discussed below, addresses this weakness of SRS.

Simple random sampling is always an EPS design (equal probability of selection), but not all EPS designs are simple random sampling.

### Systematic sampling

Systematic sampling
Systematic sampling
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 equal-probability method, in which every kth element in the frame is selected, where k, the sampling interval , is calculated as:k =...

relies on arranging the target 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 kth 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 kth 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
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 ....

, systematic sampling is a type of probability sampling. It is easy to implement and the stratification
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...

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.

Example: Suppose we wish to sample people from a long street that starts in a poor district (house #1) and ends in an expensive district (house #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 unrepresentative of the overall population, making the scheme less accurate than simple random sampling.

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.

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 {4,14,24,...,994} has a one-in-ten probability of selection, but the set {4,13,24,34,...} 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

Where 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. 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:
1. Variability within strata are minimized
2. Variability between strata are maximized
3. The variables upon which the population is stratified are strongly correlated with the desired dependent variable.

1. Focuses on important subpopulations and ignores irrelevant ones.
2. Allows use of different sampling techniques for different subpopulations.
3. Improves the accuracy/efficiency of estimation.
4. Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size.

1. Requires selection of relevant stratification variables which can be difficult.
2. Is not useful when there are no homogeneous subgroups.
3. 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. 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
Poisson sampling
In the theory of finite population sampling, Poisson sampling is a sampling process where each element of the population that is sampled is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample during the drawing of a single sample.Each element...

. 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. To address this problem, PPS may be combined with a systematic approach.

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
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....

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 generally increases the variability of sample estimates above that of simple random sampling, depending on how the clusters differ between themselves, as compared with 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
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...

is commonly implemented as multistage sampling
Multistage sampling
Multistage 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...

. 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.

### Quota sampling

In quota sampling, the population is first segmented into mutually exclusive
Mutually exclusive
In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both....

sub-groups, just as in 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...

. 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. 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 many years.

### Convenience sampling or Accidental Sampling

Convenience sampling (sometimes known as grab 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 he/she 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:
1. 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?
2. 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?
3. Is the question being asked by the research one that can adequately be answered using a convenience sample?

In social science research, snowball sampling
Snowball sampling
In sociology and statistics research, snowball sampling is a non-probability sampling technique where existing study subjects recruit future subjects from among their acquaintances. Thus the sample group appears to grow like a rolling snowball...

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.

### Line-intercept sampling

Line-intercept sampling
Line-intercept sampling
In statistics, line-intercept sampling 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 ....

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 the same information again several times over a period of time. Therefore, each participant is given the same survey or interview at two or more time points; each period of data collection is called a "wave". This longitudinal
Longitudinal study
A longitudinal study is a correlational research study that involves repeated observations of the same variables over long periods of time — often many decades. It is a type of observational study. Longitudinal studies are often used in psychology to study developmental trends across the...

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. There have been several proposed methods of analyzing panel data
Panel data
In statistics and econometrics, the term panel data refers to multi-dimensional data. Panel data contains observations on multiple phenomena observed over multiple time periods for the same firms or individuals....

, including MANOVA
MANOVA
Multivariate analysis of variance is a generalized form of univariate analysis of variance . It is used when there are two or more dependent variables. It helps to answer : 1. do changes in the independent variable have significant effects on the dependent variables; 2. what are the interactions...

, growth curves, and structural equation modeling
Structural equation modeling
Structural equation modeling is a statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions...

with lagged effects.

## 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 (e.g. if we eat the fish), this becomes a WOR design.

## Sample size

Formulas, tables, and power function charts are well known approaches to determine sample size
Sample size
Sample size determination is the act of choosing the number of observations 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...

.

### Steps for using sample size tables

1. Postulate the effect size of interest, α, and β.
2. Check sample size table
1. Select the table corresponding to the selected α
2. Locate the row corresponding to the desired power
3. Locate the column corresponding to the estimated effect size.
4. 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

Most sampling books and papers written by non-statisticians focus only in the data collection aspect, which is just a small though important part of the sampling process.

## 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:
1. Selection bias
Selection bias
Selection bias is a statistical bias in which there is an error in choosing the individuals or groups to take part in a scientific study. It is sometimes referred to as the selection effect. The term "selection bias" most often refers to the distortion of a statistical analysis, resulting from the...

: When the true selection probabilities differ from those assumed in calculating the results.
2. Random sampling error
Sampling error
-Random sampling:In statistics, sampling error or estimation error is the error caused by observing a sample instead of the whole population. The sampling error can be found by subtracting the value of a parameter from the value of a statistic...

: Random variation in the results due to the elements in the sample being selected at random.

### Non-sampling error

Non-sampling errors are caused by other problems in data collection and processing. They include:
1. Overcoverage: Inclusion of data from outside of the population.
2. Undercoverage: Sampling frame does not include elements in the population.
3. Measurement error: E.g. when respondents misunderstand a question, or find it difficult to answer.
4. Processing error: Mistakes in data coding.
5. Non-response: 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 is that of non-response.
Two major types of nonresponse exist: unit nonresponse (referring to lack of completion of any part of the survey) and item nonresponse (submission or participation in survey but failing to complete one or more components/questions of the survey).
In survey sampling
Survey sampling
In statistics, survey sampling describes the process of selecting a sample of elements from a target population in order to conduct a survey.A survey may refer to many different types or techniques of observation, but in the context of survey sampling it most often involves a questionnaire used to...

, 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 include improperly designed surveys, over-surveying (or survey fatigue), and the fact that potential participants hold 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 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.

## History

Random sampling by using lots is an old idea, mentioned several times in the Bible. In 1786 Pierre Simon Laplace estimated the population of France by using a sample, along with ratio estimator. He also computed probabilistic estimates of the error. These were not expressed as modern confidence interval
Confidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...

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
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....

with a uniform prior probability
Prior probability
In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...

and assumed that his sample was random.

In the USA the 1936 Literary Digest
Literary Digest
The Literary Digest was an influential 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 Opinion.-History:...

prediction of a Republican win in the presidential election went badly awry, due to severe bias
Bias
Bias is an inclination to present or hold a partial perspective at the expense of alternatives. Bias can come in many forms.-In judgement and decision making:...

http://online.wsj.com/public/article/SB115974322285279370-_rk13XDUHmIcnA8DYs5VUscZG94_20071001.html?mod=rss_free. 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.

• Acceptance sampling
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 and particularly the military for contracts and procurement. It is usually done as products leave the factory, or in...

• Data collection
Data collection
Data collection is a term used to describe a process of preparing and collecting data, for example, as part of a process improvement or similar project. The purpose of data collection is to obtain information to keep on record, to make decisions about important issues, to pass information on to...

• Official statistics
Official statistics
Official statistics are statistics published by government agencies or other public bodies such as international organizations. They provide quantitative or qualitative information on all major areas of citizens' lives, such as economic and social development, living conditions, health, education,...

• Replication (statistics)
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 set of all the treatment combinations to be compared in...

• Sample (statistics)
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...

• Sampling (case studies)
• Sampling error
Sampling error
-Random sampling:In statistics, sampling error or estimation error is the error caused by observing a sample instead of the whole population. The sampling error can be found by subtracting the value of a parameter from the value of a statistic...

• Gy's sampling theory
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 2000s in articles and books including:* Sampling nomogram* Sampling of particulate materials; theory and practice...

• Horvitz–Thompson estimator
Horvitz–Thompson estimator
In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the mean of a superpopulation in a stratified sample. Inverse probability weighting is applied to account for different proportions of observations within strata...

• Chambers, R L, and Skinner, C J (editors) (2003), Analysis of Survey Data, Wiley, ISBN 0-471-89987-9
• Deming, W. Edwards
W. Edwards Deming
William Edwards Deming was an American statistician, professor, author, lecturer and consultant. He is perhaps best known for his work in Japan...

(1975) On probability as a basis for action, The American Statistician, 29(4), pp146–152.
• Gy, P (1992) Sampling of Heterogeneous and Dynamic Material Systems: Theories of Heterogeneity, Sampling and Homogenizing
• Korn, E L, and Graubard, B I (1999) Analysis of Health Surveys, Wiley, ISBN 0-471-13773-1
• Stuart, Alan (1962) Basic Ideas of Scientific Sampling, Hafner Publishing Company, New York (Portrait of T. M. F. Smith on page 144)

#### 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

#### U.S. federal and military standards

• MIL-STD-105
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 Dodge sampling inspection theories and mathematical formulas...

• MIL-STD-1916