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Weighted Arithmetic Mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. Examples Basic example Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a Survey (statistics), survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric mean, geometric and harmonic mean, harmonic. Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's Human population, population. While the arithmetic mean is often used to report central tendency, central tendencies, it is not a robust statistic: it is greatly influenced by outliers (Value (mathematics), values much larger or smaller than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Estimator
The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symmetrical tests such as the t test should not be used to generate confidence intervals. The bias is of the order ''O''(1/''n'') (see big O notation) so as the sample size (''n'') increases, the bias will asymptotically approach 0. Therefore, the estimator is approximately unbiased for large sample sizes. Definition Assume there are two characteristics – ''x'' and ''y'' – that can be observed for each sampled element in the data set. The ratio ''R'' is : R = \bar_y / \bar_x The ratio estimate of a value of the ''y'' variate (''θ''''y'') is : \theta_y = R \theta_x where ''θ''''x'' is the corresponding value of the ''x'' variate. ''θ''''y'' is known to be asymptotically normally distributed.Scott AJ, Wu CFJ (1981) On the asym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimand
An estimand is a quantity that is to be estimated in a statistical analysis. The term is used to distinguish the target of inference from the method used to obtain an approximation of this target (i.e., the estimator) and the specific value obtained from a given method and dataset (i.e., the estimate). For instance, a normally distributed random variable X has two defining parameters, its mean \mu and variance \sigma^. A variance estimator: s^ = \sum_^ \left. \left( x_ - \bar \right)^ \right/ (n-1), yields an estimate of 7 for a data set x = \left\; then s^ is called an estimator of \sigma^, and \sigma^ is called the estimand. Definition In relation to an estimator, an estimand is the outcome of different treatments of interest. It can formally be thought of as any quantity that is to be estimated in any type of experiment. Overview An estimand is closely linked to the purpose or objective of an analysis. It describes what is to be estimated based on the question of interes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horvitz–Thompson Estimator
In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample by applying inverse probability weighting to account for the difference in the sampling distribution between the collected data and the target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data, as well as many sources of unequal selection probabilities. The method Formally, let Y_i, i = 1, 2, \ldots, n be an independent sample from n of N \ge n distinct strata with an overall mean \mu. Suppose further that \pi_i is the inclusion 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 dra ... that a ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability-proportional-to-size Sampling
In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size ''N'') has some ( independent) chance p_i to be selected to the sample when performing one draw. This p_i is proportional to some known quantity x_i so that p_i = \frac. One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943,Hansen, Morris H., and William N. Hurwitz. "On the theory of sampling from finite populations." The Annals of Mathematical Statistics 14.4 (1943): 333-362. is when we have several clusters of units, each with a different (known upfront) number of units, then each cluster can be selected with a probability that is proportional to the number of units inside it.Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Nashville, TN: John Wiley & Sons. So, for example, if we have 3 clusters with 10, 20 and 30 units each, then the chance of selecting the first cluster will be 1/6, the second would be 1/3, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Sampling
In statistics, cluster sampling is a sampling plan used when mutually homogeneous yet internally heterogeneous groupings are evident in a statistical population. It is often used in marketing research. In this sampling plan, the total population is divided into these groups (known as clusters) and a simple random sample of the groups is selected. The elements in each cluster are then sampled. If all elements in each sampled cluster are sampled, then this is referred to as a "one-stage" cluster sampling plan. If a simple random subsample of elements is selected within each of these groups, this is referred to as a "two-stage" cluster sampling plan. A common motivation for cluster sampling is to reduce the total number of interviews and costs given the desired accuracy. For a fixed sample size, the expected random error is smaller when most of the variation in the population is present internally within the groups, and not between the groups. Cluster elemental The population wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Sampling
In survey methodology, Poisson sampling (sometimes denoted as ''PO sampling'') is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample.Ghosh, Dhiren, and Andrew Vogt. "Sampling methods related to Bernoulli and Poisson Sampling." Proceedings of the Joint Statistical Meetings. American Statistical Association Alexandria, VA, 2002(pdf)/ref> Each element of the population may have a different probability of being included in the sample (\pi_i). The probability of being included in a sample during the drawing of a single sample is denoted as the ''first-order inclusion probability'' of that element (p_i). If all first-order inclusion probabilities are equal, Poisson sampling becomes equivalent to Bernoulli sampling, which can therefore be considered to be a special case of Poisson sampling. A mathematical consequence of Poisson sampling Mathematically, the first-order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcome (probability), outcomes that are Boolean-valued function, Boolean-valued: a single bit whose value is success/yes and no, yes/Truth value, true/Binary code, one with probability ''p'' and failure/no/false (logic), false/Binary code, zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survey Sampling
In statistics, survey sampling describes the process of selecting a sample of elements from a target population to conduct a survey. The term " survey" may refer to many different types or techniques of observation. In survey sampling it most often involves a questionnaire used to measure the characteristics and/or attitudes of people. Different ways of contacting members of a sample once they have been selected is the subject of survey data collection. The purpose of sampling is to reduce the cost and/or the amount of work that it would take to survey the entire target population. A survey that measures the entire target population is called a census. A sample refers to a group or section of a population from which information is to be obtained. Survey samples can be broadly divided into two types: probability samples and super samples. Probability-based samples implement a sampling plan with specified probabilities (perhaps adapted probabilities specified by an adaptive proc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survey Methodology
Survey methodology is "the study of survey methods". As a field of applied statistics concentrating on human-research surveys, survey methodology studies the sampling of individual units from a population and associated techniques of survey data collection, such as questionnaire construction and methods for improving the number and accuracy of responses to surveys. Survey methodology targets instruments or procedures that ask one or more questions that may or may not be answered. Researchers carry out statistical surveys with a view towards making statistical inferences about the population being studied; such inferences depend strongly on the survey questions used. Polls about public opinion, public-health surveys, market-research surveys, government surveys and censuses all exemplify quantitative research that uses survey methodology to answer questions about a population. Although censuses do not include a "sample", they do include other aspects of survey methodology, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
