Sampling (statistics)
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample of individuals from within a statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population, and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.
Each observation measures one or more properties of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. Acceptance sampling is used to determine if a production lot of material meets the governing specifications.
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 assumed that his sample was random. Alexander Ivanovich Chuprov introduced sample surveys to Imperial Russia in the 1870s.In the US, the 1936 Literary Digest prediction of a Republican win in the presidential election went badly awry, due to severe bias . 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.
Elections in Singapore have adopted this practice since the 2015 election, also known as the sample counts, whereas according to the Elections Department, their country's election commission, sample counts help reduce speculation and misinformation, while helping election officials to check against the election result for that electoral division. While the reported sample counts yield a fairly accurate indicative result with a 4% margin of error at a 95% confidence interval, ELD reminded the public that sample counts are separate from official results, and only the returning officer will declare the official results once vote counting is complete.
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 characteristics 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 of that population.Sometimes what 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 scrapped or reworked due to poor quality. In this case, the batch is the population.
Although the population of interest often consists of physical objects, sometimes it is necessary to sample over time, space, or some combination of these dimensions. For instance, an investigation of supermarket staffing could examine checkout line length at various times, or a study on endangered penguins might aim to understand their usage of various hunting grounds over time. For the time dimension, the focus may be on periods or discrete occasions.
In other cases, the examined 'population' may be even less tangible. For example, Joseph Jagger 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, while his 'sample' was formed from observed results from that wheel. Similar considerations arise when taking repeated measurements of properties of materials such as the electrical conductivity of copper.
This situation often arises when seeking 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 that does not yet exist since the program is not yet available to all.
The population from which the sample is drawn may not be the same as the population from which information is desired. Often there is a large but not complete overlap between these two groups due to frame issues etc.. Sometimes they may be entirely separate – for instance, one might study rats in order to get a better understanding of human health, or one might study records from people born in 2008 in order to make predictions about people born in 2009.
Time spent in making the sampled population and population of concern precise is often well spent because it raises many issues, ambiguities, and questions that would otherwise have been overlooked at this stage.
Sampling frame
In the most straightforward case, such as the sampling of a batch of material from production, it would be most desirable to identify and measure every single item in the population and to include any one of them in our sample. However, in the more general case this is not usually possible or practical. There is no way to identify all rats in the set of all rats. Where voting is not compulsory, there is no way to identify which people will vote at a forthcoming election. 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 with appropriate contact information. For example, in an opinion poll, possible sampling frames include an electoral register and a telephone directory.
A probability sample is a sample in which every unit in the population has a chance 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.. 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.
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' 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, systematic sampling, 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 sampling
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 and it's not practical to calculate these probabilities.
Nonprobability sampling methods include convenience 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.