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Normality Test
In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed. More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on one's interpretations of probability: * In descriptive statistics terms, one measures a goodness of fit of a normal model to the data – if the fit is poor then the data are not well modeled in that respect by a normal distribution, without making a judgment on any underlying variable. * In frequentist statistics statistical hypothesis testing, data are tested against the null hypothesis that it is normally distributed. * In Bayesian statistics, one does not "test normality" per se, but rather computes the likelihood that the data come from a normal distribution with given parameters ''μ'',''σ'' (for all ''μ'',''σ''), and compares that with the likelihood that the ... [...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] |
Normal Probability Plot
The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw data, residuals from model fits, and estimated parameters. In a normal probability plot (also called a "normal plot"), the sorted data are plotted vs. values selected to make the resulting image look close to a straight line if the data are approximately normally distributed. Deviations from a straight line suggest departures from normality. The plotting can be manually performed by using a special graph paper, called ''normal probability paper''. With modern computers normal plots are commonly made with software. The normal probability plot is a special case of the Q–Q probability plot for a normal distribution. The theoretical quantiles are generally chosen to approximate either the mean or the median of the corresponding or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramér–von Mises Criterion
In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F^* compared to a given empirical distribution function F_n, or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as :\omega^2 = \int_^ _n(x) - F^*(x)2\,\mathrmF^*(x) In one-sample applications F^* is the theoretical distribution and F_n is the empirically observed distribution. Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case. The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928–1930. The generalization to two samples is due to Anderson. The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test (1933). Cramér–von Mises test (one sample) Let x_1,x_2,\cdots,x_n be the observed values, in increasing order. Then th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anderson–Darling Test
The Anderson–Darling test is a statistical test of whether a given sample of data is drawn from a given probability distribution. In its basic form, the test assumes that there are no parameters to be estimated in the distribution being tested, in which case the test and its set of critical values is distribution-free. However, the test is most often used in contexts where a family of distributions is being tested, in which case the parameters of that family need to be estimated and account must be taken of this in adjusting either the test-statistic or its critical values. When applied to testing whether a normal distribution adequately describes a set of data, it is one of the most powerful statistical tools for detecting most departures from normal distribution, normality. ''K''-sample Anderson–Darling tests are available for testing whether several collections of observations can be modelled as coming from a single population, where the cumulative distribution function, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jarque–Bera Test
In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution. The test statistic ''JB'' is defined as : \mathit = \frac \left( S^2 + \frac14 (K-3)^2 \right) where ''n'' is the number of observations (or degrees of freedom in general); ''S'' is the sample skewness, ''K'' is the sample kurtosis : : S = \frac = \frac , : K = \frac = \frac , where \hat_3 and \hat_4 are the estimates of third and fourth central moments, respectively, \bar is the sample mean, and \hat^2 is the estimate of the second central moment, the variance. If the data comes from a normal distribution, the ''JB'' statistic asymptotically has a chi-squared distribution with two degrees of freedom, so the statistic can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D'Agostino's K-squared Test
In statistics, D'Agostino's ''K''2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to gauge the compatibility of given data with the null hypothesis that the data is a realization of independent, identically distributed Gaussian random variables. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic. Skewness and kurtosis In the following, denotes a sample of ''n'' observations, ''g''1 and ''g''2 are the sample skewness and kurtosis, ''mj''’s are the ''j''-th sample central moments, and \bar is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as and ''β''2 respectively. Such notation can be inconvenient since, for example, can be a negative quantity. The sample skewness and kurtosis are defined as : \begin & g_1 = \frac = \frac\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurtosis Risk
In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations that have a tendency to occasionally be much farther (in terms of number of standard deviations) from the average than is expected for a normal distribution. Overview Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal distribution for certain of its independent variables when the latter may in fact have kurtosis much greater than does the normal distribution. Kurtosis risk is commonly referred to as "fat tail" risk. The "fat tail" metaphor explicitly describes the situation of having more observations at either extreme than the tails of the normal distribution would suggest; therefore, the tails are "fatter". Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance, Long-Term Capital Management, a hedge fund cofounded by Myron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-statistic
In statistics, the ''t''-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's ''t''-test. The ''t''-statistic is used in a ''t''-test to determine whether to support or reject the null hypothesis. It is very similar to the z-score but with the difference that ''t''-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the ''t''-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis tests where the p-value tells us what the odds are of the results to have happened. Definition and features Let \hat\beta be an estimator of parameter ''β'' in some statistical model. Then a ''t''-statistic for this parameter is any quantity of the form : t_ = \frac, whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-score
In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores. It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization for more). Standard scores are most commonly called ''z''-scores; the two terms may be used interchangeably, as they are in this article. Other equivalent terms in use include z-values, normal scores, standardized variables and pull in high energy physics. Computing a z-score requires knowledge of the mean and standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Maximum And Minimum
In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot. The minimum and the maximum value are the first and last order statistics (often denoted ''X''(1) and ''X''(''n'') respectively, for a sample size of ''n''). If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not be outliers, if they are not unusually far from other observations. Robustness The sample maximum and minimum are the ''least'' robust statistics: they are maximally sensitive to outliers. This can either be an advantage or a drawback: if extreme values are real (not measurement errors), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Back-of-the-envelope
A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions. A similar phrase in the U.S. is "back of a napkin", also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet". History In the natural sciences, ''back-of-the-envelope calculation'' is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations, which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems. Fermi was known f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
