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The most well-known example of a problem that allows for a polynomial quantum speedup is "unstructured search", which involves finding a marked item out of a list of formula_31 items in a database. This can be solved by Grover's algorithm using formula_33 queries to the database, quadratically fewer than the formula_34...
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For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied is Boolea...
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Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing. Quantum simulation could also be used to simulate the behavior of...
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Quantum simulations might be used to predict future paths of particles and protons under superposition in the double-slit experiment.
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About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry while naturally occurring organisms also produce ammonia. Quantum simulations might be used to understand this process of increasing production.
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Quantum annealing relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution i...
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Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.
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For example, the quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts. Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines a...
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In the field of computational biology, quantum computing has the potential to play a big role in solving many biological problems. Given how computational biology is using generic data modeling and storage, its applications to computational biology are expected to arise as well.
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Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solv...
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There are a number of technical challenges in building a large-scale quantum computer. Physicist David DiVincenzo has listed these requirements for a practical quantum computer:
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Sourcing parts for quantum computers is also very difficult. Superconducting quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.
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The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers which enable interfacing with the qubits. Scaling these systems to support a growing number of qubits...
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One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples includ...
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As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.
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These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more qui...
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As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than d...
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Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between "L" and "L", where "L" i...
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A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.
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"Quantum supremacy" is a term coined by John Preskill referring to the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers. The problem need not be useful, so some view the quantum supremacy test only as a potential fut...
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In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer. This claim has been s...
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In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer Jiuzhang to demonstrate quantum supremacy. The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their q...
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On November 16, 2021 at the quantum computing summit IBM presented a 127-qubit microprocessor named IBM Eagle.
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Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.
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Bill Unruh doubted the practicality of quantum computers in a paper published in 1994. Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle. Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved. Phys...
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For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):
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The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.
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Any computational problem solvable by a classical computer is also solvable by a quantum computer. Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.
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Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a Turing machine. In other words, quantu...
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While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical...
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The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of p...
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The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that formula_37; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer ca...
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It is also known that BQP is contained in the complexity class (or more precisely in the associated class of decision problems ), which is a subclass of PSPACE.
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It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an -item database in at most formula_40 steps, a slight speedup over Grover's algorithm, w...
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In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who...
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The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the us...
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The Taylor series of a real or complex-valued function that is infinitely differentiable at a real or complex number is the power series
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where denotes the th derivative of evaluated at the point . (The derivative of order zero of is defined to be itself and and are both defined to be 1.)
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By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm:
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The above expansion holds because the derivative of with respect to is also , and equals 1. This leaves the terms in the numerator and in the denominator of each term in the infinite sum.
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The ancient Greek philosopher Zeno of Elea considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was Zeno's paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved unti...
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In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonome...
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In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, after whom the series are now named.
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The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid-18th century.
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If is given by a convergent power series in an open disk centred at in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for in this region, is given by a convergent power series
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and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centred at if and only if its Taylor series converges to the value of the function at each point of the disk.
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If is equal to the sum of its Taylor series for all in the complex plane, it is called entire. The polynomials, exponential function , and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric fu...
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Pictured is an accurate approximation of around the point . The pink curve is a polynomial of degree seven:
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The error in this approximation is no more than . For a full cycle centered at the origin () the error is less than 0.08215. In particular, for , the error is less than 0.000003.
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In contrast, also shown is a picture of the natural logarithm function and some of its Taylor polynomials around . These approximations converge to the function only in the region ; outside of this region the higher-degree Taylor polynomials are "worse" approximations for the function.
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The "error" incurred in approximating a function by its th-degree Taylor polynomial is called the "remainder" or "residual" and is denoted by the function . Taylor's theorem can be used to obtain a bound on the size of the remainder.
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In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. And even if the Taylor series of a function does converge, its limit need not in general be equal to the value of the function . For example...
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is infinitely differentiable at , and has all derivatives zero there. Consequently, the Taylor series of about is identically zero. However, is not the zero function, so does not equal its Taylor series around the origin. Thus, is an example of a non-analytic smooth function.
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In real analysis, this example shows that there are infinitely differentiable functions whose Taylor series are "not" equal to even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which m...
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More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable fun...
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A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable ; see Laurent series. For example, can be written as a Laurent series.
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There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded continuous function on , using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hille, that for any ,
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Here is the th finite difference operator with step size . The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function is analytic at , the terms in the series converge to the terms of the Taylor se...
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The series on the right is the expectation value of , where is a Poisson-distributed random variable that takes the value with probability . Hence,
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Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments .
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The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:
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They converge for formula_21. (In addition, the series for converges for , and the series for converges for .)
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All are convergent for formula_21. These are special cases of the binomial series given in the next section.
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(If , this product is an empty product and has value 1.) It converges for formula_21 for any real or complex number .
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When , this is essentially the infinite geometric series mentioned in the previous section. The special cases and give the square root function and its inverse:
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All angles are expressed in radians. The numbers appearing in the expansions of are the Bernoulli numbers. The in the expansion of are Euler numbers.
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The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:
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The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:
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Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations such as substituti...
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The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big notation:
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The values of formula_53 can be found by comparison of coefficients with the top expression for formula_17, yielding:
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Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand as a Taylor series in , we use the known Taylor series of function :
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Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example, the exponential function is the function which is equal to its own derivative everywhere, ...
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Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential ...
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In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation "as" a power series which, one hopes to prove, is the Taylor series of the desired solution.
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For example, for a function formula_59 that depends on two variables, and , the Taylor series to second order about the point is
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A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
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where is the gradient of evaluated at and is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes
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which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.
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The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval ) as an infinite sum of trigonometric functions (sines and cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite s...
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IQ classification is the practice by Intelligence quotient (IQ) test publishers of labeling IQ score ranges with category names such as "superior" or "average".
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The current scoring method for all IQ tests is the "deviation IQ". In this method, an IQ score of 100 means that the test-taker's performance on the test is at the median level of performance in the sample of test-takers of about the same age as was used to norm the test. An IQ score of 115 means performance one standa...
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Lewis Terman and other early developers of IQ tests noticed that most child IQ scores come out to approximately the same number regardless of testing procedure. All IQ tests show a slight variation in scores even when the same person takes the same test over and over again. IQ scores also slightly differ for a test-tak...
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Even before IQ tests were invented, there were attempts to classify people into intelligence categories by observing their behavior in daily life. Those other forms of behavioral observation are still important for validating classifications based primarily on IQ test scores. Both intelligence classification by observa...
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IQ tests generally are reliable enough that most people 10 years of age and older have similar IQ scores throughout life. Still, some individuals score very differently when taking the same test at different times or when taking more than one kind of IQ test at the same age. For example, many children in the famous lon...
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Terman recruited school pupils based on referrals from teachers, and gave them his Stanford–Binet IQ test. Children with an IQ above 140 by that test were included in the study. There were 643 children in the main study group. When the students who could be contacted again (503 students) were retested at high school ag...
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Because all IQ tests have error of measurement in the test-taker's IQ score, a test-giver should always inform the test-taker of the confidence interval around the score obtained on a given occasion of taking each test. IQ scores are ordinal scores and are not expressed in an interval measurement unit. Besides the inhe...
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IQ classifications for individuals also vary because category labels for IQ score ranges are specific to each brand of test. The test publishers do not have a uniform practice of labeling IQ score ranges, nor do they have a consistent practice of dividing up IQ score ranges into categories of the same size or with the ...
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IQ classifications from IQ testing are not the last word on how a test-taker will do in life, nor are they the only information to be considered for placement in school or job-training programs. There is still a dearth of information about how behavior differs between persons with differing IQ scores. For placement in ...
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There are a variety of individually administered IQ tests in use in the English-speaking world. Not all report test results as "IQ", but most now report a standard score with a median score level of 100. When a test-taker scores higher or lower than the median score, the score is indicated as 15 standard score points h...
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The Wechsler intelligence scales were originally developed from earlier intelligence scales by David Wechsler. The first Wechsler test published was the Wechsler–Bellevue Scale in 1939. The Wechsler IQ tests for children and for adults are the most frequently used individual IQ tests in the English-speaking world and i...
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Psychologists have proposed alternative language for Wechsler IQ classifications. The term "borderline", which implies being very close to being intellectually disabled, is replaced in the alternative system by a term that doesn't imply a medical diagnosis.
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The current fifth edition of the Stanford–Binet scales (SB5) was developed by Gale H. Roid and published in 2003 by Riverside Publishing. Unlike scoring on previous versions of the Stanford–Binet test, SB5 IQ scoring is deviation scoring in which each standard deviation up or down from the norming sample median score i...
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The Woodcock–Johnson a III NU Tests of Cognitive Abilities (WJ III NU) was developed by Richard W. Woodcock, Kevin S. McGrew and Nancy Mather and published in 2007 by Riverside. The WJ III classification terms are not applied.
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The Kaufman Adolescent and Adult Intelligence Test was developed by Alan S. Kaufman and Nadeen L. Kaufman and published in 1993 by American Guidance Service. Kaufman test scores "are classified in a symmetrical, nonevaluative fashion", in other words the score ranges for classification are just as wide above the median...
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The Kaufman Assessment Battery for Children, Second Edition was developed by Alan S. Kaufman and Nadeen L. Kaufman and published in 2004 by American Guidance Service.
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The Das-Naglieri Cognitive Assessment System test was developed by Jack Naglieri and J. P. Das and published in 1997 by Riverside.
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The Differential Ability Scales Second Edition (DAS–II) was developed by Colin D. Elliott and published in 2007 by Psychological Corporation. The DAS-II is a test battery given individually to children, normed for children from ages two years and six months through seventeen years and eleven months. It was normed on 3,...
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Reynolds Intellectual Ability Scales (RIAS) were developed by Cecil Reynolds and Randy Kamphaus. The RIAS was published in 2003 by Psychological Assessment Resources.
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Lewis Terman, developer of the Stanford–Binet Intelligence Scales, based his English-language Stanford–Binet IQ test on the French-language Binet–Simon test developed by Alfred Binet. Terman believed his test measured the "general intelligence" construct advocated by Charles Spearman (1904). Terman differed from Binet ...
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