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Samuelson's Inequality
In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre–Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection ''x''1, ..., ''x''''n'', is within uncorrected sample standard deviations of their sample mean. Statement of the inequality If we let : \overline = \frac be the sample mean and : s = \sqrt be the standard deviation of the sample, then : \overline - s\sqrt \le x_j \le \overline + s\sqrt\qquad \text j = 1,\dots,n. Equality holds on the left (or right) for x_j if and only if all the ''n'' − 1 x_is other than x_j are equal to each other and greater (smaller) than x_j. If you instead define s = \sqrt then the inequality becomes \overline - s\sqrt \le x_j \le \overline + s\sqrt\qquad \text j = 1,\dots,n. Comparison to Chebyshev's inequality Chebyshev's inequality locates a certain fraction of the data within certain bounds, while Sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "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.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Samuelson
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory". "In a career that spanned seven decades, he transformed his field, influenced millions of students and turned MIT into an economics powerhouse" Economic historian Randall E. Parker has called him the "Father of Modern Economics", and ''The New York Times'' considers him to be the "foremost academic economist of the 20th century". Samuelson was likely the most influential economist of the latter half of the 20th century."Paul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Statistical Association
The ''Journal of the American Statistical Association (JASA)'' is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States. It is published four times a year in March, June, September and December by Taylor & Francis, Ltd on behalf of the American Statistical Association. As a statistics journal it publishes articles primarily focused on the application of statistics, statistical theory and methods in economic, social, physical, engineering, and health sciences. The journal also includes reviews of academic books which are important to the advancement of the field. It had an impact factor of 2.063 in 2010, tenth highest in the "Statistics and Probability" category of ''Journal Citation Reports''. In a 2003 survey of statisticians, the ''Journal of the American Statistical Association'' was ranked first, among all journals, for "Applications of Statistics" and second (after ''Annals of Statistics'') f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McGill University
McGill University (french: link=no, Université McGill) is an English-language public research university located in Montreal, Quebec, Canada. Founded in 1821 by royal charter granted by King George IV,Frost, Stanley Brice. ''McGill University, Vol. I. For the Advancement of Learning, 1801–1895.'' McGill-Queen's University Press, 1980. the university bears the name of James McGill, a Scottish merchant whose bequest in 1813 formed the university's precursor, University of McGill College (or simply, McGill College); the name was officially changed to McGill University in 1885. McGill's main campus is on the slope of Mount Royal in downtown Montreal in the borough of Ville-Marie, with a second campus situated in Sainte-Anne-de-Bellevue, west of the main campus on Montreal Island. The university is one of two members of the Association of American Universities located outside the United States, alongside the University of Toronto, and is the only Canadian member of the Glob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmond Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions. Works Selection * * * * Théorie des équations numériques', Paris: Gauthier-Villars. 1884 on Google Books * * Oeuvres de Laguerrepubl. sous les auspices de l'Académie des sciences par MM. Charles Hermite, Henri Poincaré, et Eugène Rouché.'' (Paris, 1898-1905) (reprint: New York : Chelsea publ., 1972 ) Extensive lists More than 80 articleson Nundam.org.p See also * Isotropic line * ''q''-Laguerre polynomials * Big ''q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter '' s'', for the sample standard deviation. The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a popu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithmetic mean'', also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the ''sample mean'' (\bar) to distinguish it from the mean, or expected value, of the underlying distribution, the ''population mean'' (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.p. 181/ref> Outside probability and statistics, a wide range of other notions of mean are o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev's Inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/''k''2 of the distribution's values can be ''k'' or more standard deviations away from the mean (or equivalently, at least 1 − 1/''k''2 of the distribution's values are less than ''k'' standard deviations away from the mean). The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Residuals
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym ''Student''. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. Motivation The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the ''residuals'' at different input variable values may differ, even if the variances of the ''errors'' at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym ''Student''. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. Motivation The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the ''residuals'' at different input variable values may differ, even if the variances of the ''errors'' at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laguerre
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials). Laguerre's method is a root-finding algorithm tailored to polynomials. He laid the foundations of a geometry of oriented spheres (Laguerre geometry and Laguerre plane), including the Laguerre transformation or transformation by reciprocal directions. Works Selection * * * * Théorie des équations numériques', Paris: Gauthier-Villars. 1884 on Google Books * * Oeuvres de Laguerrepubl. sous les auspices de l'Académie des sciences par MM. Charles Hermite, Henri Poincaré, et Eugène Rouché.'' (Paris, 1898-1905) (reprint: New York : Chelsea publ., 1972 ) Extensive lists More than 80 articleson Nundam.org.p See also * Isotropic line * ''q''-Laguerre polynomials * Big ''q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
