Duncan's test has been criticised as being too liberal by many statisticians including Henry Scheffé, and John W. Tukey.

In fact, the root mean square ( rms ) errors are much better than these upper bounds, being only O ( ε √ log N ) for Cooley – Tukey and O ( ε √ N ) for the naïve DFT ( Schatzman, 1996 ).

However, for composite sizes such as prime powers, the Cooley – Tukey FFT algorithm is much simpler and more practical to implement, so Rader's algorithm is typically only used for large-prime base cases of Cooley – Tukey's recursive decomposition of the DFT ( Frigo and Johnson, 2005 ).

Bruun's algorithm has not seen widespread use, however, as approaches based on the ordinary Cooley – Tukey FFT algorithm have been successfully adapted to real data with at least as much efficiency.

His results, however, have been disputed, especially in 1954 by a team consisting of John Tukey, Frederick Mosteller and William G. Cochran, who stated much of Kinsey's work was based on convenience samples rather than random samples, and thus would have been vulnerable to bias.

He suggested the idea of how Fourier transforms could be programmed to be much faster to both Cooley and Tukey.

These statistical developments, all championed by Tukey, were designed to complement the analytic theory of testing statistical hypotheses, particularly the Laplacian tradition's emphasis on exponential families.

* Tukey honestly significant difference test, a multiple comparisons method in statistics

This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

The most well-known use of the Cooley – Tukey algorithm is to divide the transform into two pieces of size at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general ( as was known to both Gauss and Cooley / Tukey ).

The Rader-Brenner algorithm ( 1976 ) is a Cooley – Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability ; it was later superseded by the split-radix variant of Cooley – Tukey ( which achieves the same multiplication count but with fewer additions and without sacrificing accuracy ).

Tukey was perhaps the most vocal critic, saying, " A random selection of three people would have been better than a group of 300 chosen by Mr. Kinsey.

( Although the PFA is distinct from the Cooley – Tukey algorithm, Good's 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their famous 1965 paper, and there was initially some confusion about whether the two algorithms were different.

Good on what is now called the prime-factor FFT algorithm ( PFA ); although Good's algorithm was initially mistakenly thought to be equivalent to the Cooley – Tukey algorithm, it was quickly realized that PFA is a quite different algorithm ( only working for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley – Tukey ).

John Wilder Tukey ForMemRS (; June 16, 1915 – July 26, 2000 ) was an American statistician best known for development of the FFT algorithm and box plot.

Tukey was born in New Bedford, Massachusetts in 1915, and obtained a B. A.

The term " software ", which Paul Niquette claims he coined in 1953, was first used in print by Tukey in a 1958 article in American Mathematical Monthly, and thus some attribute the term to him.

Exploratory data analysis was promoted by John Tukey to encourage statisticians visually to examine their data sets, to formulate hypotheses that could be tested on new data-sets.

Dr. Kurtz graduated from Knox College in 1950, and was awarded a Ph. D. degree from Princeton University in 1956, where his advisor was John Tukey, and joined the Mathematics Department of Dartmouth College that same year.

According to, this name was coined by John Tukey in the 1950s.

City Marshal Francis Tukey resisted mayor John Prescott Bigelow's appointment of McGinniskin, expressing the predominant anti-Irish sentiments in the city by arguing it was done at " the expense of an American.

He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted " binary digit " to simply " bit ".

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

For the case of power-of-two, Papadimitriou ( 1979 ) argued that the number of complex-number additions achieved by Cooley – Tukey algorithms is optimal under certain assumptions on the graph of the algorithm ( his assumptions imply, among other things, that no additive identities in the roots of unity are exploited ).

The upper bound on the relative error for the Cooley – Tukey algorithm is O ( ε log N ), compared to O ( εN < sup > 3 / 2 </ sup >) for the naïve DFT formula ( Gentleman and Sande, 1966 ), where ε is the machine floating-point relative precision.

During the 1970s, Edward Tufte developed a course on statistical graphics, which he further developed in joint seminars with John Tukey, a pioneer in the field of information design.

Nevertheless, Bruun's algorithm illustrates an alternative algorithmic framework that can express both itself and the Cooley – Tukey algorithm, and thus provides an interesting perspective on FFTs that permits mixtures of the two algorithms and other generalizations.

FFTs became popular after James Cooley of IBM and John Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer.

Among many contributions to civil society, Tukey served on a committee of the American Statistical Association that produced a report challenging the conclusions of the Kinsey Report, Statistical Problems of the Kinsey Report on Sexual Behavior in the Human Male.

While working with John von Neumann on early computer designs, Tukey introduced the word " bit " as a contraction of " binary digit ".

* Exploratory data analysis by John W. Tukey, based on a course taught at Princeton.

* An interview with John Tukey at Princeton University on 11 April 1984 Includes references to a earlier Ong's Hat story

Data flow diagram connecting the inputs x ( left ) to the outputs y that depend on them ( right ) for a " butterfly " step of a radix-2 Cooley – Tukey FFT.

Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science.

* Uncomfortable science, due to statistician John Tukey: Inference from a limited sample of data, where further samples influenced by the same causality, a finite natural phenomenon for which it is difficult to overcome the problem of using a common sample of data for both exploratory data analysis and confirmatory data analysis.

Another participant at that meeting, Richard Garwin of IBM, recognized the potential of the method and put Tukey in touch with Cooley, who implemented it for a different ( and less-classified ) problem: analyzing 3d crystallographic data ( see also: multidimensional FFTs ).

Tukey promoted the use of five number summary of numerical data — the two extremes ( maximum and minimum ), the median, and the quartiles — because these median and quartiles, being functions of the empirical distribution are defined for all distributions, unlike the mean and standard deviation ; moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries ( the mean and standard deviation ).

Uncomfortable science is the term coined by statistician John Tukey for cases in which there is a need to draw an inference from a limited sample of data, where further samples influenced by the same cause system will not be available.

Methods particularly useful in analysis of variance, and in constructing simultaneous confidence bands for regressions involving basis functions are the Scheffé's method and, if the researcher has in mind only pairwise comparisons, the Tukey method.

If the log is taken base 2, the unit of information is the binary digit or bit ( so named by John Tukey ); if we use a natural logarithm instead, we might call the resulting unit the " nat.

Therefore, for N even the convolution is cyclic, but in this case N is composite and one would normally use a more efficient FFT algorithm such as Cooley – Tukey.

A more modern reference is, which is the basis of the name " Stone – Tukey theorem ".