1.
Chi-squared distribution
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In probability theory and statistics, the chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. When it is being distinguished from the more general noncentral chi-squared distribution, many other statistical tests also use this distribution, like Friedmans analysis of variance by ranks. Zk are independent, standard normal variables, then the sum of their squares. This is usually denoted as Q ∼ χ2 or Q ∼ χ k 2, the chi-squared distribution has one parameter, k — a positive integer that specifies the number of degrees of freedom The chi-squared distribution is used primarily in hypothesis testing. Unlike more widely known such as the normal distribution and the exponential distribution. It arises in the following tests, among others. The primary reason that the distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t statistic in a t-test, for these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution. Testing hypotheses using a distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a normal distribution. So wherever a normal distribution could be used for a hypothesis test, specifically, suppose that Z is a standard normal random variable, with mean =0 and variance =1. A sample drawn at random from Z is a sample from the shown in the graph of the standard normal distribution. Define a new random variable Q, to generate a random sample from Q, take a sample from Z and square the value. The distribution of the values is given by the random variable Q = Z2. The distribution of the random variable Q is an example of a chi-squared distribution, the subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single normal distribution is said to have 1 degree of freedom. Just as extreme values of the distribution have low probability. An additional reason that the distribution is widely used is that it is a member of the class of likelihood ratio tests
2.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
3.
Hypothesis test
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A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference, commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. Hypothesis tests are used in determining outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance. The most common techniques are based on either Akaike information criterion or Bayes factor. Confirmatory data analysis can be contrasted with exploratory data analysis, which may not have pre-specified hypotheses, Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inference, although the two types of inference have notable differences. Statistical hypothesis tests define a procedure that controls the probability of incorrectly deciding that a position is incorrect. The procedure is based on how likely it would be for a set of observations to occur if the hypothesis were true. Note that this probability of making a decision is not the probability that the null hypothesis is true. This contrasts with other techniques of decision theory in which the null. One naïve Bayesian approach to hypothesis testing is to base decisions on the posterior probability, a number of other approaches to reaching a decision based on data are available via decision theory and optimal decisions, some of which have desirable properties. Hypothesis testing, though, is a dominant approach to analysis in many fields of science. Extensions to the theory of testing include the study of the power of tests. Such considerations can be used for the purpose of sample size determination prior to the collection of data, in the statistics literature, statistical hypothesis testing plays a fundamental role. The usual line of reasoning is as follows, There is an initial hypothesis of which the truth is unknown. The first step is to state the relevant null and alternative hypotheses and this is important, as mis-stating the hypotheses will muddy the rest of the process. This is equally important as invalid assumptions will mean that the results of the test are invalid, decide which test is appropriate, and state the relevant test statistic T. Derive the distribution of the test statistic under the hypothesis from the assumptions. In standard cases this will be a well-known result, for example, the test statistic might follow a Students t distribution or a normal distribution
4.
Null hypothesis
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In inferential statistics, the term null hypothesis is a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. The null hypothesis is generally assumed to be true until evidence indicates otherwise, in statistics, it is often denoted H0. The concept of a hypothesis is used differently in two approaches to statistical inference. In the significance testing approach of Ronald Fisher, a hypothesis is rejected if the observed data are significantly unlikely to have occurred if the null hypothesis were true. In this case the null hypothesis is rejected and a hypothesis is accepted in its place. If the data are consistent with the hypothesis, then the null hypothesis is not rejected. In neither case is the hypothesis or its alternative proven, the null hypothesis is tested with data. This is analogous to a trial, in which the defendant is assumed to be innocent until proven guilty beyond a reasonable doubt. Proponents of each approach criticize the other approach, nowadays, though, a hybrid approach is widely practiced and presented in textbooks. The hybrid is in turn criticized as incorrect and incoherent—for details, hypothesis testing requires constructing a statistical model of what the data would look like given that chance or random processes alone were responsible for the results. The hypothesis that chance alone is responsible for the results is called the null hypothesis, the model of the result of the random process is called the distribution under the null hypothesis. The obtained results are compared with the distribution under the null hypothesis. The null hypothesis assumes no relationship between variables in the population from which the sample is selected, If the data-set of a randomly selected representative sample is very unlikely relative to the null hypothesis, the experimenter rejects the null hypothesis concluding it is false. This class of data-sets is usually specified via a test statistic which is designed to measure the extent of apparent departure from the null hypothesis. If the data do not contradict the hypothesis, then only a weak conclusion can be made, namely. For instance, a drug may reduce the chance of having a heart attack. Possible null hypotheses are this drug does not reduce the chances of having an attack or this drug has no effect on the chances of having a heart attack. The test of the consists of administering the drug to half of the people in a study group as a controlled experiment
5.
Pearson's chi-squared test
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Pearsons chi-squared test is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is suitable for unpaired data from large samples and it is the most widely used of many chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900, in contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to Pearson χ-squared test or statistic are used. It tests a hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution. The events considered must be exclusive and have total probability 1. A common case for this is where the events each cover an outcome of a categorical variable, a simple example is the hypothesis that an ordinary six-sided die is fair. Pearsons chi-squared test is used to three types of comparison, goodness of fit, homogeneity, and independence. A test of goodness of fit establishes whether a frequency distribution differs from a theoretical distribution. A test of homogeneity compares the distribution of counts for two or more using the same categorical variable. A test of independence assesses whether unpaired observations on two variables, expressed in a table, are independent of each other. Determine the degrees of freedom, df, of that statistic, for a test of goodness-of-fit, this is essentially the number of categories reduced by the number of parameters of the fitted distribution. For test of homogeneity, df = *, where Rows corresponds to the number of categories, and Cols corresponds the number of independent groups. For test of independence, df = *, where in case, Rows corresponds to number of categories in one variable. Select a desired level of confidence for the result of the test, accept or reject the null hypothesis that the observed frequency distribution is the same as the theoretical distribution based on whether the test statistic exceeds the critical value of χ2. If the test statistic exceeds the value of χ2, the null hypothesis can be rejected. In this case N observations are divided among n cells, a simple application is to test the hypothesis that, in the general population, values would occur in each cell with equal frequency. The reduction in the degrees of freedom is calculated as p = s +1, for instance, when checking a three-co-variate Weibull distribution, p =4, and when checking a normal distribution, p =3, and when checking a Poisson distribution, p =2. Thus, there will be n − p degrees of freedom and it should be noted that the degrees of freedom are not based on the number of observations as with a Students t or F-distribution
6.
Variance
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The variance has a central role in statistics. It is used in statistics, statistical inference, hypothesis testing, goodness of fit. This makes it a central quantity in numerous such as physics, biology, chemistry, cryptography, economics. The variance of a random variable X is the value of the squared deviation from the mean of X, μ = E . This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, the variance can also be thought of as the covariance of a random variable with itself, Var = Cov . The variance is also equivalent to the second cumulant of a probability distribution that generates X, the variance is typically designated as Var , σ X2, or simply σ2. On computational floating point arithmetic, this equation should not be used, if a continuous distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either. Many other distributions for which the value does exist also do not have a finite variance because the integral in the variance definition diverges. An example is a Pareto distribution whose index k satisfies 1 < k ≤2. e, the normal distribution with parameters μ and σ is a continuous distribution whose probability density function is given by f =12 π σ2 e −22 σ2. In this distribution, E = μ and the variance Var is related with σ via Var = ∫ − ∞ ∞22 π σ2 e −22 σ2 d x = σ2. The role of the distribution in the central limit theorem is in part responsible for the prevalence of the variance in probability. The exponential distribution with parameter λ is a distribution whose support is the semi-infinite interval. Its probability density function is given by f = λ e − λ x, the variance is equal to Var = ∫0 ∞2 λ e − λ x d x = λ −2. So for an exponentially distributed random variable, σ2 = μ2, the Poisson distribution with parameter λ is a discrete distribution for k =0,1,2, …. Its probability mass function is given by p = λ k k, E − λ, and it has expected value μ = λ. The variance is equal to Var = ∑ k =0 ∞ λ k k, E − λ2 = λ, So for a Poisson-distributed random variable, σ2 = μ. The binomial distribution with n and p is a discrete distribution for k =0,1,2, …, n. Its probability mass function is given by p = p k n − k, the variance is equal to Var = ∑ k =0 n p k n − k 2 = n p
7.
Central limit theorem
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If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution. The central limit theorem has a number of variants, in its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, in more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory. When the variance of the i. i. d, Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i. i. d, Random variables with power law tail distributions decreasing as | x |−α −1 where 0 < α <2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Suppose we are interested in the sample average S n, = X1 + ⋯ + X n n of these random variables, by the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n → ∞. The classical central limit theorem describes the size and the form of the stochastic fluctuations around the deterministic number µ during this convergence. For large enough n, the distribution of Sn is close to the distribution with mean µ. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi, formally, the theorem can be stated as follows, Lindeberg–Lévy CLT. Suppose is a sequence of i. i. d, Random variables with E = µ and Var = σ2 < ∞. Then as n approaches infinity, the random variables √n converge in distribution to a normal N, n → d N. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | =0, the theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the limit theorem the random variables Xi have to be independent. The theorem also requires that random variables | Xi | have moments of order. Suppose is a sequence of independent random variables, each with finite expected value μi, in practice it is usually easiest to check Lyapunov’s condition for δ =1. If a sequence of random variables satisfies Lyapunov’s condition, then it also satisfies Lindeberg’s condition, the converse implication, however, does not hold. In the same setting and with the notation as above. Suppose that for every ε >0 lim n → ∞1 s n 2 ∑ i =1 n E =0 where 1 is the indicator function
8.
Normal distribution
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In probability theory, the normal distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The normal distribution is useful because of the limit theorem. Physical quantities that are expected to be the sum of independent processes often have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are normally distributed, the normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped, the probability density of the normal distribution is, f =12 π σ2 e −22 σ2 Where, μ is mean or expectation of the distribution. σ is standard deviation σ2 is variance A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate. The simplest case of a distribution is known as the standard normal distribution. The factor 1 /2 in the exponent ensures that the distribution has unit variance and this function is symmetric around x =0, where it attains its maximum value 1 /2 π and has inflection points at x = +1 and x = −1. Authors may differ also on which normal distribution should be called the standard one, the probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a normal deviate, then X = Zσ + μ will have a normal distribution with expected value μ. Conversely, if X is a normal deviate, then Z = /σ will have a standard normal distribution. Every normal distribution is the exponential of a function, f = e a x 2 + b x + c where a is negative. In this form, the mean value μ is −b/, for the standard normal distribution, a is −1/2, b is zero, and c is − ln /2. The standard Gaussian distribution is denoted with the Greek letter ϕ. The alternative form of the Greek phi letter, φ, is used quite often. The normal distribution is often denoted by N. Thus when a random variable X is distributed normally with mean μ and variance σ2, some authors advocate using the precision τ as the parameter defining the width of the distribution, instead of the deviation σ or the variance σ2
9.
George Biddell Airy
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Sir George Biddell Airy KCB PRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His reputation has been tarnished by allegations that, through his inaction, Airy was born at Alnwick, one of a long line of Airys who traced their descent back to a family of the same name residing at Kentmere, in Westmorland, in the 14th century. The branch to which he belonged, having suffered in the English Civil War, moved to Lincolnshire, Airy was educated first at elementary schools in Hereford, and afterwards at Colchester Royal Grammar School. An introverted child, Airy gained popularity with his schoolmates through his skill in the construction of peashooters. From the age of 13, Airy stayed frequently with his uncle, Arthur Biddell at Playford, Biddell introduced Airy to his friend Thomas Clarkson, the slave trade abolitionist who lived at Playford Hall. As a result, he entered Trinity in 1819, as a sizar, meaning that he paid a reduced fee, here he had a brilliant career, and seems to have been almost immediately recognised as the leading man of his year. In 1822 he was elected scholar of Trinity, and in the year he graduated as senior wrangler. On 1 October 1824 he was elected fellow of Trinity, and this chair he held for little more than a year, being elected in February 1828 Plumian professor of astronomy and director of the new Cambridge Observatory. In 1836 he was elected a Fellow of the Royal Society and in 1840, in 1859 he became foreign member of the Royal Netherlands Academy of Arts and Sciences. At the Cambridge Observatory Airy soon showed his power of organisation, the only telescope in the establishment when he took charge was the transit instrument, and to this he vigorously devoted himself. Before long a mural circle was installed, and regular observations were instituted with it in 1833, Airys writings during this time are divided between mathematical physics and astronomy. In 1831 the Copley Medal of the Royal Society was awarded to him for these researches. One of the sections of his able and instructive report was devoted to A Comparison of the Progress of Astronomy in England with that in other Countries and this reproach was subsequently to a great extent removed by his own labours. One of the most remarkable of Airys researches was his determination of the density of the Earth. In 1826, the idea occurred to him of attacking this problem by means of experiments at the top. His first attempt, made in the year, at the Dolcoath mine in Cornwall. A second attempt in 1828 was defeated by a flooding of the mine, the experiments eventually took place at the Harton pit near South Shields in 1854.566. The currently accepted value for Earths density is 5.5153 g/cm³, in 1830, Airy calculated the lengths of the polar radius and equatorial radius of the earth using measurements taken in the UK
10.
Karl Pearson
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Karl Pearson FRS was an influential English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics, Pearson was also a protégé and biographer of Sir Francis Galton. Pearson was born in Islington, London to William and Fanny and he then travelled to Germany to study physics at the University of Heidelberg under G H Quincke and metaphysics under Kuno Fischer. He next visited the University of Berlin, where he attended the lectures of the famous physiologist Emil du Bois-Reymond on Darwinism, Pearson also studied Roman Law, taught by Bruns and Mommsen, medieval and 16th century German Literature, and Socialism. He became an historian and Germanist and spent much of the 1880s in Berlin, Heidelberg, Vienna, Saig bei Lenzkirch. He wrote on Passion plays, religion, Goethe, Werther, as well as sex-related themes, Pearson was offered a Germanics post at Kings College, Cambridge. Comparing Cambridge students to those he knew from Germany, Karl found German students inathletic and he wrote his mother, I used to think athletics and sport was overestimated at Cambridge, but now I think it cannot be too highly valued. Have you ever attempted to conceive all there is in the world worth knowing—that not one subject in the universe is unworthy of study, mankind seems on the verge of a new and glorious discovery. What Newton did to simplify the planetary motions must now be done to unite in one whole the various isolated theories of mathematical physics, Pearson then returned to London to study law, emulating his father. His next career move was to the Inner Temple, where he read law until 1881, after this, he returned to mathematics, deputising for the mathematics professor at Kings College, London in 1881 and for the professor at University College, London in 1883. In 1884, he was appointed to the Goldsmid Chair of Applied Mathematics and Mechanics at University College, Pearson became the editor of Common Sense of the Exact Sciences when William Kingdon Clifford died. The collaboration, in biometry and evolutionary theory, was a fruitful one, Weldon introduced Pearson to Charles Darwins cousin Francis Galton, who was interested in aspects of evolution such as heredity and eugenics. Pearson became Galtons protégé—his statistical heir as some have put it—at times to the verge of hero worship, in 1890 Pearson married Maria Sharpe. Maria died in 1928 and in 1929 Karl married Margaret Victoria Child and he and his family lived at 7 Well Road in Hampstead, now marked with a blue plaque. He predicted that Galton, rather than Charles Darwin, would be remembered as the most prodigious grandson of Erasmus Darwin, when Galton died, he left the residue of his estate to the University of London for a Chair in Eugenics. Pearson was the first holder of this chair — the Galton Chair of Eugenics and he formed the Department of Applied Statistics, into which he incorporated the Biometric and Galton laboratories. He remained with the department until his retirement in 1933, Pearson was a zealous atheist and a freethinker. This book covered several themes that were later to become part of the theories of Einstein, Pearson asserted that the laws of nature are relative to the perceptive ability of the observer
11.
Skewness
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In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined, the qualitative interpretation of the skew is complicated and unintuitive. Skew must not be thought to refer to the direction the curve appears to be leaning, in fact, conversely, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the tail is fat. Further, in multimodal distributions and discrete distributions, skewness is also difficult to interpret, importantly, the skewness does not determine the relationship of mean and median. In cases where it is necessary, data might be transformed to have a normal distribution, consider the two distributions in the figure just below. Within each graph, the values on the side of the distribution taper differently from the values on the left side. A left-skewed distribution usually appears as a right-leaning curve, positive skew, The right tail is longer, the mass of the distribution is concentrated on the left of the figure. A right-skewed distribution usually appears as a left-leaning curve, Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the sequence, whose values are evenly distributed around a central value of 50. If the distribution is symmetric, then the mean is equal to the median, if, in addition, the distribution is unimodal, then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4, note, however, that the converse is not true in general, i. e. zero skewness does not imply that the mean is equal to the median. Paul T. von Hippel points out, Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew and this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long, most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they contradict the textbook interpretation of the median. It is sometimes referred to as Pearsons moment coefficient of skewness, or simply the moment coefficient of skewness, the last equality expresses skewness in terms of the ratio of the third cumulant κ3 to the 1. 5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant, the skewness is also sometimes denoted Skew. Starting from a standard cumulant expansion around a distribution, one can show that skewness =6 /standard deviation + O
12.
Pearson distribution
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The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics, the Pearson system was originally devised in an effort to model visibly skewed observations. Except in pathological cases, a family can be made to fit the observed mean. However, it was not known how to construct probability distributions in which the skewness and this need became apparent when trying to fit known theoretical models to observed data that exhibited skewness. Pearsons examples include data, which are usually asymmetric. In his original paper, Pearson identified four types of distributions in addition to the normal distribution, a second paper fixed two omissions, it redefined the type V distribution and introduced the type VI distribution. Together the first two cover the five main types of the Pearson system. In a third paper, Pearson introduced further special cases and subtypes, rhind devised a simple way of visualizing the parameter space of the Pearson system, which was subsequently adopted by Pearson. The Pearson types are characterized by two quantities, commonly referred to as β1 and β2, the first is the square of the skewness, β1 = γ12 where γ1 is the skewness, or third standardized moment. The second is the kurtosis, or fourth standardized moment. The diagram on the shows which Pearson type a given concrete distribution belongs to. Many of the skewed and/or non-mesokurtic distributions familiar to us today were still unknown in the early 1890s, what is now known as the beta distribution had been used by Thomas Bayes as a posterior distribution of the parameter of a Bernoulli distribution in his 1763 work on inverse probability. The Beta distribution gained prominence due to its membership in Pearsons system and was known until the 1940s as the Pearson type I distribution. The gamma distribution originated from Pearsons work and was known as the Pearson type III distribution, Pearsons 1895 paper introduced the type IV distribution, which contains Students t-distribution as a special case, predating William Sealy Gossets subsequent use by several years. His 1901 paper introduced the distribution and the beta prime distribution. A Pearson density p is defined to be any valid solution to the equation p ′ p + a + x − λ b 22 + b 1 + b 0 =0. In Equation, the parameter a determines a point, and hence under some conditions a mode of the distribution. Since we are confronted with a first order differential equation with variable coefficients, its solution is straightforward
13.
Fisher's exact test
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Fishers exact test is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample sizes are small, Fisher is said to have devised the test following a comment from Dr. Muriel Bristol, who claimed to be able to detect whether the tea or the milk was added first to her cup. He tested her claim in the lady tasting tea experiment, the test is useful for categorical data that result from classifying objects in two different ways, it is used to examine the significance of the association between the two kinds of classification. We want to know whether these two classifications are associated—that is, whether Dr. Bristol really can tell whether milk or tea was poured in first, most uses of the Fisher test involve, like this example, a 2 ×2 contingency table. As pointed out by Fisher, this leads under a null hypothesis of independence to a distribution of the numbers in the cells of the table. With large samples, a chi-squared test can be used in this situation, in fact, for small, sparse, or unbalanced data, the exact and asymptotic p-values can be quite different and may lead to opposite conclusions concerning the hypothesis of interest. It becomes difficult to calculate with large samples or well-balanced tables, for hand calculations, the test is only feasible in the case of a 2 ×2 contingency table. However the principle of the test can be extended to the case of an m × n table. For example, a sample of teenagers might be divided into male and female on the one hand, and those that are and are not currently studying for a statistics exam on the other. If we were to choose 10 of the teenagers at random, what is the probability that 9 or more of them would be among the 12 women, before we proceed with the Fisher test, we first introduce some notation. We represent the cells by the letters a, b, c and d, call the totals across rows and columns marginal totals, with the data above, this gives, p = / =10. This remains true even if men enter our sample with different probabilities than women, the requirement is merely that the two classification characteristics—gender, and studier —are not associated. For example, suppose we knew probabilities P, Q, p, q with P + Q = p + q =1 such that had respective probabilities for each individual encountered under our sampling procedure. Then still, were we to calculate the distribution of cell entries conditional given marginals, in the example, there are 11 such cases. Of these only one is more extreme in the direction as our data, it looks like this. This gives a one-tailed test, with p approximately 0.001346076 +0.000033652 =0.001379728, for example, in the R statistical computing environment, this value can be obtained as fisher. test$p. value. This value can be interpreted as the sum of evidence provided by the observed data—or any more extreme table—for the null hypothesis. The smaller the value of p, the greater the evidence for rejecting the null hypothesis, for a two-tailed test we must also consider tables that are equally extreme, but in the opposite direction
14.
Probability distribution
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For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. In more technical terms, the probability distribution is a description of a phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey, a probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of numbers or a higher-dimensional vector space, or it may be a list of non-numerical values, for example. Probability distributions are divided into two classes. A discrete probability distribution can be encoded by a discrete list of the probabilities of the outcomes, on the other hand, a continuous probability distribution is typically described by probability density functions. The normal distribution represents a commonly encountered continuous probability distribution, more complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A probability distribution whose sample space is the set of numbers is called univariate. Important and commonly encountered univariate probability distributions include the distribution, the hypergeometric distribution. The multivariate normal distribution is a commonly encountered multivariate distribution, to define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. For example, the probability that an object weighs exactly 500 g is zero. Continuous probability distributions can be described in several ways, the cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. As probability theory is used in diverse applications, terminology is not uniform. The following terms are used for probability distribution functions, Distribution. Probability distribution, is a table that displays the probabilities of outcomes in a sample. Could be called a frequency distribution table, where all occurrences of outcomes sum to 1. Distribution function, is a form of frequency distribution table. Probability distribution function, is a form of probability distribution table
15.
Binomial distribution
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The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the distribution is a hypergeometric distribution. However, for N much larger than n, the distribution remains a good approximation. In general, if the random variable X follows the distribution with parameters n ∈ ℕ and p ∈. The probability of getting exactly k successes in n trials is given by the probability mass function, N, where = n. k. is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows, K successes occur with probability pk and n − k failures occur with probability n − k. However, the k successes can occur anywhere among the n trials, in creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as f = f. The probability mass function satisfies the recurrence relation, for every n, p, Looking at the expression ƒ as a function of k. This k value can be found by calculating f f = p and comparing it to 1. There is always an integer M that satisfies p −1 ≤ M < p. ƒ is monotone increasing for k < M and monotone decreasing for k > M, in this case, there are two values for which ƒ is maximal, p and p −1. M is the most probable outcome of the Bernoulli trials and is called the mode, note that the probability of it occurring can be fairly small. It can also be represented in terms of the incomplete beta function, as follows. Some closed-form bounds for the distribution function are given below. Suppose a biased coin comes up heads with probability 0.3 when tossed, what is the probability of achieving 0,1. For example, if n =100, and p =1/4, P k −1 − = n p ∑ k =1 n. Since Var = p, we get, Var = Var = Var + ⋯ + Var = n Var = n p
16.
P-value
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Their misuse has been a matter of considerable controversy. The p-value is defined informally as the probability of obtaining an equal to or more extreme than what was actually observed. This ignores the distinction between two-tailed and one-tailed tests which is discussed below, in frequentist inference, the p-value is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing. If the p-value is less than or equal to the significance level. However, that does not prove that the hypothesis is true. When the p-value is calculated correctly, this test guarantees that the Type I error rate is at most α. For typical analysis, using the standard α =0.05 cutoff, the p-value does not, in itself, support reasoning about the probabilities of hypotheses but is only a tool for deciding whether to reject the null hypothesis. In statistics, a hypothesis refers to a probability distribution that is assumed to govern the observed data. However, if X is a random variable and an instance x is observed. Thus, this definition is inadequate and needs to be changed so as to accommodate the continuous random variables. The p-value is defined as the probability, under the assumption of hypothesis H, depending on how it is looked at, the more extreme than what was actually observed can mean or or the smaller of and. Thus, the p-value is given by Pr for right tail event, the smaller the p-value, the larger the significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. The hypothesis H is rejected if any of these probabilities is less than or equal to a small, fixed but arbitrarily pre-defined threshold value α, which is referred to as the level of significance. Unlike the p-value, the α level is not derived from any observational data and does not depend on the underlying hypothesis, thus, the p-value is not fixed. This implies that p-value cannot be given a frequency counting interpretation since the probability has to be fixed for the frequency counting interpretation to hold. In other words, if the same test is repeated independently bearing upon the same null hypothesis. Nevertheless, these different p-values can be combined using Fishers combined probability test, the fixed pre-defined α level can be interpreted as the rate of falsely rejecting the null hypothesis, since Pr = Pr = α. Usually, instead of the observations, X is instead a test statistic
17.
Time series
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A time series is a series of data points indexed in time order. Most commonly, a series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data, examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. Time series are very frequently plotted via line charts, Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values, Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in there is no natural ordering of the observations. Time series analysis is also distinct from data analysis where the observations typically relate to geographical locations. A stochastic model for a series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. Methods for time series analysis may be divided into two classes, frequency-domain methods and time-domain methods, the former include spectral analysis and wavelet analysis, the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the stationary stochastic process has a certain structure which can be described using a small number of parameters. In these approaches, the task is to estimate the parameters of the model describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure, Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate. A time series is one type of Panel data, Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel. A data set may exhibit characteristics of both data and time series data. One way to tell is to ask what makes one data record unique from the other records, if the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a data field and an additional identifier which is unrelated to time. If the differentiation lies on the identifier, then the data set is a cross-sectional data set candidate
18.
Autocorrelation
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Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the lag between them. It is often used in processing for analyzing functions or series of values. Unit root processes, trend stationary processes, autoregressive processes, different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance, in statistics, the autocorrelation of a random process is the correlation between values of the process at different times, as a function of the two times or of the time lag. Let X be a process, and t be any point in time. Then Xt is the value produced by a run of the process at time t. Suppose that the process has mean μt and variance σt2 at time t, then the definition of the autocorrelation between times s and t is R = E σ t σ s, where E is the expected value operator. Note that this expression is not well-defined for all-time series or processes, because the mean may not exist, if the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. This further implies that the autocorrelation can be expressed as a function of the time-lag, and this gives the more familiar form R = E σ2, and the fact that this is an even function can be stated as R = R. It is common practice in some disciplines, other statistics and time series analysis, to drop the normalization by σ2. In signal processing, the definition is often used without the normalization. When the autocorrelation function is normalized by mean and variance, it is referred to as the autocorrelation coefficient or autocovariance function. Given a signal f, the continuous autocorrelation R f f is most often defined as the continuous cross-correlation integral of f with itself, for a real function, f ¯ = f. Note that the u in the integral is a dummy variable and is only necessary to calculate the integral. The discrete autocorrelation R at lag l for a discrete signal y is R y y = ∑ n ∈ Z y y ¯, the above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that last forever are treated instead as random processes, in which case different definitions are needed, based on expected values, for wide-sense-stationary random processes, the autocorrelations are defined as R f f = E R y y = E . For processes that are not stationary, these will also be functions of t, for processes that are also ergodic, the expectation can be replaced by the limit of a time average
19.
Cryptanalysis
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Cryptanalysis is the study of analyzing information systems in order to study the hidden aspects of the systems. Cryptanalysis is used to breach cryptographic security systems and gain access to the contents of encrypted messages, methods for breaking modern cryptosystems often involve solving carefully constructed problems in pure mathematics, the best-known being integer factorization. Given some encrypted data, the goal of the cryptanalyst is to gain as much information as possible about the original, attacks can be classified based on what type of information the attacker has available. Ciphertext-only, the cryptanalyst has access only to a collection of ciphertexts or codetexts, known-plaintext, the attacker has a set of ciphertexts to which he knows the corresponding plaintext. Chosen-plaintext, the attacker can obtain the corresponding to an arbitrary set of plaintexts of his own choosing. Adaptive chosen-plaintext, like an attack, except the attacker can choose subsequent plaintexts based on information learned from previous encryptions. Related-key attack, Like a chosen-plaintext attack, except the attacker can obtain ciphertexts encrypted under two different keys, the keys are unknown, but the relationship between them is known, for example, two keys that differ in the one bit. Attacks can also be characterised by the resources they require and those resources include, Time — the number of computation steps which must be performed. Memory — the amount of required to perform the attack. Data — the quantity and type of plaintexts and ciphertexts required for a particular approach and its sometimes difficult to predict these quantities precisely, especially when the attack isnt practical to actually implement for testing. But academic cryptanalysts tend to provide at least the estimated order of magnitude of their attacks difficulty, saying, for example, the results of cryptanalysis can also vary in usefulness. Global deduction — the attacker discovers a functionally equivalent algorithm for encryption and decryption, instance deduction — the attacker discovers additional plaintexts not previously known. Information deduction — the attacker gains some Shannon information about plaintexts not previously known, distinguishing algorithm — the attacker can distinguish the cipher from a random permutation. Academic attacks are often against weakened versions of a cryptosystem, such as a cipher or hash function with some rounds removed. In academic cryptography, a weakness or a break in a scheme is defined quite conservatively, it might require impractical amounts of time, memory. Furthermore, it might reveal a small amount of information, enough to prove the cryptosystem imperfect. Finally, an attack might only apply to a version of cryptographic tools, like a reduced-round block cipher. In practice, they are viewed as two sides of the coin, secure cryptography requires design against possible cryptanalysis
20.
Ciphertext
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In cryptography, ciphertext or cyphertext is the result of encryption performed on plaintext using an algorithm, called a cipher. Ciphertext is also known as encrypted or encoded information because it contains a form of the original plaintext that is unreadable by a human or computer without the proper cipher to decrypt it, decryption, the inverse of encryption, is the process of turning ciphertext into readable plaintext. Ciphertext is not to be confused with codetext because the latter is a result of a code, let m be the plaintext message that Alice wants to secretly transmit to Bob and let E k be the encryption cipher, where k is a cryptographic key. Alice must first transform the plaintext into ciphertext, c, in order to send the message to Bob, as follows. In a symmetric-key system, Bob knows Alices encryption key, once the message is encrypted as ciphertext, Alice can safely transmit it to Bob. In order to read Alices message, Bob must decrypt the ciphertext using E k −1 which is known as the decryption cipher, D k, D k = D k = m. Alternatively, in a key system, everyone, not just Alice and Bob, knows the encryption key. Only Bob knows the decryption key D k, and decryption proceeds as D k = m, the history of cryptography began thousands of years ago. Cryptography uses a variety of different types of encryption, earlier algorithms were performed by hand and are substantially different from modern algorithms, which are generally executed by a machine. Historical pen and paper used in the past are sometimes known as classical ciphers. Many of the ciphers, with the exception of the one-time pad. Modern ciphers are more secure than classical ciphers and are designed to withstand a range of attacks. An attacker should not be able to find the key used in a cipher, even if he knows any amount of plaintext. Symmetric key ciphers can be divided into block ciphers and stream ciphers, block ciphers operate on fixed-length groups of bits, called blocks, with an unvarying transformation. Stream ciphers encrypt plaintext digits one at a time on a stream of data. Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, typically, this involves knowing how the system works and finding a secret key. Cryptanalysis is also referred to as codebreaking or cracking the code, ciphertext is generally the easiest part of a cryptosystem to obtain and therefore is an important part of cryptanalysis. Depending on what information is available and what type of cipher is being analyzed and this is often the meaning of an unqualified use of chosen-plaintext attack
21.
Nomogram
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Nomograms use a parallel coordinate system invented by dOcagne rather than standard Cartesian coordinates. A nomogram consists of a set of n scales, one for each variable in an equation. Knowing the values of n-1 variables, the value of the variable can be found, or by fixing the values of some variables. The result is obtained by laying a straightedge across the values on the scales. The virtual or drawn line created by the straightedge is called a line or isopleth. Nomograms flourished in different contexts for roughly 75 years because they allowed quick. Results from a nomogram are obtained very quickly and reliably by simply drawing one or more lines, the user does not have to know how to solve algebraic equations, look up data in tables, use a slide rule, or substitute numbers into equations to obtain results. The user does not even need to know the underlying equation the nomogram represents, in addition, nomograms naturally incorporate implicit or explicit domain knowledge into their design. For example, to create larger nomograms for greater accuracy the nomographer usually includes only scale ranges that are reasonable, many nomograms include other useful markings such as reference labels and colored regions. All of these provide useful guideposts to the user, while the slide rule is intended to be a general-purpose device, a nomogram is designed to perform a specific calculation, with tables of values effectively built into the construction of the scales. Nomograms are typically used in applications where the level of accuracy they offer is sufficient, alternatively, a nomogram can be used to check an answer obtained from another, more exact but possibly error-prone calculation. Other types of graphical calculators such as charts, trilinear diagrams. These do not meet the definition of a nomogram as a graphical calculator whose solution is found by the use of one or more linear isopleths. A nomogram for a three-variable equation typically has three scales, although there exist nomograms in which two or even all three scales are common, here two scales represent known values and the third is the scale where the result is read off. The simplest such equation is u1 + u2 + u3 =0 for the three variables u1, u2 and u3, an example of this type of nomogram is shown on the right, annotated with terms used to describe the parts of a nomogram. More complicated equations can sometimes be expressed as the sum of functions of the three variables. For example, the nomogram at the top of this article could be constructed as a parallel-scale nomogram because it can be expressed as such a sum after taking logarithms of both sides of the equation. The scale for the variable can lie between the other two scales or outside of them
22.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR
23.
Ronald A. Fisher
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Sir Ronald Aylmer Fisher FRS, who published as R. A. Fisher, was an English statistician and biologist who used mathematics to combine Mendelian genetics and natural selection. This helped to create the new Darwinist synthesis of evolution known as the evolutionary synthesis. He was also a prominent eugenicist in the part of his life. He is known as one of the three founders of population genetics. He outlined Fishers principle as well as the Fisherian runaway and sexy son hypothesis theories of sexual selection and he also made important contributions to statistics, including the maximum likelihood, fiducial inference, the derivation of various sampling distributions among many others. Anders Hald called him a genius who almost single-handedly created the foundations for modern statistical science, not only was he the most original and constructive of the architects of the neo-Darwinian synthesis, Fisher also was the father of modern statistics and experimental design. He therefore could be said to have provided researchers in biology and medicine with their most important research tools, geoffrey Miller said of him To biologists, he was an architect of the modern synthesis that used mathematical models to integrate Mendelian genetics with Darwins selection theories. To psychologists, Fisher was the inventor of various tests that are still supposed to be used whenever possible in psychology journals. To farmers, Fisher was the founder of agricultural research. Fisher was born in East Finchley in London, England, one of twins with the other being still-born, from 1896 until 1904 they lived at Inverforth House in London, where English Heritage installed a blue plaque in 2002, before moving to Streatham. He entered Harrow School age 14 and won the schools Neeld Medal in mathematics, in 1909, he won a scholarship to Gonville and Caius College, Cambridge. In 1919 he began working at Rothamsted Research and his fame grew and he began to travel and lecture widely. In 1937, he visited the Indian Statistical Institute in Calcutta, mahalanobis, often returning to encourage its development, being the guest of honour at its 25th anniversary in 1957 when it had 2000 employees. His marriage disintegrated during World War II and his oldest son George and his daughter and one of his biographers, Joan, married the noted statistician George E. P. Box. Fisher gained a scholarship to study Mathematics at the University of Cambridge in 1909, in 1915 he published a paper The evolution of sexual preference on sexual selection and mate choice. He published The Correlation Between Relatives on the Supposition of Mendelian Inheritance in 1918, in which he introduced the term variance, Joan Box, Fishers biographer and daughter says that Fisher had resolved this problem in 1911. Between 1912 and 1922 Fisher recommended, analyzed and vastly popularized Maximum likelihood, in 1928 Joseph Oscar Irwin began a three-year stint at Rothamsted and became one of the first people to master Fishers innovations. His first application of the analysis of variance was published in 1921 and he pioneered the principles of the design of experiments and the statistics of small samples and the analysis of real data
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker