Central limit theorem
A theorem on the convergence of sample means to normality.
The central limit theorem (CLT) is a fundamental concept in probability theory. It states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution, even if the original variables themselves are not normally distributed. The theorem is a key concept because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
The theorem has a long history, with early versions dating back to 1811, though its modern, precise formulation was only established in the 1920s. The earliest known version was the de Moivre–Laplace theorem, which showed that the normal distribution could approximate the binomial distribution. In its classical form, the CLT applies to a sequence of independent and identically distributed (i.i.d.) random variables with a finite expected value and variance. As the sample size grows, the distribution of the sample average, when scaled by the square root of the sample size, approaches a normal distribution. This convergence is uniform, meaning the maximum difference between the cumulative distribution functions of the normalized mean and the standard normal distribution tends to zero.
Several variants relax the i.i.d. requirement. The Lyapunov CLT allows for independent but not identically distributed variables, provided a condition on the growth of their moments is met. The Lindeberg–Feller CLT provides an even weaker condition for convergence. There is also a CLT for sums of a random number of random variables, and a multidimensional version where sums of independent random vectors converge to a multivariate normal distribution. A generalized central limit theorem has also been developed by multiple mathematicians, extending the concept further.
- field
- Probability theory, statistics
- known_for
- Describing the convergence of the sample mean distribution to a normal distribution
- modern_form_stated
- 1920s
Lore & Background
The central limit theorem (CLT) states that, given appropriate conditions, the distribution of a normalized sample mean converges to a standard normal distribution, even when the original variables are not normally distributed. This theorem is fundamental in probability theory because it allows statistical methods designed for normal distributions to be applied to problems involving other types of distributions. The earliest version of the theorem, which approximated the binomial distribution with a normal distribution, is known as the de Moivre–Laplace theorem. The theorem underwent significant development, with previous versions dating back to 1811, but its modern form was precisely stated in the 1920s. In its common form, the random variables must be independent and identically distributed (i.i.d.), though this requirement can be weakened for non-identical or non-independent observations if certain conditions are met. The classical CLT describes the distributional form of fluctuations around the expected value as the sample size grows: the distribution of the normalized mean approaches a normal distribution with mean zero and variance one, regardless of the original distribution’s shape. Variants include the Lyapunov CLT, which requires independent but not identically distributed variables with limited moment growth, and the Lindeberg–Feller CLT, which uses a weaker condition. A multidimensional CLT exists for independent and identically distributed random vectors, where sums converge to a multivariate normal distribution. The generalized central limit theorem represents a further effort to extend the theorem’s applicability.
Reader's Guide
The central limit theorem is a cornerstone of statistical inference. In its classical form, it states that for a sequence of independent and identically distributed random variables with finite mean and variance, the distribution of the normalized sample mean approaches a normal distribution as the sample size increases. This holds regardless of the shape of the original distribution. The theorem's significance lies in its justification for using normal distribution-based methods on data from other distributions, provided the sample size is large enough. The requirement of independence and identical distribution can be weakened under certain conditions. The theorem describes the size and distributional form of stochastic fluctuations around the population mean during convergence, with the scaled difference between sample mean and population mean converging to a normal distribution with mean zero and variance equal to the original variance.
Did You Know?
- The central limit theorem states that the distribution of a normalized version of the sample mean converges to a standard normal distribution.
- The earliest version of the theorem is the de Moivre–Laplace theorem, which approximates the binomial distribution with a normal distribution.
- The requirement for the random variables to be independent and identically distributed can be weakened under certain conditions.
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