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Sampling Probability
In statistics, in the theory relating to sampling from finite populations, the sampling probability (also known as inclusion probability) of an element or member of the population, is its probability of becoming part of the sample during the drawing of a single sample. For example, in simple random sampling the probability of a particular unit i to be selected into the sample is :p_ = \frac = \frac where n is the sample size and N is the population size. Each element of the population may have a different probability of being included in the sample. The inclusion probability is also termed the "first-order inclusion probability" to distinguish it from the "second-order inclusion probability", i.e. the probability of including a pair of elements. Generally, the first-order inclusion probability of the ''i''th element of the population is denoted by the symbol π''i'' and the second-order inclusion probability that a pair consisting of the ''i''th and ''j''th element of the popula ...
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Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ...
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Sampling (statistics)
In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a population (statistics), 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 (in many cases, collecting the whole population is impossible, like getting sizes of all stars in the universe), and thus, it can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified samplin ...
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Statistical Population
In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothetical and potentially infinite group of objects conceived as a generalization from experience (e.g. the set of all possible hands in a game of poker). A population with finitely many values N in the support of the population distribution is a finite population with population size N. A population with infinitely many values in the support is called infinite population. A common aim of statistical analysis is to produce information about some chosen population. In statistical inference, a subset of the population (a statistical '' sample'') is chosen to represent the population in a statistical analysis. Moreover, the statistical sample must be unbiased and accurately model the population. The ratio of the size of this statistical ...
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Element (statistics)
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called containing the first four positive integers (A = \), one could say that "3 is an element of ", expressed notationally as 3 \in A . Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets of . Sets can themselves be elements. For example, consider the set B = \. The elements of are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \. The elements of a set can be anything. For example the elements of the set C = \ are the color red, the number 12, and the set . In logical terms, (x \in y) \leftrightarrow \forall x_x = y x \in \mathfrak D y. Notation and terminology The binary relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing :x \in A means that ...
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Probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ...
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Simple Random Sample
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way. In SRS, each subset of ''k'' individuals has the same probability of being chosen for the sample as any other subset of ''k'' individuals. Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods. Introduction The principle of simple random sampling is that every set with the same number of items has the same probability of being chosen. For example, suppose ''N'' college students want to get a ticket for a basketball game, but there are only ''X'' < ''N'' tickets for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number in the range from 0 to ''N''-1, and random numbers are generated, either electronically or from a t ...
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Sampling Bias
In statistics, sampling bias is a bias (statistics), bias in which a sample is collected in such a way that some members of the intended statistical population, population have a lower or higher sampling probability than others. It results in a biased sample of a population (or non-human factors) in which all individuals, or instances, were not equally likely to have been selected. If this is not accounted for, results can be erroneously attributed to the phenomenon under study rather than to the method of sampling (statistics), sampling. Medical sources sometimes refer to sampling bias as ascertainment bias. Ascertainment bias has basically the same definition, but is still sometimes classified as a separate type of bias. Distinction from selection bias Sampling bias is usually classified as a subtype of selection bias, sometimes specifically termed sample selection bias, but some classify it as a separate type of bias. A distinction, albeit not universally accepted, of samplin ...
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Sampling Design
In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn. Mathematical formulation Mathematically, a sampling design is denoted by the function P(S) which gives the probability of drawing a sample S. An example of a sampling design During Bernoulli sampling, P(S) is given by : P(S) = q^ \times (1-q)^ where for each element q is the probability of being included in the sample and N_\text(S) is the total number of elements in the sample S and N_\text is the total number of elements in the population (before sampling commenced). Sample design for managerial research In business research, companies must often generate samples of customers, clients, employees, and so forth to gather their opinions. Sample design is also a critical component of marketing research and employee research for many organizations. During sample design, firms must answer questions such as: * What is the relevant population, sampl ...
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Sampling Frame
In statistics, a sampling frame is the source material or device from which a Sampling (statistics), sample is drawn. It is a list of all those within a Statistical population, population who can be sampled, and may include individuals, households or institutions. Importance of the sampling frame is stressed by Jessen and Salant and Dillman.Salant, Priscilla, and Don A. Dillman. "How to Conduct your own Survey: Leading professional give you proven techniques for getting reliable results" (1995) A slightly more general concept of sampling frame includes Area sampling frame, area sampling frames, whose elements have a geographic nature. Area sampling frames can be useful for example in agricultural statistics when a suitable and updated agricultural census is not available. In Environmental surveying, environmental surveys, area sampling frames may be the only option. Obtaining and organizing a sampling frame In the most straightforward cases, such as when dealing with a batch of ...
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