# Quality Engineering And Management Systems Education Essay

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SamplingÂ is that part ofÂ statisticalÂ practice concerned with the selection of an unbiased orÂ randomÂ subset of individual observations within a population of individuals intended to yield some knowledge about theÂ populationÂ of concern, especially for the purposes of making predictions based onÂ statistical inference. Sampling is an important aspect ofÂ data collection.AL

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.

EachÂ observationÂ measures one or more properties (such as weight, location, color) of observable bodies distinguished as independent objects or individuals. InÂ survey sampling, survey weights can be applied to the data to adjust for theÂ sample design. Results fromÂ probability theoryÂ andÂ statistical theoryÂ are employed to guide practice.

Specifying aÂ sampling frame, aÂ setÂ of items or events possible to measure

Specifying aÂ sampling methodÂ for selecting items or events from the frame

Successful statistical practice is based on focused problem definition. In sampling, this includes defining theÂ populationÂ from which our sample is drawn. A population can be defined as including all people or items with the characteristic one wish 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.

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.

## 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. Not all frames explicitly list population elements. For example, a street map can be used as a frame for a door-to-door survey; although it doesn't show individual houses, we can select streets from the map and then visit all houses on those streets.

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.

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.

## Probability and non probability sampling

AÂ probability samplingÂ 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.

Probability sampling includes: Simple Random Sampling, Systematic Sampling, and Stratified Sampling, Probability Proportional to Size Sampling, and Cluster or Multistage Sampling. These various ways of probability sampling have two things in common:

Every element has a known nonzero probability of being sampled and

Involves random selection at some point.

Nonprobability samplingÂ is any sampling method where some elements of the population haveÂ noÂ chance of selection, 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.

Nonprobability Sampling includes:Â Accidental Sampling,Â Quota SamplingÂ andÂ Purposive Sampling. In addition, nonresponse effects may turnÂ anyÂ 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 aÂ simple random sampleÂ ('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 givenÂ pairÂ 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 willÂ on averageÂ produce five men and five women, but any given trial is likely to overrepresent one sex and underrepresent the other.Â

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.

## Systematic sampling

Systematic samplingÂ 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 everyÂ kth element from then onwards. In this case,Â 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 theÂ kth element in the list.

As long as the starting point isÂ randomized, systematic sampling is a type ofÂ probability sampling. It is easy to implement and theÂ stratificationÂ induced can make it efficient,Â ifÂ the variable by which the list is ordered is correlated with the variable of interest.

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 beÂ unrepresentative of the overall population, making the scheme less accurate than simple random sampling.

Another drawback of systematic sampling is that even in scenarios where it is more accurate than SRS, its theoretical properties make it difficult toÂ quantifyÂ that accuracy. Systematic sampling is an EPS method, because all elements have the same probability of selection.

## 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.

## A stratified sampling approach is most effective when three conditions are met

Variability within strata are minimized

Variability between strata are maximized

The variables upon which the population is stratified are strongly correlated with the desired dependent variable.

## Advantages over other sampling methods

Focuses on important subpopulations and ignores irrelevant ones.

Allows use of different sampling techniques for different subpopulations.

Improves the accuracy/efficiency of estimation.

Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size.

## Disadvantages

Requires selection of relevant stratification variables which can be difficult.

Is not useful when there are no homogeneous subgroups.

Can be expensive to implement.

## 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 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 forÂ Poisson sampling. However, this has the drawbacks 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.

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 cheaper to 'cluster' the sample in some way e.g. by selecting respondents from certain areas only, or certain time-periods only. (Nearly all samples are in some sense 'clustered' in time - although this is rarely taken into account in the analysis.)

Cluster samplingÂ is an example of 'two-stage sampling' or 'multistage sampling': in the first stage a sample of areas is chosen; in the second stage a sample of respondentsÂ withinÂ those areas is selected.

This can reduce travel and other administrative costs. It also means that one does not need aÂ sampling frameÂ 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. 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.

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.

## 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.

The procedure for matched random sampling can be briefed with the following contexts,

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.

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.

## Quota sampling

InÂ quota sampling, the population is first segmented intoÂ mutually exclusiveÂ sub-groups, just as inÂ 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 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

Convenience samplingÂ 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 sample population selected because it is readily available and convenient. 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 was 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 was to be conducted at different times of day and several times per week. This type of sampling is most useful for pilot testing. Several important considerations for researchers using convenience samples include:

Are there controls within the research design or experiment which can serve to lessen the impact of a non-random, convenience sample whereby ensuring the results will be more representative of the population?

Is there good reason to believe that a particular convenience sample would or should respond or behave differently than a random sample from the same population?

Is the question being asked by the research one that can adequately be answered using a convenience sample?

## Panel sampling

Panel samplingÂ 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 nation-wide 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 within-person 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.

## Replacement of selected units

Sampling schemes may beÂ without replacementÂ orÂ with replacement. 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.

## Formulas

Where the frame and population are identical, statistical theory yields exact recommendations onÂ sample size. However, where it is not straightforward to define a frame representative of the population, it is more important to understand theÂ cause systemÂ of which the population are outcomes and to ensure that all sources of variation are embraced in the frame. Large numbers 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. Research Information Technology, Learning, and Performance JournalÂ that provides an explanation of Cochran's 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.

## Steps for using sample size tables

Postulate the effect size of interest, Î±, and Î².

Check sample size table

Select the table corresponding to the selected Î±

Locate the row corresponding to the desired power

Locate the column corresponding to the estimated effect size

The intersection of the column and row is the minimum sample size required.

## Sampling and data collection

Good data collection involves:

Following the defined sampling process

Keeping the data in time order

Noting comments and other contextual events

Recording non-responses

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

## Errors in research

There are always errors in a research. By sampling, the total errors can be classified into sampling errors and non-sampling errors.

## Sampling error

Sampling errors are caused by sampling design. It includes:

(1)Â Selection error: Incorrect selection probabilities are used.

(2)Â Estimation error: Biased parameter estimate because of the elements in these samples.

## Non-sampling error

Non-sampling errors are caused by the mistakes in data processing. It includes:

(1)Â Overcoverage: Inclusion of data from outside of the population.

(2)Â Undercoverage: Sampling frame does not include elements in the population.

(3)Â Measurement error: The respondents misunderstand the question.

(4)Â Processing error: Mistakes in data coding.

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.