Bell curve

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the. In its most general form, under some conditions (which include finite ), it states that averages of samples of observations of independently drawn from independent distributions

The of the normal distribution is



f(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2} } $$

where
 * $$\mu$$ is the or  of the distribution (and also its  and ),
 * $$\sigma$$ is the, and
 * $$\sigma^2$$ is the.

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where $$X_1,\ldots ,X_n$$ are random variables with the same arbitrary distribution, zero mean, and variance $$\sigma^2$$ and $$Z$$ is their mean scaled by $$\sqrt{n}$$
 * $$ Z = \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i\right) $$

Then, as $$n$$ increases, the probability distribution of $$Z$$ will tend to the normal distribution with zero mean and variance $$\sigma^2$$.

The theorem can be extended to variables $$(X_i)$$ that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many s,, and s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
 * The $$B(n,p)$$ is  with mean $$np$$ and variance $$np(1-p)$$ for large $$n$$ and for $$p$$ not too close to 0 or 1.
 * The with parameter $$\lambda$$ is approximately normal with mean $$\lambda$$ and variance $$\lambda$$, for large values of $$\lambda$$.
 * The $$\chi^2(k)$$ is approximately normal with mean $$k$$ and variance $$2k$$, for large $$k$$.
 * The $$t(\nu)$$ is approximately normal with mean 0 and variance 1 when $$\nu$$ is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the, improvements of the approximation are given by the s.

Standard deviation and coverage
About 68% of values drawn from a normal distribution are within one standard deviation s away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the, or the 3-sigma rule.

More precisely, the probability that a normal deviate lies in the range between $$\mu-n\sigma$$ and $$\mu+n\sigma$$ is given by

F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname{erf} \left(\frac{n}{\sqrt{2}}\right). $$ To 12 significant figures, the values for $$n=1,2,\ldots, 6$$ are: