query: the underlying statistical distribution for the  p-chart is the

pos: P-chart the p-chart has a positive lower control limit or formula_6. In the case of 100% inspection, variation in the production rate (e.g., due to maintenance or shift changes) conspires to produce different sample sizes for each observation plotted on the p-chart. There are three ways to deal with this: Some practitioners have pointed out that the p-chart is sensitive to the underlying assumptions, using control limits derived from the binomial distribution rather than from the observed sample variance. Due to this sensitivity to the underlying assumptions, p-charts are often implemented incorrectly, with control limits that are either too wide or

pos: P-chart P-chart In statistical quality control, the p-chart is a type of control chart used to monitor the proportion of nonconforming units in a sample, where the sample proportion nonconforming is defined as the ratio of the number of nonconforming units to the sample size, n. The p-chart only accommodates "pass"/"fail"-type inspection as determined by one or more go-no go gauges or tests, effectively applying the specifications to the data "before" they are plotted on the chart. Other types of control charts display the magnitude of the quality characteristic under study, making troubleshooting possible directly from those charts. The binomial distribution

neg: Control chart attribute data, particularly when the assumptions of either binomially distributed data (p- and np-charts) or Poisson-distributed data (u- and c-charts) are violated. Two primary justifications are given for this practice. First, normality is not necessary for statistical control, so the Individuals chart may be used with non-normal data. Second, attribute charts derive the measure of dispersion directly from the mean proportion (by assuming a probability distribution), while Individuals charts derive the measure of dispersion from the data, independent of the mean, making Individuals charts more robust than attributes charts to violations of the assumptions about the distribution of the underlying

neg: Mixture distribution mixture models, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions. Given a finite set of probability density functions "p"("x"), …, "p"("x"), or corresponding cumulative distribution functions "P"("x"), …, "P"("x") and weights "w", …, "w" such that and the mixture distribution can be represented by writing either the density, "f", or the distribution function, "F", as a sum (which in both cases is a convex combination): This type of mixture, being a finite sum, is called a finite mixture, and in applications, an unqualified reference

neg: Directional statistics show some relevant circular distributions. The von Mises distribution is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable; however, for statistical purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the wrapped normal distribution, which, analogously to

neg: Probability box family of the underlying distribution, which can often result in tighter confidence bands. Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous. Confidence bands at the (1 − α)% confidence level are defined such that, (1 − α)% of the time they are constructed, they will completely enclose the distribution from which the data were randomly sampled. A confidence band about a distribution function is sometimes used as a p-box even though it represents statistical rather than rigorous or sure bounds. This use

neg: Process performance index the specification limits, though this interpretation is controversial. Strictly speaking, from a statistical standpoint, P is meaningless if the process under study is not in control because one cannot reliably estimate the process underlying probability distribution, let alone parameters like formula_1 and formula_2. Furthermore, using this metric of past process performance to predict future performance is highly suspect. From a management standpoint, when an organization is under pressure to set up a new process quickly and economically, P is a convenient metric to gauge how set-up is progressing (increasing P being interpreted as "the process capability is improving"). The risk

neg: History of information theory particles would cause the "H"-function to inevitably increase from any initial configuration until equilibrium was reached; and further identified it as an underlying microscopic rationale for the macroscopic thermodynamic entropy of Clausius. Boltzmann's definition was soon reworked by the American mathematical physicist J. Willard Gibbs into a general formula for statistical-mechanical entropy, no longer requiring identical and non-interacting particles, but instead based on the probability distribution "p" for the complete microstate "i" of the total system: This (Gibbs) entropy, from statistical mechanics, can be found to directly correspond to the Clausius's classical thermodynamic definition. Shannon himself was apparently not particularly

neg: Levene's test good statistical power. If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a chi-squared distribution with four degrees of freedom (a heavily skewed distribution). Using the mean provided the best power for symmetric, moderate-tailed, distributions. Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances

neg: Nonparametric skew sample's standard deviation. Statistical tests of "D" have assumed that the null hypothesis being tested is that the distribution is symmetric . Gastwirth estimated the asymptotic variance of "n""D". If the distribution is unimodal and symmetric about 0, the asymptotic variance lies between 1/4 and 1. Assuming a conservative estimate (putting the variance equal to 1) can lead to a true level of significance well below the nominal level. Assuming that the underlying distribution is symmetric Cabilio and Masaro have shown that the distribution of "S" is asymptotically normal. The asymptotic variance depends on the underlying distribution: for the normal

neg: Control chart population. It is sometimes noted that the substitution of the Individuals chart works best for large counts, when the binomial and Poisson distributions approximate a normal distribution. i.e. when the number of trials for p- and np-charts or for u- and c-charts. Critics of this approach argue that control charts should not be used when their underlying assumptions are violated, such as when process data is neither normally distributed nor binomially (or Poisson) distributed. Such processes are not in control and should be improved before the application of control charts. Additionally, application of the charts in the presence of such

neg: P-value AMA style uses ""P" value," APA style uses ""p" value," and the American Statistical Association uses ""p"-value." In statistics, every conjecture concerning the unknown distribution formula_1 of a random variable formula_2 is called a "statistical hypothesis". If we state one hypothesis only and the aim of the statistical test is to verify whether this hypothesis is not false, but not, at the same time, to investigate other hypotheses, then such a test is called a "significance test". A statistical hypothesis that refers only to the numerical values of unknown parameters of a distribution is called a "parametric hypothesis". Methods of

neg: Brown–Forsythe test that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a χ distribution with four degrees of freedom (a sharply skewed distribution). Using the mean provided the best power for symmetric, moderate-tailed, distributions. Brown–Forsythe test The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an ANOVA on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is

neg: Occupancy–abundance relationship can expect to detect rare species occupying fewer sites than common ones, even if the underlying occupancy distribution is the same. However, this explanation makes only one prediction, that is, that with sufficient sampling, no relationship will be found to exist. This prediction is readily falsified, given that exceptionally well studied taxa such as breeding birds (e.g. Zuckerberg et al. 2009, Gaston) show well documented O-A relationships. A second statistical explanation involves the use of statistical distributions such as the Poisson or negative-binomial. This explanation suggests that due to the underlying distribution of aggregation and density, and observed O–A relationship

neg: Ronald Fisher analysis of real data. In 1925 he published "Statistical Methods for Research Workers", one of the 20th century's most influential books on statistical methods. Fisher's method is a technique for data fusion or "meta-analysis" (analysis of analyses). This book also popularized the p-value, and plays a central role in his approach. Fisher proposes the level p=0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, and applies this to a normal distribution (as a two-tailed test), thus yielding the rule of two standard deviations (on a normal distribution) for statistical significance. The

neg: P-chart too narrow, leading to incorrect decisions regarding process stability. A p-chart is a form of the Individuals chart (also referred to as "XmR" or "ImR"), and these practitioners recommend the individuals chart as a more robust alternative for count-based data. P-chart In statistical quality control, the p-chart is a type of control chart used to monitor the proportion of nonconforming units in a sample, where the sample proportion nonconforming is defined as the ratio of the number of nonconforming units to the sample size, n. The p-chart only accommodates "pass"/"fail"-type inspection as determined by one or more go-no go gauges

neg: Dragon King Theory and are discarded, or statistical methods are chosen that are somehow insensitive to outliers. Contrarily, DK are outliers that are highly informative, and should be the focus of much statistical attention. Thus a first step is identifying DK in historical data. Existing tests are either based on the asymptotic properties of the empirical distribution function (EDF) or on an assumption about the underlying cumulative distribution function (CDF) of the data. It turns out that testing for outliers relative to an exponential distribution is very general. The latter follows from the Pickands–Balkema–de Haan theorem of Extreme value theory which states that

neg: P-value underlying random variable is continuous, then the probability distribution of the "p"-value is uniform on the interval [0,1]. By contrast, if the alternative hypothesis is true, the distribution is dependent on sample size and the true value of the parameter being studied. The distribution of "p"-values for a group of studies is called a "p"-curve. The curve is affected by four factors: the proportion of studies that examined false null hypotheses, the power of the studies that investigated false null hypotheses, the alpha levels, and publication bias. A "p"-curve can be used to assess the reliability of scientific literature, such

neg: P-value The "p"-values for the chi-squared distribution (for various values of "χ" and degrees of freedom), now notated as "P," was calculated in , collected in . The use of the "p"-value in statistics was popularized by Ronald Fisher, and it plays a central role in his approach to the subject. In his influential book "Statistical Methods for Research Workers" (1925), Fisher proposed the level "p" = 0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, and applied this to a normal distribution (as a two-tailed test), thus yielding the rule of

neg: X̅ and s chart plus the standard deviation of the quality characteristic across all units in the sample as follows: The normal distribution is the basis for the charts and requires the following assumptions: The control limits for this chart type are: As with the formula_7 and R and individuals control charts, the formula_7 chart is only valid if the within-sample variability is constant. Thus, the s chart is examined before the formula_7 chart; if the s chart indicates the sample variability is in statistical control, then the formula_7 chart is examined to determine if the sample mean is also in statistical control. If

neg: Statistical inference Statistical inference Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. Statistical inference makes propositions about a population, using data drawn from the population with some form

neg: Elżbieta Pleszczyńska comparable results only for comparable distribution types (in practice multivariate normal distribution is being assumed). Parametric statistical tests are derived from distribution assumptions. Classic methods fail if the input data contain strong outliers, and interpretation of their results should be different for different distribution types. In practice, the underlying assumptions are often not checked, moreover they are always violated – there is no normal distribution in the real world, because every real variable is limited (for example people cannot be –170 cm or +2 km tall), and the normal distribution implies positive probability density for every real number. In most

neg: Pearson distribution II distribution is a special case of type I, but is usually no longer singled out.) The gamma distribution originated from Pearson's work (Pearson 1893, p. 331; Pearson 1895, pp. 357, 360, 373–376) and was known as the Pearson type III distribution, before acquiring its modern name in the 1930s and 1940s. A Pearson density "p" is defined to be any valid solution to the differential equation (cf. Pearson 1895, p. 381) with : According to Ord, Pearson devised the underlying form of Equation (1) on the basis of, firstly, the formula for the derivative of the logarithm of the

neg: C-chart setup. C-chart In statistical quality control, the c-chart is a type of control chart used to monitor "count"-type data, typically total number of nonconformities per unit. It is also occasionally used to monitor the total number of events occurring in a given unit of time. The c-chart differs from the p-chart in that it accounts for the possibility of more than one nonconformity per inspection unit, and that (unlike the p-chart and u-chart) it requires a fixed sample size. The p-chart models "pass"/"fail"-type inspection only, while the c-chart (and u-chart) give the ability to distinguish between (for example) 2 items

neg: C-chart C-chart In statistical quality control, the c-chart is a type of control chart used to monitor "count"-type data, typically total number of nonconformities per unit. It is also occasionally used to monitor the total number of events occurring in a given unit of time. The c-chart differs from the p-chart in that it accounts for the possibility of more than one nonconformity per inspection unit, and that (unlike the p-chart and u-chart) it requires a fixed sample size. The p-chart models "pass"/"fail"-type inspection only, while the c-chart (and u-chart) give the ability to distinguish between (for example) 2 items which

neg: Statistical arbitrage statistical relationship on which the model is based may be spurious, or may break down due to changes in the distribution of returns on the underlying assets. Factors, which the model may not be aware of having exposure to, could become the significant drivers of price action in the markets, and the inverse applies also. The existence of the investment based upon model itself may change the underlying relationship, particularly if enough entrants invest with similar principles. The exploitation of arbitrage opportunities themselves increases the efficiency of the market, thereby reducing the scope for arbitrage, so continual updating of models

neg: Sampling distribution simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values. The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size formula_1. It may be considered as the distribution of the statistic for "all possible samples from the same population" of a given sample size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, the

neg: Charles Legéndy of burst detection. (Poisson surprise is defined as –log P, where P is the probability of a spike pattern under the assumed baseline distribution of spiking — in this case the Poisson distribution.) The test can quickly detect and quantify bursts in neuronal spike trains; and it is used, for instance, as a way to detect loss of functionality in certain brain regions. The critique of the Poisson surprise test is that actual spike distributions differ from the Poisson distribution to varying degrees (for instance for interspike intervals smaller than the refractory period); the critique generally suggests changing the underlying

neg: Probability box tighter p-boxes than could be obtained by treating the interval estimates of the parameters as unrelated as is done for distributional p-boxes. There may be uncertainty about the shape of a probability distribution because the sample size of the empirical data characterizing it is small. Several methods in traditional statistics have been proposed to account for this sampling uncertainty about the distribution shape, including Kolmogorov–Smirnov and similar confidence bands, which are distribution-free in the sense that they make no assumption about the shape of the underlying distribution. There are related confidence-band methods that do make assumptions about the shape or

neg: Tukey lambda distribution and PDF can be tabulated for any number of cumulative probabilities, "p", using the quantile function "Q" to calculate the value "x", for each cumulative probability "p", with the probability density given by , the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table. The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for and is given by the formula (except when "λ" = 0) More

neg: BKL singularity reservations) an approximate method based on the assumption of statistical independence of the random quantity δ and of the random quantities "k", "x". For the function "P"(δ) an integral equation was set up which expressed the fact that the quantities δ and δ interconnected by the relation have the same distribution; this equation was solved numerically. In a later work, KL et al. showed that the distribution "P"(δ) can actually be found exactly by an analytical method. For statistical properties in the stationary limit, it is reasonable to introduce the so-called natural extension of the transformation "Tx" = {1/"x"} by

neg: Median and on the underlying population distribution. For a sample of size formula_90 from the normal distribution, the efficiency for large N is The efficiency tends to formula_92 as formula_93 tends to infinity. For univariate distributions that are "symmetric" about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median. If data are represented by a statistical model specifying a particular family of probability distributions, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution. Pareto interpolation is

neg: Generalized inverse Gaussian distribution Generalized inverse Gaussian distribution In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function where "K" is a modified Bessel function of the second kind, "a" > 0, "b" > 0 and "p" a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen. It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. It is also known as the Sichel distribution, after Herbert Sichel. Its statistical properties are discussed in

neg: Statistical coupling analysis distribution observed in all proteins, they are said to have some degree of conservation. In statistical coupling analysis, the conservation (ΔG) at each site ("i") is defined as: formula_1. Here, P describes the probability of finding amino acid "x" at position "i", and is defined by a function in binomial form as follows: where N is 100, n is the percentage of sequences with residue "x" (e.g. methionine) at position "i", and p corresponds to the approximate distribution of amino acid "x" in all positions among all sequenced proteins. The summation runs over all 20 amino acids. After ΔG is

neg: Independent and identically distributed random variables of statistical inference. The assumption (or requirement) that observations be IID tends to simplify the underlying mathematics of many statistical methods (see mathematical statistics and statistical theory). However, in practical applications of statistical modeling the assumption may or may not be realistic. To test how realistic the assumption is on a given data set, the autocorrelation can be computed, lag plots drawn or turning point test performed. The generalization of exchangeable random variables is often sufficient and more easily met. The assumption is important in the classical form of the central limit theorem, which states that the probability distribution of

neg: Brown–Forsythe test be used instead of the usual ANOVA. Good [1994,2005], noting that the deviations are linearly dependent, has modified the test so as to drop the redundant deviations. Levene's test uses the mean instead of the median. Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good statistical power. If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed Monte Carlo studies that indicated

neg: Overdispersion modeled with a normal distribution with the exact variance – the normal distribution is a two-parameter model, with mean and variance. Thus, in the absence of an underlying model, there is no notion of data being overdispersed relative to the normal model, though the fit may be poor in other respects (such as the higher moments of skew, kurtosis, etc.). However, in the case that the data is modeled by a normal distribution with an expected variation, it can be over- or under-dispersed relative to that prediction. For example, in a statistical survey, the margin of error (determined by sample

neg: Fisher's method exponential distribution. Scaling a value that follows an exponential distribution by a factor of two yields a quantity that follows a chi-squared distribution with two degrees of freedom. Finally, the sum of "k" independent chi-squared values, each with two degrees of freedom, follows a chi-squared distribution with 2"k" degrees of freedom. Dependence among statistical tests is generally positive, which means that the p-value of "X" is too small (anti-conservative) if the dependency is not taken into account. Thus, if Fisher's method for independent tests is applied in a dependent setting, and the p-value is not small enough to reject the

neg: Spaceflight radiation carcinogenesis can be used The result of either of these equations is inserted into the expression for the REID. Related probability distribution functions (PDFs) are grouped together into a combined probability distribution function, "P(x)". These PDFs are related to the risk coefficient of the normal form (dosimetry, bias and statistical uncertainties). After a sufficient number of trials have been completed (approximately 10), the results for the REID estimated are binned and the median values and confidence intervals are found. The chi-squared (χ) test is used for determining whether two separate PDFs are significantly different (denoted "p(R)" and "p(R)", respectively). Each "p(R)"

neg: Stationary ergodic process a measured process which is the superposition of two underlying processes, each with its own statistical properties. Although the measured process may be stationary in the long term, it is not appropriate to consider the sampled distribution to be the reflection of a single (ergodic) process: The ensemble average is meaningless. Also see ergodic theory and ergodic process. Stationary ergodic process In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such

neg: Hyperprior Hyperprior In Bayesian statistics, a hyperprior is a prior distribution on a hyperparameter, that is, on a parameter of a prior distribution. As with the term "hyperparameter," the use of "hyper" is to distinguish it from a prior distribution of a parameter of the model for the underlying system. They arise particularly in the use of conjugate priors. For example, if one is using a beta distribution to model the distribution of the parameter "p" of a Bernoulli distribution, then: In principle, one can iterate the above: if the hyperprior itself has hyperparameters, these may be called hyperhyperparameters, and so

neg: Probability integral transform the exponential distribution. One use for the probability integral transform in statistical data analysis is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of whether a uniform distribution is appropriate for the constructed dataset. Examples of this are P-P plots and Kolmogorov-Smirnov tests. A second use for the transformation is in the theory related to copulas which are a means of both defining and working with distributions

neg: Hyperparameter Hyperparameter In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. For example, if one is using a beta distribution to model the distribution of the parameter "p" of a Bernoulli distribution, then: One may take a single value for a given hyperparameter, or one can iterate and take a probability distribution on the hyperparameter itself, called a hyperprior. One often uses a prior which comes from a parametric family of probability distributions – this is done partly for explicitness

neg: Maximum likelihood sequence estimation That is, the estimate of {"x"("t")} is defined to be sequence of values which maximize the functional where "p"("r" | "x") denotes the conditional joint probability density function of the observed series {"r"("t")} given that the underlying series has the values {"x"("t")}. In contrast, the related method of maximum a posteriori estimation is formally the application of the maximum a posteriori (MAP) estimation approach. This is more complex than maximum likelihood sequence estimation and requires a known distribution (in Bayesian terms, a prior distribution) for the underlying signal. In this case the estimate of {"x"("t")} is defined to be sequence

neg: U-chart with the c-chart, the Poisson distribution is the basis for the chart and requires the same assumptions. The control limits for this chart type are formula_3 where formula_4 is the estimate of the long-term process mean established during control-chart setup. The observations formula_5 are plotted against these control limits, where x is the number of nonconformities for the ith subgroup and n is the number of inspection units in the ith subgroup. U-chart In statistical quality control, the u-chart is a type of control chart used to monitor "count"-type data where the sample size is greater than one, typically the

neg: Cosmic variance of a statistical process, the observer can only view one realization of that process, so our observation is statistically insignificant for saying much about the model, unless the observer is careful to include the variance. This variance is called the "cosmic variance" and is separate from other sources of experimental error: a very accurate measurement of only one value drawn from a distribution still leaves considerable uncertainty about the underlying model. Variance is normally plotted separately from other sources of uncertainty. Because it is necessarily a large fraction of the signal, workers must be very careful in interpreting the statistical

neg: Observational equivalence Observational equivalence Observational equivalence is the property of two or more underlying entities being indistinguishable on the basis of their observable implications. Thus, for example, two scientific theories are observationally equivalent if all of their empirically testable predictions are identical, in which case empirical evidence cannot be used to distinguish which is closer to being correct; indeed, it may be that they are actually two different perspectives on one underlying theory. In econometrics, two parameter values (or two "structures," from among a class of statistical models) are considered observationally equivalent if they both result in the same probability distribution of

neg: Winsorized mean corresponds to replacing parts with the corresponding quantiles. The winsorized mean is a useful estimator because it is less sensitive to outliers than the mean but will still give a reasonable estimate of central tendency or mean for almost all statistical models. In this regard it is referred to as a robust estimator. The winsorized mean uses more information from the distribution or sample than the median. However, unless the underlying distribution is symmetric, the winsorized mean of a sample is unlikely to produce an unbiased estimator for either the mean or the median. Winsorized mean A winsorized mean is

neg: Probability distribution distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function formula_23 defined on the real numbers such that Discrete distributions and some continuous distributions (like the Cantor distribution) do not admit such a density. In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function "X" from a probability space formula_25 to measurable space formula_26. A probability distribution of "X" is the pushforward measure "X"P of "X" , which is a probability measure on formula_26 satisfying "X"P = P"X". A frequent problem in statistical simulations (the Monte Carlo method) is the

neg: Information geometry change as function of formula_30. Every formula_30 determines one probability distribution for formula_6. Basically all distributions for which there exists an explicit analytical formula fall into this category (Binomial, Normal, Poisson, ...). The parameters in these cases have a concrete meaning in the underlying setup, which is a statistical model for the context of formula_6. The parameters are quite different in nature from formula_6 itself, because they do not describe formula_6, but the observation context for formula_6. A parameterization of the form with that mixes different distributions formula_40, is called a mixture distribution, "mixture" or formula_41-parameterization or "mixture" for short.

neg: X̅ and R chart across all units in the sample, formula_7, plus the range of the quality characteristic across all units in the sample as follows: The normal distribution is the basis for the charts and requires the following assumptions: The control limits for this chart type are: As with the formula_3 and s and individuals control charts, the formula_3 chart is only valid if the within-sample variability is constant. Thus, the R chart is examined before the formula_3 chart; if the R chart indicates the sample variability is in statistical control, then the formula_3 chart is examined to determine if the sample mean

neg: Kolmogorov–Smirnov test a step function formula_32 that has a discontinuity at formula_33. In other words, the limit formula_34, if it exists, is different from formula_35. Thus, when computing the statistic it is unclear how to replace the limit, unless we know the limiting value of the underlying distribution. In SAS, the Kolmogorov–Smirnov test is implemented in codice_1. The discretized KS test is implemented in the codice_2 function in the dgof package of the R project for statistical computing. In Stata, the command codice_3 performs a Kolmogorov–Smirnov test. The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions

neg: Chauvenet's criterion criterion provides an objective and quantitative method for data rejection, it does not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known. Chauvenet's criterion In statistical theory, Chauvenet's criterion (named for William Chauvenet) is a means of assessing whether one piece of experimental data — an outlier — from a set of observations, is likely to be spurious. The idea

neg: Generalization error formula_14 data points, the empirical error is: The generalization error is the difference between the expected and empirical error. This is the difference between error on the training set and error on the underlying joint probability distribution. It is defined as: An algorithm is said to generalize if: Since formula_4 cannot be computed for an unknown probability distribution, the generalization error cannot be computed either. Instead, the aim of many problems in statistical learning theory is to bound or characterize the generalization error in probability: That is, the goal is to characterize the probability formula_20 that the generalization error is

neg: Resampling (statistics) distribution is known. Thus one is always free to choose the statistic which best discriminates between hypothesis and alternative and which minimizes losses. Permutation tests can be used for analyzing unbalanced designs and for combining dependent tests on mixtures of categorical, ordinal, and metric data (Pesarin, 2001). They can also be used to analyze qualitative data that has been quantitized (i.e., turned into numbers). Permutation tests may be ideal for analyzing quantitized data that do not satisfy statistical assumptions underlying traditional parametric tests (e.g., t-tests, ANOVA) (Collingridge, 2013). Before the 1980s, the burden of creating the reference distribution was overwhelming

neg: Evaluation measures (information retrieval) an interpolated precision that takes the maximum precision over all recalls greater than formula_22: An alternative is to derive an analytical formula_21 function by assuming a particular parametric distribution for the underlying decision values. For example, a "binormal precision-recall curve" can be obtained by assuming decision values in both classes to follow a Gaussian distribution. For modern (web-scale) information retrieval, recall is no longer a meaningful metric, as many queries have thousands of relevant documents, and few users will be interested in reading all of them. Precision at k documents (P@k) is still a useful metric (e.g., P@10 or "Precision

neg: Information theory the product of the marginal distributions to the actual joint distribution: Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution. The "Kullback–Leibler divergence" (or "information divergence", "information gain", or "relative entropy") is a way of comparing two distributions: a "true" probability distribution "p(X)", and an arbitrary probability distribution "q(X)". If we compress data in a manner that assumes "q(X)" is the distribution underlying

neg: Prediction interval 100(1 − "p"/2) percentile of Student's t-distribution with "n" − 1 degrees of freedom. Therefore, the numbers are the endpoints of a 100(1 − "p")% prediction interval for formula_16. One can compute prediction intervals without any assumptions on the population; formally, this is a non-parametric method. Suppose one randomly draws a sample of two observations "X" and "X" from a population in which values are assumed to have a continuous probability distribution The answer is exactly 50%, "regardless" of the underlying population – the probability of picking 3 and then 7 is the same as picking 7 and then 3,

neg: Data dredging Data dredging Data dredging (also data fishing, data snooping, data butchery, and "p"-hacking) is the misuse of data analysis to find patterns in data that can be presented as statistically significant when in fact there is no real underlying effect. This is done by performing many statistical tests on the data and only paying attention to those that come back with significant results, instead of stating a single hypothesis about an underlying effect before the analysis and then conducting a single test for it. The process of data dredging involves automatically testing huge numbers of hypotheses about a single data

neg: One- and two-tailed tests so seen in this light, the specific but not extreme unlikely data should not be considered significant. If the test statistic follows a Student's "t"-distribution in the null hypothesis – which is common where the underlying variable follows a normal distribution with unknown scaling factor, then the test is referred to as a one-tailed or two-tailed "t"-test. If the test is performed using the actual population mean and variance, rather than an estimate from a sample, it would be called a one-tailed or two-tailed "Z"-test. The statistical tables for "t" and for "Z" provide critical values for both one- and

neg: Box plot Box plot In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes ("whiskers") indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram. Outliers may be plotted as individual points. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). The spacings between the different parts of

neg: Box plot be a useful tool for understanding the box plot (Figure 5). Box plot In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes ("whiskers") indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram. Outliers may be plotted as individual points. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution (though Tukey's boxplot assumes symmetry for the whiskers and

neg: P-value the "p" = 1/2 significance level. This is and other work by Arbuthnot is credited as "… the first use of significance tests …" the first example of reasoning about statistical significance, and "… perhaps the first published report of a nonparametric test …", specifically the sign test; see details at . The same question was later addressed by Pierre-Simon Laplace, who instead used a "parametric" test, modeling the number of male births with a binomial distribution: The "p"-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test, using the chi-squared distribution and notated as capital P.

neg: Chi-squared distribution "less extreme" than this point, subtracting the CDF value from 1 gives the "p"-value. A low "p"-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and not-significant results. The table below gives a number of "p"-values matching to "χ" for the first 10 degrees of freedom. These values can be calculated evaluating the quantile function (also known as “inverse CDF” or “ICDF”) of the chi-squared distribution; e. g., the ICDF for and yields as in the table above. This

neg: Control limits the plot points on the chart, as long as the process is and remains in statistical control. A false-detection rate of at least 0.27% is therefore expected. It is often not known whether a particular process generates data that conform to particular distributions, but the Chebyshev's inequality and the Vysochanskij–Petunin inequality allow the inference that for any unimodal distribution at least 95% of the data will be encapsulated by limits placed at 3 sigma. Control limits Control limits, also known as natural process limits, are horizontal lines drawn on a statistical process control chart, usually at a distance of ±3

neg: Size effect on structural strength fiber-polymer composites, wind turbines, large geotechnical excavations, earth and rock slopes, ﬂoating sea ice carrying loads, oil platforms under ice forces, etc. Their design depends on the material properties measured on much smaller laboratory specimens. These properties must be extrapolated to sizes greater by one or two orders of magnitude. Even if an expensive full-scale failure test, for example a failure test of the rudder of a very large aircraft, can be carried out, it is financially prohibitive to repeat it thousand times to obtain the statistical distribution of load capacity. Such statistical information, underlying the safety factors, is obtainable

neg: BKL singularity formula_100 (in its calculation one must note that formula_101). [[File:Bkl fig3a.svg|250px|left|thumb|Figure 5. The probability distribution function "P"(δ). Red line: exact function . Blue line: approximate solution of the integral equation in. Both curves appear to be strikingly similar and the means of both distributions are 0.50.]] Since by δ is expressed in terms of the random quantities "x" and "x", the knowledge of their joint distribution makes it possible to calculate the statistical distribution "P"(δ) by integrating "P"("x", "x") over one of the variables at a constant value of δ. Due to symmetry of the function with respect to the

neg: Maximum spacing estimation Maximum spacing estimation In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of "spacings" in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points. The concept underlying the method is based on the probability integral transform, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random

neg: Multimodal distribution Another bimodality index has been proposed by de Michele and Accatino. Their index ("B") is where "μ" is the arithmetic mean of the sample and where "m" is number of data points in the "i" bin,"x" is the center of the "i" bin and "L" is the number of bins. The authors suggested a cut off value of 0.1 for "B" to distinguish between a bimodal ("B" > 0.1)and unimodal ("B" < 0.1) distribution. No statistical justification was offered for this value. A further index ("B") has been proposed by Sambrook Smith "et al" formula_38 where "p" and "p" are

neg: Energy distance null hypothesis that two random variables, "X" and "Y", have the same probability distributions: formula_17 . For statistical samples from "X" and "Y": the following arithmetic averages of distances are computed between the X and the Y samples: The E-statistic of the underlying null hypothesis is defined as follows: One can prove that formula_22 and that the corresponding population value is zero if and only if "X" and "Y" have the same distribution (formula_17). Under this null hypothesis the test statistic converges in distribution to a quadratic form of independent standard normal random variables. Under the alternative hypothesis "T" tends

neg: Conditional independence 0 corresponding, respectively, to whether or not the "i"th chosen voter will or will not vote "yes". In a frequentist approach to statistical inference one would not attribute any probability distribution to "p" (unless the probabilities could be somehow interpreted as relative frequencies of occurrence of some event or as proportions of some population) and one would say that "X", ..., "X" are independent random variables. By contrast, in a Bayesian approach to statistical inference, one would assign a probability distribution to "p" regardless of the non-existence of any such "frequency" interpretation, and one would construe the probabilities as degrees

neg: Binomial proportion confidence interval close to "p" = 0.5. The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion . The Jeffreys prior for this problem is a Beta distribution with parameters . After observing successes in trials, the posterior distribution for is a Beta distribution with parameters . When and , the Jeffreys interval is taken to be the equal-tailed posterior probability interval, i.e., the and quantiles of a Beta distribution with parameters . These quantiles need to be computed numerically, although this is reasonably simple with modern statistical software. In order to avoid

neg: BKL singularity thus represent τ as The variance of this sum in the limit of large "s" is </math> depend only on the differences |"p" − "p"′|. The mean value formula_117; the mean square formula_118 By taking into account also the values of correlations formula_119 with "p" = 1, 2, 3 (calculated numerically) the final result "D" = (3.5 ± 0.1)"s" is obtained. With increasing "s" the relative fluctuation formula_120 The respective statistical distribution is given by the same Gaussian distribution in which the random variable is now "s" at a given τ: From this point of view, the source of the

neg: Boundary problem in geography multiple boundary units found in geographical data (Griffith 1980). Martin (1987) also argued that some of the underlying assumptions of the statistical techniques are unrealistic or unreasonably strict. Moreover, Griffith (1985) himself also identified the inferiority of the techniques through simulation analysis. As particularly applicable using GIS technologies (Haslett et al. 1990; Openshaw et al. 1987), a possible solution for addressing both edge and shape effects is to an re-estimation of the spatial or process under repeated random realizations of the boundary. This solution provides an experimental distribution that can be subjected to statistical tests (Fotheringham and Rogerson 1993). As

neg: Disparity filter algorithm of weighted network weight distribution of a node with degree "k" follows formula_1. The disparity filter algorithm is based on p-value statistical significance test of the null model: For a given normalized weight "p", the p-value "α" of "p" based on the null model is given by formula_2 which reduces to formula_3. The meaning of "α" is the probability of having normalized weight larger or equal to "p" in the framework of the given null model. By setting a significance level "α" (between 0 and 1), for any link of normalized weight "p", if "α" is larger than "α", it will be filtered

neg: Rank product determine how likely a given RP value or better is observed in a random experiment. Permutation re-sampling requires a computationally demanding number of permutations to get reliable estimates of the "p"-values for the most differentially expressed genes, if "n" is large. Eisinga, Breitling and Heskes (2013) provide the exact probability mass distribution of the rank product statistic. Calculation of the exact "p"-values offers a substantial improvement over permutation approximation, most significantly for that part of the distribution rank product analysis is most interested in, i.e., the thin right tail. However, exact statistical significance of large rank products may take unacceptable

neg: ANOVA on ranks 'quantile normalization' in which a further transformation is applied to the ranks such that the resulting values have some defined distribution (often a normal distribution with a specified mean and variance). Further analyses of quantile-normalized data may then assume that distribution to compute significance values. However, two specific types of secondary transformations, the random normal scores and expected normal scores transformation, have been shown to greatly inflate Type I errors and severely reduce statistical power. The ANOVA on ranks has never been recommended when the underlying assumption of homogeneous variances has been violated, either by itself, or in conjunction with

neg: Structured expert judgment: the classical model Figure 1 shows the distribution of experts over the number of assessed calibration variables. The p-values are sensitive to the power of the statistical test, and hence to the number of calibration variables. These numbers are roughly comparable for experts in the post-2006 data. Figure 2 shows the p-values of all post-2006 experts, arranged from best to worst. In this summary, 227 of the 320 experts have a statistical accuracy score less than 0.05, which is the traditional rejection threshold for simple hypothesis testing. Half of the experts score below 0.005, and roughly one third fall into the abysmal range

neg: Student's t-distribution to construct the "p"-value. Student's t distribution can be generalized to a three parameter location-scale family, introducing a location parameter formula_56 and a scale parameter formula_26, through the relation or This means that formula_118 has a classic Student's t distribution with formula_119 degrees of freedom. The resulting non-standardized Student's "t"-distribution has a density defined by Here, formula_26 does "not" correspond to a standard deviation: it is not the standard deviation of the scaled "t" distribution, which may not even exist; nor is it the standard deviation of the underlying normal distribution, which is unknown. formula_26 simply sets the overall scaling

neg: Statistics kinds of statistical statements may have truth values which are not invariant under some transformations. Whether or not a transformation is sensible to contemplate depends on the question one is trying to answer" (Hand, 2004, p. 82). Consider independent identically distributed (IID) random variables with a given probability distribution: standard statistical inference and estimation theory defines a random sample as the random vector given by the column vector of these IID variables. The population being examined is described by a probability distribution that may have unknown parameters. A statistic is a random variable that is a function of the random

neg: Statistical model As a different example, the set of positive-mean Gaussian distributions, which has dimension 2, is nested within the set of all Gaussian distributions. It is assumed that there is a "true" probability distribution underlying the observed data, induced by the process that generated the data. The main goal of model selection is to make statements about which elements of formula_3 are most likely to adequately approximate the true distribution. Models can be compared to each other by exploratory data analysis or confirmatory data analysis. In exploratory analysis, a variety of models are formulated and an assessment is performed of how

neg: Information projection formula_10 to make sure KL divergence stays finite. For M-projection, formula_7 will typically over-estimate the support of formula_8. This is due to formula_13 whenever formula_14 to make sure KL divergence stays finite. The concept of information projection can be extended to arbitrary statistical f-divergences and other divergences. Information projection In information theory, the information projection or I-projection of a probability distribution "q" onto a set of distributions "P" is where formula_2 is the Kullback–Leibler divergence from "q" to "p". Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection formula_3 is the "closest" distribution to "q" of all the

neg: Student's t-test exactly follows or closely approximates a "t"-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test. Once the "t" value and degrees of freedom are determined, a "p"-value can be found using a table of values from Student's "t"-distribution. If the calculated "p"-value is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis. In testing the null hypothesis

neg: One- and two-tailed tests zero values, and has only one tail, the upper one. It measures goodness of fit of data with a theoretical distribution, with zero corresponding to exact agreement with the theoretical distribution; the "p"-value thus measures how likely the fit would be this bad or worse. The distinction between one-tailed and two-tailed tests was popularized by Ronald Fisher in the influential book Statistical Methods for Research Workers, where he applied it especially to the normal distribution, which is a symmetric distribution with two equal tails. The normal distribution is a common measure of location, rather than goodness-of-fit, and has two tails,

neg: Sato–Tate conjecture Sato–Tate conjecture In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves "E" over the finite field with "p" elements, with "p" a prime number, obtained from an elliptic curve "E" over the rational number field, by the process of reduction modulo a prime for almost all "p". If "N" denotes the number of points on "E" and defined over the field with "p" elements, the conjecture gives an answer to the distribution of the second-order term for "N". That is, by Hasse's theorem on elliptic curves we have as "p" → ∞, and the

neg: Sato–Tate conjecture holds in most cases. Sato–Tate conjecture In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves "E" over the finite field with "p" elements, with "p" a prime number, obtained from an elliptic curve "E" over the rational number field, by the process of reduction modulo a prime for almost all "p". If "N" denotes the number of points on "E" and defined over the field with "p" elements, the conjecture gives an answer to the distribution of the second-order term for "N". That is, by Hasse's theorem on elliptic curves we have as "p"

neg: Dirichlet distribution a model then the following holds: This relationship is used in Bayesian statistics to estimate the underlying parameter p of a categorical distribution given a collection of "N" samples. Intuitively, we can view the hyperprior vector α as pseudocounts, i.e. as representing the number of observations in each category that we have already seen. Then we simply add in the counts for all the new observations (the vector c) in order to derive the posterior distribution. In Bayesian mixture models and other hierarchical Bayesian models with mixture components, Dirichlet distributions are commonly used as the prior distributions for the categorical

neg: Cumulative distribution function the discrete values 0 and 1, with equal probability. Then the CDF of formula_1 is given by Sometimes, it is useful to study the opposite question and ask how often the random variable is "above" a particular level. This is called the complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic "at least" as extreme as the one observed. Thus, provided that the test statistic, "T", has a continuous distribution, the one-sided

neg: Yoix facilitate interaction with the GraphViz layout engines. The YChart data visualization toolkit was added to the Yoix distribution with release 2.2.0. YChart allows one to organize and display cells of data. Two interactive YChart applications contained in the Yoix distribution are a Periodic Table of the Elements and a Unicode Chart. A program to demonstrate using YChart with variable width cells, as might occur with a schedule, is also available in the Yoix distribution. The Yoix distribution also includes a Yoix package, called "Byzgraf", for rendering basic data plots such as line charts, histograms and statistical box plots. As currently

neg: Mean from the mean of the underlying distribution, the population mean (denoted formula_3 or formula_4). In probability and statistics, the population mean, or expected value, are a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable "X", the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value "x" of "X" and its probability "p"("x"), and then adding all

neg: EWMA chart EWMA chart In statistical quality control, the EWMA chart (or exponentially weighted moving average chart) is a type of control chart used to monitor either variables or attributes-type data using the monitored business or industrial process's entire history of output. While other control charts treat rational subgroups of samples individually, the EWMA chart tracks the exponentially-weighted moving average of all prior sample means. EWMA weights samples in geometrically decreasing order so that the most recent samples are weighted most highly while the most distant samples contribute very little. Although the normal distribution is the basis of the EWMA chart, the

neg: P-value two standard deviations (on a normal distribution) for statistical significance (see 68–95–99.7 rule). He then computed a table of values, similar to Elderton but, importantly, reversed the roles of "χ" and "p." That is, rather than computing "p" for different values of "χ" (and degrees of freedom "n"), he computed values of "χ" that yield specified "p"-values, specifically 0.99, 0.98, 0.95, 0,90, 0.80, 0.70, 0.50, 0.30, 0.20, 0.10, 0.05, 0.02, and 0.01. That allowed computed values of "χ" to be compared against cutoffs and encouraged the use of "p"-values (especially 0.05, 0.02, and 0.01) as cutoffs, instead of computing and

neg: Qualitative variation of "r" finite probability distributions. Let these distributions have values that are divided into "k" categories. Then the distance "D" is where "r" is the number of discrete probability distributions in each population, "k" is the number of categories in distributions "P" and "Q" and "p" (respectively "q") is the theoretical probability of category "i" in distribution "P" ("Q") in population "P"("Q"). Its statistical properties were examined by Sanchez "et al." who recommended a bootstrap procedure to estimate confidence intervals when testing for differences between samples. Let where min("x","y") is the lesser value of the pair "x" and "y". Then

neg: Cross entropy Cross entropy In information theory, the cross entropy between two probability distributions formula_1 and formula_2 over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set, if a coding scheme is used that is optimized for an "artificial" probability distribution formula_2, rather than the "true" distribution formula_1. The cross entropy for the distributions formula_1 and formula_2 over a given set is defined as follows: where formula_8 is the entropy of formula_1, and formula_10 is the Kullback–Leibler divergence of formula_2 from formula_1 (also known as the "relative entropy" of "p"

neg: P-value its cumulative distribution function (CDF) is often a difficult problem. Today, this computation is done using statistical software, often via numeric methods (rather than exact formulae), but, in the early and mid 20th century, this was instead done via tables of values, and one interpolated or extrapolated "p"-values from these discrete values. Rather than using a table of "p"-values, Fisher instead inverted the CDF, publishing a list of values of the test statistic for given fixed "p"-values; this corresponds to computing the quantile function (inverse CDF). When the null hypothesis is true, if it takes the form formula_33, and the

neg: Continuity correction variable with the same expected value and the same variance as "X", i.e., E("Y") = "np" and var("Y") = "np"(1 − "p"). This addition of 1/2 to "x" is a continuity correction. A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if "X" has a Poisson distribution with expected value λ then the variance of "X" is also λ, and if "Y" is normally distributed with expectation and variance both λ. Before the ready availability of statistical software having the ability to evaluate probability distribution functions

neg: Project networks identical to such networks. One of key characteristics of accidental networks, which is important for understanding of properties and processes which in them happen is such statistical characteristic of an accidental network as distribution of nodes on number of connections (DD, degree distribution). The characteristic of DD, distribution of knots on number of connections of P(q) is probability that incidentally chosen knot in a casual network has the power of q: Here {N(q)} — average of knots of degree of q in networks, at the same time averaging undertakes on the whole ensemble. The total number of knots at all

neg: Ramanathan Gnanadesikan to work in the nascent statistics group at Bell Telephone Laboratories and became a department head shortly afterwards. He worked closely with Martin Wilk, with whom he wrote a number of papers on graphical methods for data analysis. The most widely cited of these (Wilk and Gnanadesikan, 1968) describes the Q-q and P-p plots, which are used to compare different statistical distributions. This work arose in part out of work in speaker recognition, as it was found that the statistical distribution of energy across frequency bands was different for different speakers. His highly-cited 1977 monograph on multivariate data analysis has

neg: Quantum logic classical case, a probability distribution of truth values T and F. Say the probabilities are "p" for T and "q" = 1 − "p" for F. By the previous section "p" = Tr("S" "E") and "q" = Tr("S" ("I" − "E")). Perhaps the most fundamental difference between classical and quantum systems is the following: regardless of what process is used to determine "E" immediately after the measurement the system will be in one of two statistical states: (We leave to the reader the handling of the degenerate cases in which the denominators may be 0.) We now form the convex