Sampling (statistics)
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Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. Each observation measures one or more properties (weight, location, etc.) of an observable entity enumerated to distinguish objects or individuals. Survey weights often need to be applied to the data to adjust for the sample design. Results from probability theory and statistical theory are employed to guide practice.
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
 Reviewing the sampling process
[edit] Population definition
Successful statistical practice is based on focused problem definition. In sampling, this includes defining the population 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 of material from production 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 studied the behaviour of roulette wheels at a casino in Monte Carlo, 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 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.
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.
[edit] 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. There is no way to identify every voter 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 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, possible sampling frames include:
Not all frames explicitly list elements of the population. For example, a street map can be used as a frame for a doortodoor survey; although it doesn't show individual houses, we can select streets from the map and then visit all houses on those streets. (One advantage of such a frame is that it would include people who have recently moved and are not yet on the list frames discussed above.)
The sampling frame must be representative of the population and this is a question outside the scope of statistical theory demanding the judgment of experts in the particular subject matter being studied. All the above frames omit some people who will vote at the next election and contain some people who will not; some frames will contain multiple records for the same person. People not in the frame have no prospect of being sampled. Statistical theory tells us about the uncertainties in extrapolating from a sample to the frame. In extrapolating from frame to population, its role is motivational and suggestive.
"To the scientist, however, representative sampling is the only justified procedure for choosing individual objects for use as the basis of generalization, and is therefore usually the only acceptable basis for ascertaining truth." (Andrew A. Marino) [1]. It is important to understand this difference to steer clear of confusing prescriptions found in many web pages.
In defining the frame, practical, economic, ethical, and technical issues need to be addressed. The need to obtain timely results may prevent extending the frame far into the future.
The difficulties can be extreme when the population and frame are disjoint. This is a particular problem in forecasting where inferences about the future are made from historical data. In fact, in 1703, when Jacob Bernoulli proposed to Gottfried Leibniz the possibility of using historical mortality data to predict the probability of early death of a living man, Gottfried Leibniz recognized the problem in replying:
"Nature has established patterns originating in the return of events but only for the most part. New illnesses flood the human race, so that no matter how many experiments you have done on corpses, you have not thereby imposed a limit on the nature of events so that in the future they could not vary."
A frame may also provide additional 'auxiliary information' about its elements; when this information is related to variables or groups of interest, it may be used to improve survey design. For instance, an electoral register might include name and sex; this information can be used to ensure that a sample taken from that frame covers all demographic categories of interest. (Sometimes the auxiliary information is less explicit; for instance, a telephone number may provide some information about location.)
Having established the frame, there are a number of ways for organizing it to improve efficiency and effectiveness.
It's at this stage that the researcher should decide whether the sample is in fact to be the whole population and would therefore be a census.
[edit] 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 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 oneintwo 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. (In effect, the person who is selected from that household is taken as 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 'selfweighting' because all sampled units are given the same weight.
Probability sampling includes: Simple Random Sampling, Systematic Sampling, Stratified Sampling, Probability Proportional to Size Sampling, and Cluster or Multistage Sampling. These various ways of probabiliy 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 place 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 includes: Accidental Sampling, Quota Sampling 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.
[edit] 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
[edit] Simple random sampling
In a simple random sample ('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, but not all EPS designs are simple random sampling.
[edit] Systematic sampling
Systematic sampling 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, systematic sampling is a type of probability sampling. It is easy to implement and the stratification 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.
For 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.
For example, consider a street where the oddnumbered houses are all on the north (expensive) side of the road, and the evennumbered 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 oddnumbered, expensive side, or they will all be from the evennumbered, 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 oneinten probability of selection, but the set {4,13,24,34,...} has zero probability of selection.
Systematic sampling can also be adapted to a nonEPS approach; for an example, see discussion of PPS samples below.
[edit] 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 subpopulation, out of which individual elements can be randomly selected (Pedhazur & Schmelkin, 1991). 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). It is important to note that 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, preexisting 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 criterionrelevant 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 costeffective) 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.
[edit] Poststratification
Stratification is sometimes introduced after the sampling phase in a process called “poststratification (Pedhazur & Schmelkin, 1991).” 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 (Pedhazur & Schmelkin, 1991).
[edit] Oversampling
Choicebased sampling is one of the stratified sampling strategies. In choicebased sampling (Scott and Wild 1986), the data are stratified on the target and a sample is taken from each strata 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 choicebased 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.
[edit] 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. This information can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, discussed above.
Another option is probabilityproportionaltosize ('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. However, this has the drawbacks of variable sample size, and different portions of the population may still be over or underrepresented due to chance variation in selections. To address this problem, PPS may be combined with a systematic approach.
For 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 guestnights 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.
[edit] Cluster sampling
Sometimes it is cheaper to 'cluster' the sample in some way e.g. by selecting respondents from certain areas only, or certain timeperiods only. (Nearly all samples are in some sense 'clustered' in time  although this is rarely taken into account in the analysis.)
Cluster sampling is an example of 'twostage sampling' or 'multistage sampling': in the first stage a sample of areas is chosen; in the second stage a sample of respondents within those areas is selected.
This can reduce travel and other administrative costs. It also means that one does not need a sampling frame listing all elements in the target population. Instead, clusters can be chosen from a clusterlevel frame, with an elementlevel frame created only for the selected clusters. 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 withincluster variation.
Nevertheless, some of the disadvantages of cluster sampling are the reliance of sample estimate precision on the actual clusters chosen. If clusters chosen are biased in a certain way, inferences drawn about population parameters from these sample estimates will be far off from being accurate.
Multistage sampling Multistage sampling is a complex form of cluster sampling in which two or more levels of units are imbedded 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 samples of preceding random samples. It is not as effective as true random sampling, but it probably solves more of the problems inherent to random sampling. Moreover, It is an effective strategy because it banks on multiple randomizations. As such, it is extremely useful.
Multistage sampling is used frequently when a complete list of all members of the population does not exist and is inappropriate. Moreover, by avoiding the use of all sample units in all selected clusters, multistage sampling avoids the large, and perhaps unnecessary, costs associated traditional cluster sampling.
[edit] Matched random sampling
A method of assigning participants to groups in which pairs of participants are first matched on some characteristic and then individually assigned randomly to groups. (Brown, Cozby, Kee, & Worden, 1999, p.371).
The Procedure for Matched random sampling can be briefed with the following contexts,
a) Two samples in which the members are clearly paired, or are matched explicitly by the researcher. For example, IQ measurements or pairs of identical twins.
b) Those samples in which the same attribute, or variable, is measured twice on each subject, under different circumstances. Commonly called repeated measures. Examples include the times of a group of athletes for 1500m before and after a week of special training; the milk yields of cows before and after being fed a particular diet.
[edit] Quota sampling
In quota sampling, the population is first segmented into mutually exclusive subgroups, 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, 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 nonprobability sampling. In quota sampling the selection of the sample is nonrandom. 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
[edit] Mechanical sampling
Mechanical sampling is typically used in sampling solids, liquids and gases, using devices such as grabs, scoops, thief probes, the COLIWASA and riffle splitter.
Care is needed in ensuring that the sample is representative of the frame. Much work in the theory and practice of mechanical sampling was developed by Pierre Gy and Jan Visman.
[edit] Convenience sampling
Sometimes called grab or opportunity sampling, this is the method of choosing items in an unstructured manner from the population frame. Though almost impossible to treat rigorously, it is the method most commonly employed in many practical situations. This is due largely to the fact that when most researchers aim to study the behaviors of human beings, very rarely does one find an ideal environment for carrying out that research. It may be that if not for the convenience sample, a particular type of research could simply not take place. 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 nonrandom, convenience sample whereby 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 is a similar technique, where existing study subjects are used to recruit more subjects into the sample.
[edit] Lineintercept sampling
Lineintercept 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.
[edit] 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 sampling methodology is often chosen for large scale or nationwide studies in order to gauge changes in the population with regard to any number of variables from chronic illness to job stress to weekly food expenditures. Panel sampling can also be used to inform researchers about withinperson health changes due to age or help explain changes in continuous dependent variables such as spousal interaction. There have been several proposed methods of analyzing panel sample data, including MANOVA, growth curves, and structural equation modeling with lagged effects. For a more thorough look at analytical techniques for panel data, see Johnson (1995).
[edit] Event Sampling Methodology
Event Sampling Methodology (ESM) is a new form of sampling method that allows researchers to study ongoing experiences and events that vary across and within days in its naturallyoccurring environment. Because of the frequent sampling of events inherent in ESM, it enables researchers to measure the typology of activity and detect the temporal and dynamic fluctuations of work experiences. Popularity of ESM as a new form of research design increased over the recent years because it addresses the shortcomings of crosssectional research, where once unable to, researchers can now detect intraindividual variances across time. In ESM, participants are asked to record their experiences and perceptions in a paper or electronic diary.
There are three types of ESM: 1) Signal contingent – random beeping notifies participants to record data. The advantage of this type of ESM is minimization of recall bias. 2) Event contingent – records data when certain events occur 3) Interval contingent – records data according to the passing of a certain period of time
ESM has several disadvantages. One of the disadvantages of ESM is it can sometimes be perceived as invasive and intrusive by participants. ESM also leads to possible selfselection bias. It may be that only certain types of individuals are willing to participate in this type of study creating a nonrandom sample. Another concern is related to participant cooperation. Participants may not be actually fill out their diaries at the specified times. Furthermore, ESM may substantively change the phenomenon being studied. Reactivity or priming effects may occur, such that repeated measurement may cause changes in the participants' experiences. This method of sampling data is also highly vulnerable to common method variance. (Alliger & Williams, 1993)
Further, it is important to think about whether or not an appropriate dependent variable is being used in an ESM design. For example, it might be logical to use ESM in order to answer research questions which involve dependent variables with a great deal of variation throughout the day. Thus, variables such as change in mood, change in stress level, or the immediate impact of particular events may be best studied using ESM methodology. However, it is not likely that utilizing ESM will yield meaningful predictions when measuring someone performing a repetitive task throughout the day or when dependent variables are longterm in nature (coronary heart problems).
[edit] 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.
[edit] Sample size
Formulas, tables, and power function charts are well known approaches to determine sample size.
[edit] Formulas
Where the frame and population are identical, statistical theory yields exact recommendations on sample size.^{[1]} However, where it is not straightforward to define a frame representative of the population, it is more important to understand the cause system of which the population are outcomes and to ensure that all sources of variation are embraced in the frame. Large number of observations are of no value if major sources of variation are neglected in the study. In other words, it is taking a sample group that matches the survey category and is easy to survey. Bartlett, Kotrlik, and Higgins (2001) published a paper titled Organizational Research: Determining Appropriate Sample Size in Survey Research Information Technology, Learning, and Performance Journal that provides an explanation of Cochran’s (1977) formulas. A discussion and illustration of sample size formulas, including the formula for adjusting the sample size for smaller populations, is included. A table is provided that can be used to select the sample size for a research problem based on three alpha levels and a set error rate.
[edit] Steps for using sample size tables
1. Postulate the effect size of interest, α, and β.
2. Check sample size table (Cohen, 1988)
a. Select the table corresponding to the selected α
b. Locate the row corresponding to the desired power
c. Locate the column corresponding to the estimated effect size
d. The intersection of the column and row is the minimum sample size required.
[edit] Computer program
G*Power [2]
[edit] Types of data
[edit] Categorical and numerical
There are two types of random variables: categorical and numerical. Categorical random variables yield responses such as 'yes' or 'no'. Categorical variables can yield more than two possible responses. For example: 'Which day of the week are you most likely to wash clothes?' Numerical random variables yield numerical responses, such as your height in centimeters.
There are two types of numerical variables: discrete and continuous. Discrete random variables produce numerical responses from a counting process. An example is 'how many times do you visit the cash machine in a typical month?' Continuous random variables produce responses from a measuring process. Height is an example of a continuous variable because the response takes on a value from an interval. Precision of the measurement instrument(s) may lead to tied observations. A tied observation occurs when the measuring device is not sensitive or sophisticated enough to detect incremental differences in the experimental or survey data.
Generally continuous random variable requires less samples than of discrete random variable. This can be justified by referring to the Central Limit Theorem
[edit] 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 nonresponses
Most sampling books and papers written by nonstatisticians focus only in the data collection aspect, which is just a small though important part of the sampling process.
[edit] Review of sampling process
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 nonresponses.
[edit] Nonresponse
In survey sampling, many of the individuals identified as part of the sample may be unwilling to participate or impossible to contact. In this case, there is a risk of differences, between (say) the willing and unwilling, leading to selection bias in conclusions. This is often addressed by followup 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. Nonresponse is particularly a problem in internet sampling. One of the main reasons for this problem could be that people may hold multiple email adresses, which they don't use anymore or don't check regularly.
The Dillman Total Design Method (TDM), developed by professor Don Dillman, is a method of reducing nonresponse for telephone, mail, and internet surveys at a low cost. TDM is based on Social Exchange, which suggests the likelihood that individuals will respond to a survey questionnaire is a function of how much effort is required to respond, and what they feel they are likely to get in exchange for completing the questionnaire. The basic elements and procedures of the TDM are:
 Minimize the burden on the respondent by designing questionnaires that are attractive in appearance and easy to complete; printing mail questionnaires in booklet format; placing personal questions at the end; creating a vertical flow of questions; and creating sections of questions based on their content.
 Personalize all communication with the respondent by printing letters and envelopes individually, using blue ball point pens for signatures and a first class stamp on outgoing and return envelopes; and constructing a persuasive letter.
 Provide information about the survey in a cover letter to respondents, interviewers, and clerical personnel. If possible, also send out letters in advance informing respondents that a survey is forthcoming.
Followup contacts of nonrespondents is essential, and TDM includes more than 30 steps intended to maximize response rate (and reduce nonresponse)
A sample of the some of the precisely laidout steps includes: 1. The survey population is sent a questionnaire booklet containing as many as 12 pages. 2. The booklet has an illustrated front cover and a specified instruction format, a means of identifying respondents to allow for removal of their names from the mailing list, and a return envelope. 3. The covering letter, which clearly describes the purpose of the study and explains why the respondent's opinion is being sought, must be signed by hand, in blue ink. 4. An optional return postcard contains the respondent's name and permits the questionnaire to be returned anonymously; alternatively, the questionnaires may be prenumbered. 5. Followup procedures adhere to the following:
One week after the initial mailout a reminder postcard is sent; three weeks and seven weeks after the initial mailout, nonresponders are sent duplicate packets.The sevenweek packet is sent by registered mail. Followup letters to nonresponders are precisely formatted, and Dillman provides detailed advice on how to construct the questionnaire.
Dillman's have been criticized on the ground that they are too rigid. While it is the nature of "how to" books to give exact instructions, experienced researchers suggest adapting the components of any method to their own situation.
Dillman's "cookbook" method has been cited in more than 1750 scientific publications
In the 2007 update to his original work, Dillman includes new and updated material that covers both the principles behind and directions for how to:
 Conduct Web surveys
 Visually design questionnaires
 Use paper mailed surveys
[edit] 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 underrepresented 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 nonresponse.
[edit] 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 intervals 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 with a uniform prior probability and it assumed his sample was random. The theory of smallsample statistics developed by William Sealy Gossett put the subject on a more rigorous basis in the 20th century. However, the importance of random sampling was not universally appreciated and in the USA the 1936 Literary Digest prediction of a Republican win in the presidential election went badly awry, due to severe bias [3]. 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.
[edit] See also
Wikiversity has learning materials about Sampling (statistics) 
 Data collection
 Official statistics
 Replication (statistics)
 Sample (statistics)
 Sample size rule of thumb for estimate of population mean
 Sampling (case studies)
 Sampling error
 Gy's Theory of Sampling (TOS)
[edit] External links
 Chapter on Sampling at the Research Methods Knowledge Base
 Survey Sampling Methods at the SatPac survey software site
 TRSL – Template Range Sampling Library is a C++ library that implements several sampling schemes behind an (STLlike) iterator interface.
 Continuous Sampling vs. Costs  Electronics Industry Example
[edit] Notes
 ^ Mathematical details are displayed in the Sample size article.
[edit] References
 Alliger, G.M. & Williams, K.J. (1993). Using signalcontingent experience sampling methodology to study work in the field: A discussion and illustration examining task perceptions and mood. Personnel Psychology, 46(3), 525549.
 Brown, K.W., Cozby, P.C., Kee, D.W., & Worden, P.E. (1999). Research Methods in Human Development, 2d ed. Mountain View, CA: Mayfield. ISBN 1559348755
 Bartlett, J. E., II, Kotrlik, J. W., & Higgins, C. (2001). Organizational research: Determining appropriate sample size for survey research. Information Technology, Learning, and Performance Journal, 19(1) 4350.
 Chambers, R L, and Skinner, C J (editors) (2003), Analysis of Survey Data, Wiley, ISBN 0471899879
 Cochran, W G (1977) Sampling Techniques, Wiley, ISBN 047116240X
 Deming, W E (1975) On probability as a basis for action, The American Statistician, 29(4), pp146–152.
 Flyvbjerg, B (2006) "Five Misunderstandings About Case Study Research." Qualitative Inquiry, vol. 12, no. 2, April 2006, pp. 219–245. [4]
 Gy, P (1992) Sampling of Heterogeneous and Dynamic Material Systems: Theories of Heterogeneity, Sampling and Homogenizing
 Johnson, D. R. (1995). Alternative methods for the quantitative analysis of panel data in family research: Pooled timeseries models. Journal of Marriage & the Family, 57(4), 10651077.
 Kish, L (1995) Survey Sampling, Wiley, ISBN 0471109495
 Korn, E L, and Graubard, B I (1999) Analysis of Health Surveys, Wiley, ISBN 0471137731
 Lohr, H (1999) Sampling: Design and Analysis, Duxbury, ISBN 0534353614
 Pedhazur, E., & Schmelkin, L. (1991). Measurement design and analysis: An integrated approach. New York: Psychology Press.
 Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, SpringerVerlag, ISBN 0387406204
 Stuart, Alan (1962) Basic Ideas of Scientific Sampling, Hafner Publishing Company, New York
 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 Indexedby AQL
