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In probability theory, the central limit theorem (CLT) states that, in many situations, when independent and identically distributed random variables are added, their properly normalized sum tends toward a normal distribution. This article gives two illustrations of this theorem.
The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem. Many psychological measurements and physical phenomena (like noise) can be approximated well by the normal distribution. An example frequently given is the intelligence quotient (IQ), seen ...
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the ...
Galton box A Galton box demonstrated. The Galton board, also known as the Galton box or quincunx or bean machine (or incorrectly Dalton board), is a device invented by Francis Galton [1] to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution.
Comparison of probability density functions for the sum of n dice to illustrate the central limit theorem: Image title: Comparison of probability density functions, p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem; illustrated by CMG Lee.
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: The binomial distribution B ( n , p ) {\textstyle B(n,p)} is approximately normal with mean n p {\textstyle np} and variance n p ( 1 − p ) {\textstyle np(1-p)} for large n {\displaystyle n} and for p ...
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Central composite design; Central limit theorem. Central limit theorem (illustration) – redirects to Illustration of the central limit theorem; Central limit theorem for directional statistics; Lyapunov's central limit theorem; Martingale central limit theorem; Central moment; Central tendency; Census; Cepstrum; CHAID – CHi-squared ...