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Wikipedia: Normal distribution

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Normal
Probability density function The red line is the standard normal distribution
Cumulative distribution function Colors match the image above
Parameters	$μ$ location (real) $σ 2 > 0$ squared scale (real)
Support	$x \in\mathbb{R}\!$
Probability density function (pdf)	$\frac{1}{\sigma \sqrt{2\pi} } \exp \left(-\frac{(x-\mu)^2}{2\sigma ^2} \right)$
Cumulative distribution function (cdf)	$\frac12 \left(1+\mathrm{erf}\left( \frac{x-\mu}{\sigma\sqrt2}\right) \right)$
Mean	$μ$
Median	$μ$
Mode	$μ$
Variance	$σ 2$
Skewness	0
Excess kurtosis	0
Entropy	$\tfrac{1}{2}\left(\ln(2\pi\sigma^2)+1\right) = \ln\left(\sigma\sqrt{2 \pi e}\right)$
Moment-generating function (mgf)	$M_X(t)= \exp\left(\mu\,t+\frac{\sigma^2 t^2}{2}\right)$
Characteristic function	$\chi_X(t)=\exp\left(\mu\,i\,t-\frac{\sigma^2 t^2}{2}\right)$

In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that describes data that clusters around a mean or average. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve.

The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches. Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data is used.

For theoretical reasons (such as the central limit theorem), any variable that is the sum of a large number of independent factors is likely to be normally distributed. For this reason, the normal distribution is used throughout statistics, natural science, and social science^[1] as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption.

The probability density function for a normal distribution is given by the formula

$p(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right),$

where μ is the mean, σ is the standard deviation (a measure of the “width” of the bell), and exp denotes the exponential function. For a mean of 0 and a standard deviation of 1, this formula simplifies to

$p(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2},$

which is known as the standard normal distribution. When properly scaled and translated, the corresponding cumulative distribution function is known as the error function.

The Gaussian distribution is named for Carl Friedrich Gauss, who used it to analyze astronomical data,^[2] and defined the formula for its probability density function.

1 History
2 Characterization
3 Properties
4 Complex normal distribution
5 Related distributions
6 Descriptive and inferential statistics
7 Occurrence
8 Use in computational statistics
- 8.1 Generating values for normal random variables
- 8.2 Numerical approximations of the normal distribution and its cdf
9 See also
10 Notes
11 References
12 External links

History

The normal distribution was first introduced by Abraham de Moivre in an article in the year 1733,^[3] which was reprinted in the second edition of his The Doctrine of Chances, 1738 in the context of approximating certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the theorem of de Moivre-Laplace.

Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors. The fact the distribution is sometimes called Gaussian is an example of Stigler's Law.

The name "bell curve" goes back to Esprit Jouffret who first used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently by Charles Sanders Peirce, Francis Galton and Wilhelm Lexis around 1875.^{[citation needed]} Despite this terminology, other probability distributions may be more appropriate in some contexts; see the discussion of occurrence, below.

Characterization

There are various ways to characterize a probability distribution. The most visual is the probability density function (PDF). Equivalent ways are the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, the cumulant-generating function, and Maxwell's theorem. See probability distribution for a discussion.

To indicate that a real-valued random variable X is normally distributed with mean μ and variance σ² ≥ 0, we write

$X \sim N(\mu, \sigma^2).\,\!$

While it is certainly useful for certain limit theorems (e.g. asymptotic normality of estimators) and for the theory of Gaussian processes to consider the probability distribution concentrated at μ (see Dirac measure) as a distribution with mean μ and variance σ² = 0. This degenerate case is often excluded from the considerations because no density with respect to the Lebesgue measure exists.

The normal distribution may also be parameterized using a precision parameter τ, defined as the reciprocal of σ². This parameterization has an advantage in numerical applications where σ² is very close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution.

Probability density function

The continuous probability density function of the normal distribution is the Gaussian function

$\varphi_{\mu,\sigma^2}(x) = \frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x- \mu)^2}{2\sigma^2}} = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right),\quad x\in\mathbb{R},$

where σ > 0 is the standard deviation, the real parameter μ is the expected value, and

$\varphi(x)=\varphi_{0,1}(x)=\frac{e^{-x^2/2}}{\sqrt{2\pi\,}}, \,\quad x\in\mathbb{R},$

is the density function of the "standard" normal distribution: i.e., the normal distribution with μ = 0 and σ = 1. The integral of $\varphi_{\mu,\sigma^2}$ over the real line is equal to one as shown in the Gaussian integral article.

As a Gaussian function with the denominator in the exponent equal to 2, the standard normal density function $\varphi_{}$ is an eigenfunction of the Fourier transform.

The probability density function has notable properties including:

symmetry about its mean μ
the mode and median both equal the mean μ
the inflection points of the curve occur one standard deviation away from the mean, i.e. at μ − σ and μ + σ.

Cumulative distribution function

Generating functions

Moment generating function

The moment generating function is defined as the expected value of exp(tX). For a normal distribution, the moment generating function is

$\begin{align} M_X(t) & {} = \mathrm{E} \left[ \exp{(tX)} \right] \\ & {} = \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi} } \exp{\left( -\frac{(x - \mu)^2}{2 \sigma^2} \right)} \exp{(tx)} \, dx \\ & {} = \exp{ \left( \mu t + \frac{\sigma^2 t^2}{2} \right)} \end{align}$

as can be seen by completing the square in the exponent.

Cumulant generating function

The cumulant generating function is the logarithm of the moment generating function: g(t) = μt + σ²t²/2. Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.

Characteristic function

The characteristic function is defined as the expected value of exp(itX), where i is the imaginary unit. So the characteristic function is obtained by replacing t with it in the moment-generating function.

For a normal distribution, the characteristic function is ^[7]

$\begin{align} \chi_X(t;\mu,\sigma) &{} = M_X(i t) = \mathrm{E} \left[ \exp(i t X) \right] \\ &{}= \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi}} \exp \left(- \frac{(x - \mu)^2}{2\sigma^2} \right) \exp(i t x) \, dx \\ &{}= \exp \left( i \mu t - \frac{\sigma^2 t^2}{2} \right). \end{align}$

Properties

Some properties of the normal distribution:

If $X \sim N(\mu, \sigma^2)$ and $a$ and $b$ are real numbers, then $a X + b \sim N(a \mu + b, (a \sigma)^2)$ (see expected value and variance).
The joint distribution of two independent normal random variables is a bivariante normal distribution.

If and are independent normal random variables, then:
- Their sum is normally distributed with $U = X + Y \sim N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ (proof). Thus the normal distribution is infinitely divisible. Interestingly, the converse holds: if two independent random variables have a normally-distributed sum, then they must be normal themselves; this is known as Cramér's theorem.
- Their difference is normally distributed with $V = X - Y \sim N(\mu_X - \mu_Y, \sigma^2_X + \sigma^2_Y)$ .
- If the variances of X and Y are equal, then U and V are independent of each other (and equal variance); in the bivariate distribution, this corresponds to a change of coordinates in the plane from $(x, y)$ to $(x + y, x - y)$ , or equivalently a change of axes (rotation and dilation).
- The Kullback-Leibler divergence is given by:
: $D_{\rm KL}( X \| Y ) = { 1 \over 2 } \left(2 \log \left( { \sigma_Y \over \sigma_X } \right) + \frac{\sigma^2_X}{\sigma^2_Y} + \frac{\left(\mu_Y - \mu_X\right)^2}{\sigma^2_Y} - 1\right).$
If and are independent normal random variables, then:
- Their product $X Y$ follows a distribution with density $p$ given by
- : $p(z) = \frac{1}{\pi\,\sigma_X\,\sigma_Y} \; K_0\left(\frac{|z|}{\sigma_X\,\sigma_Y}\right),$ where $K 0$ is a modified Bessel function of the second kind.
- Their ratio follows a Cauchy distribution with $X/Y \sim \mathrm{Cauchy}(0, \sigma_X/\sigma_Y)$ . Thus the Cauchy distribution is a special kind of ratio distribution.
If $X_1, \dots, X_n$ are independent standard normal variables, then $X_1^2 + \cdots + X_n^2$ has a chi-square distribution with n degrees of freedom.
If $X_1,\dots,X_n$ are independent standard normal variables, then the sample mean $\bar{X}=(X_1+\cdots+X_n)/n$ and sample variance $S^2=((X_1-\bar{X})^2+\cdots+(X_n-\bar{X})^2)/(n-1)$ are independent. This can be proven using Basu's theorem or Cochran's theorem. This property characterizes normal distributions (and helps to explain why the F-test is non-robust with respect to non-normality!) See also Student's t-distribution, which uses a ratio derived from these.

Standardizing normal random variables

As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.

If $X$ ~ $N (μ,σ 2)$ , then

$Z = \frac{X - \mu}{\sigma} \!$

is a standard normal random variable: $Z$ ~ $N (0,1)$ . An important consequence is that the cdf of a general normal distribution is therefore

$\Pr(X \le x) = \Phi \left( \frac{x-\mu}{\sigma} \right) = \frac{1}{2} \left( 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right) .$

Conversely, if $Z$ is a standard normal distribution, $Z$ ~ $N (0,1)$ , then

X = σ Z + μ

is a normal random variable with mean $μ$ and variance $σ 2$ .

The standard normal distribution has been tabulated (usually in the form of value of the cumulative distribution function Φ), and the other normal distributions are the simple transformations, as described above, of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.

Moments

The first few moments of the normal distribution are:

Number	Raw moment	Central moment	Cumulant
0	1	1
1	$μ$	0	$μ$
2	$μ 2 + σ 2$	$σ 2$	$σ 2$
3	$μ 3 + 3μσ 2$	0	0
4	$μ 4 + 6μ 2 σ 2 + 3σ 4$	$3σ 4$	0
5	$μ 5 + 10μ 3 σ 2 + 15μσ 4$	0	0
6	$μ 6 + 15μ 4 σ 2 + 45μ 2 σ 4 + 15σ 6$	$15σ 6$	0
7	$μ 7 + 21μ 5 σ 2 + 105μ 3 σ 4 + 105μσ 6$	0	0
8	$μ 8 + 28μ 6 σ 2 + 210μ 4 σ 4 + 420μ 2 σ 6 + 105σ 8$	$105σ 8$	0

All cumulants of the normal distribution beyond the second are zero.

Higher central moments (of order 2k) are given by the formula

$E\left[(X-\mu)^{2k}\right]=\frac{(2k)!}{2^k k!} \sigma^{2k}.$

The general p-th raw moment (p not necessarily an integer) can be expressed as

$\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right]= \left(2 \sigma^2\right)^\frac p 2 \frac {\Gamma\left(\frac{1+p}2\right)}{\sqrt \pi}\, _1F_1\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2}\right),$

$\operatorname{E} \left[N\left(\mu, \sigma^2 \right)^p \right]=(-2 \sigma^2)^\frac p 2 \cdot U\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2} \right),$

where $1 F 1$ and $U$ are confluent hypergeometric functions (the function's second branch cut can be chosen by multiplying with $( - 1) p$ ).

Central limit theorem

Main article: central limit theorem

Plot of the pdf of a normal distribution with μ = 12 and σ = 3, approximating the pdf of a binomial distribution with n = 48 and p = 1/4

Under certain conditions (such as being independent and identically-distributed with finite variance), the sum of a large number of random variables is approximately normally distributed; this is the central limit theorem.

The practical importance of the central limit theorem is that the normal cumulative distribution function can be used as an approximation to some other cumulative distribution functions, for example:

A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 (some books recommend using this approximation only if np and n(1 − p) are both at least 5; in this case, a continuity correction should be applied).
The approximating normal distribution has parameters μ = np, σ² = np(1 − p).
A Poisson distribution with parameter λ is approximately normal for large λ.
The approximating normal distribution has parameters μ = σ² = λ.

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 of the approximation error of the cumulative distribution function is given by the Berry–Esséen theorem.

Infinite divisibility

The normal distributions are infinitely divisible probability distributions: Given a mean μ, a variance σ ² ≥ 0, and a natural number n, the sum X₁ + . . . + X_n of n independent random variables

$X_1,X_2,\dots,X_n \sim N(\mu/n, \sigma^2\!/n)\,$

has this specified normal distribution (to verify this, use characteristic functions or convolution and mathematical induction).

Stability

The normal distributions are strictly stable probability distributions.

Standard deviation and confidence intervals

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue) while two standard deviations from the mean (medium and dark blue) account for about 95% and three standard deviations (light, medium, and dark blue) account for about 99.7%.

About 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. This is known as the "68-95-99.7 rule" or the "empirical rule" or the "3-sigma rule."

To be more precise, the area under the bell curve between μ − nσ and μ + nσ in terms of the cumulative normal distribution function is given by

$\begin{align}&\Phi_{\mu,\sigma^2}(\mu+n\sigma)-\Phi_{\mu,\sigma^2}(\mu-n\sigma)\\ &=\Phi(n)-\Phi(-n)=2\Phi(n)-1=\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr),\end{align}$

where erf is the error function. To 12 decimal places, the values for the 1-, 2-, up to 6-sigma points are:

$n\,$	$\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr)\,$
1	0.682689492137
2	0.954499736104
3	0.997300203937
4	0.999936657516
5	0.999999426697
6	0.999999998027

The next table gives the reverse relation of sigma multiples corresponding to a few often used values for the area under the bell curve. These values are useful to determine (asymptotic) confidence intervals of the specified levels based on normally distributed (or asymptotically normal) estimators:

$\mathrm{erf}\bigl(n/\sqrt{2}\,\bigr)$	$n\,$
0.80	1.28155
0.90	1.64485
0.95	1.95996
0.98	2.32635
0.99	2.57583
0.995	2.80703
0.998	3.09023
0.999	3.29052
0.9999	3.8906
0.99999	4.4172

where the value on the left of the table is the proportion of values that will fall within a given interval and n is a multiple of the standard deviation that specifies the width of the interval.

Exponential family form

The Normal distribution is a two-parameter exponential family form with natural parameters μ and 1/σ², and natural statistics X and X². The canonical form has parameters ${\mu \over \sigma^2}$ and ${1 \over \sigma^2}$ and sufficient statistics $\sum x$ and $-{1 \over 2} \sum x^2.$

Complex normal distribution

Consider the complex Gaussian random variable,

$Z=X+iY,\,$

where X and Y are real and independent Gaussian variables with equal variances $\sigma_r^2$ . The pdf of the joint variables is then

$\frac{1}{2\,\pi\,\sigma_r^2} e^{-(x^2+y^2)/(2 \sigma_r ^2)}.$

Because $\sigma_Z =\sqrt{2}\sigma_r$ , the resulting pdf for the complex Gaussian variable Z is

$\frac{1}{\pi\,\sigma_Z^2} e^{-|Z|^2\!/\sigma_Z^2}.$

Related distributions

$R \sim \mathrm{Rayleigh}(\sigma)$ is a Rayleigh distribution if $R = \sqrt{X^2 + Y^2}$ where $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are two independent normal distributions.
$Y \sim \chi_{\nu}^2$ is a chi-square distribution with $ν$ degrees of freedom if $Y = \sum_{k=1}^{\nu} X_k^2$ where $X_k \sim N(0,1)$ for $k=1,\dots,\nu$ and are independent.
$Y \sim \mathrm{Cauchy}(\mu = 0, \theta = 1)$ is a Cauchy distribution if $Y = X 1 / X 2$ for $X_1 \sim N(0,1)$ and $X_2 \sim N(0,1)$ are two independent normal distributions.
$Y \sim \mbox{Log-N}(\mu, \sigma^2)$ is a log-normal distribution if $Y = e X$ and $X \sim N(\mu, \sigma^2)$ .
Relation to stable distribution: if $X\sim \textrm{stable}(2,\beta,\sigma/\sqrt{2},\mu)$ then $X \sim N(\mu,\sigma^2)$ .
Truncated normal distribution. If $X \sim N(\mu, \sigma^2),\!$ then truncating X below at $A$ and above at $B$ will lead to a random variable with mean $E(X)=\mu + \frac{\sigma(\varphi_1-\varphi_2)}{T},\!$ where $T=\Phi\left(\frac{B-\mu}{\sigma}\right)-\Phi\left(\frac{A-\mu}{\sigma}\right), \; \varphi_1 = \varphi\left(\frac{A-\mu}{\sigma}\right), \; \varphi_2 = \varphi\left(\frac{B-\mu}{\sigma}\right)$ and $\varphi$ is the probability density function of a standard normal random variable.
If $X$ is a random variable with a normal distribution, and $Y = | X |$ , then $Y$ has a folded normal distribution.

Descriptive and inferential statistics

Scores

Many scores are derived from the normal distribution, including percentile ranks ("percentiles" or "quantiles"), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, a number of behavioral statistical procedures are based on the assumption that scores are normally distributed; for example, t-tests and ANOVAs (see below). Bell curve grading assigns relative grades based on a normal distribution of scores.

This section requires expansion.

Normality tests

Main article: normality test

Normality tests check a given set of data for similarity to the normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small P-value indicates non-normal data.

Estimation of parameters

Estimators

Main articles: Standard error of the mean and Unbiased estimation of standard deviation

For a normal distribution with mean μ and variance σ², the sample mean $\overline{X}$ :

is the UMVU estimator for the population mean μ, by the Lehmann–Scheffé theorem (because it is an unbiased, complete, sufficient statistic),
is the maximum likelihood estimator for the population mean μ,
has sampling distribution $N (μ,σ 2 / n),$ since the sum of n i.i.d. random variables has distribution $N (n μ, n σ 2),$ and thus
the standard error of the sample mean is $\sigma/\sqrt{n}.$

As the number of samples grows, the standard error of the sample mean decays as

$1/\sqrt{n}$

so if one wishes to decrease the standard error by a factor of 10, one must increase the number of samples by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and number of trials in Monte Carlo simulation.

The sample distribution of the mean depends on the standard deviation σ; it is not an ancillary statistic, and thus to estimate the error of the sample mean, one must estimate the standard deviation.

The sample standard deviation, defined as:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}\,,$

is a common estimator for the population standard deviation:

$s 2$ is an unbiased estimator for the variance σ²,
the sampling distribution of s² is, up to scale, a chi-square distribution with n–1 degrees of freedom;
:properly, $(n-1)s_n^2/\sigma^2 \sim \chi_{n-1}^2;$
s is an ancillary statistic of the population mean μ; its sampling distribution does not depend on the population mean, and
s² (and thus s) are independent of the sample mean $\overline{X},$ by Cochran's theorem.

Correction factor c₄ in unbiased estimation of standard deviation of a normal distribution.

Note that:

There is a factor of n–1, not n, in the definition; this corresponds to the number of degrees of freedom (the residuals sum to 1, which removes one degree of freedom), and is known as Bessel's correction.
The normal distribution is the only distribution whose sample mean and sample variance are independent.
While $s 2$ is an unbiased estimator for the variance σ², s is a biased estimator for the standard deviation σ; see unbiased estimation of standard deviation.

For the normal distribution, one can compute a correction factor, which depends on n, to arrive at an unbiased estimator of the standard deviation. This is denoted by c₄, and the corrected (unbiased) estimator is $s / c 4$ . For n=2 $c_4 \approx 0.80$ , while for n=10 $c_4 \approx 0.97,$ so this correction is rarely used outside of high-precision estimation of small samples.

The standard error of the uncorrected (biased) sample standard deviation s is^[8]^[9] $\sigma\sqrt{1-c_4^{2}} \approx \sigma/\sqrt{2n},$ thus it also decays as $1/\sqrt{n}$ .

Unbiased estimation of parameters

The maximum likelihood estimator of the population mean μ from a sample is an unbiased estimator of the mean. The maximum likelihood estimator of the variance is unbiased if we assume the population is known a priori, but in practice that does not happen. However, if we are faced with a sample and have no knowledge of the mean or the variance of the population from which it is drawn, as assumed in the maximum likelihood derivation above, then the maximum likelihood estimator of the variance is biased. An unbiased estimator of the variance σ² is:

$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2.$

This "sample variance" follows a Gamma distribution if all X_i are independent and identically-distributed:

$S^2 \sim \Gamma\left(\frac{n-1}{2},\frac{2 \sigma^2}{n-1}\right),$

with mean $\operatorname{E}(S^2)=\sigma^2$ and variance $\operatorname{Var}(S^2)=2\sigma^4/(n-1).$

The maximum likelihood estimate of the standard deviation is the square root of the maximum likelihood estimate of the variance. However, neither this nor the square root of the sample variance provides an unbiased estimate for standard deviation: see unbiased estimation of standard deviation for formulae particular to the normal distribution.

Maximum likelihood estimation of parameters

Suppose

$X_1,\dots,X_n$

are independent and each is normally distributed with expectation μ and variance σ² > 0. In the language of statisticians, the observed values of these n random variables make up a "sample of size n from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on the observed values of this sample. The continuous joint probability density function of these n independent random variables is

$\begin{align}f(x_1,\dots,x_n;\mu,\sigma) &= \prod_{i=1}^n \varphi_{\mu,\sigma^2}(x_i)\\ &=\frac1{(\sigma\sqrt{2\pi})^n}\prod_{i=1}^n \exp\biggl(-{1 \over 2} \Bigl({x_i-\mu \over \sigma}\Bigr)^2\biggr), \quad(x_1,\ldots,x_n)\in\mathbb{R}^n. \end{align}$

As a function of μ and σ, the likelihood function based on the observations X₁, ..., X_n is

$L(\mu,\sigma) = \dfrac C{\sigma^n} \exp\left(-{\sum_{i=1}^n (X_i-\mu)^2 \over 2\sigma^2}\right), \quad\mu\in\mathbb{R},\ \sigma>0,$

with some constant C > 0 (which in general would be even allowed to depend on X₁, ..., X_n, but will vanish anyway when partial derivatives of the log-likelihood function with respect to the parameters are computed, see below).

In the method of maximum likelihood, the values of μ and σ that maximize the likelihood function are taken as estimates of the population parameters μ and σ.

Usually in maximizing a function of two variables, one might consider partial derivatives. But here we will exploit the fact that the value of μ that maximizes the likelihood function with σ fixed does not depend on σ. Therefore, we can find that value of μ, then substitute it for μ in the likelihood function, and finally find the value of σ that maximizes the resulting expression.

It is evident that the likelihood function is a decreasing function of the sum

$\sum_{i=1}^n (X_i-\mu)^2. \,\!$

So we want the value of μ that minimizes this sum. Let

$\overline{X}_n=(X_1+\cdots+X_n)/n$

be the "sample mean" based on the n observations. Observe that

$\begin{align} \sum_{i=1}^n (X_i-\mu)^2 &=\sum_{i=1}^n\bigl((X_i-\overline{X}_n)+(\overline{X}_n-\mu)\bigr)^2\\ &=\sum_{i=1}^n(X_i-\overline{X}_n)^2 + 2(\overline{X}_n-\mu)\underbrace{\sum_{i=1}^n (X_i-\overline{X}_n)}_{=\,0} + \sum_{i=1}^n (\overline{X}_n-\mu)^2\\ &=\sum_{i=1}^n(X_i-\overline{X}_n)^2 + n(\overline{X}_n-\mu)^2. \end{align}$

Only the last term depends on μ and it is minimized by

$\widehat{\mu}_n=\overline{X}_n.$

That is the maximum-likelihood estimate of μ based on the n observations X₁, ..., X_n. When we substitute that estimate for μ into the likelihood function, we get

$L(\overline{X}_n,\sigma) = \frac C{\sigma^n} \exp\biggl(-{\sum_{i=1}^n (X_i-\overline{X}_n)^2 \over 2\sigma^2}\biggr), \quad\sigma>0.$

It is conventional to denote the "log-likelihood function", i.e., the logarithm of the likelihood function, by a lower-case ℓ, and we have

$\ell(\overline{X}_n,\sigma)=\log C-n\log\sigma-{\sum_{i=1}^n(X_i-\overline{X}_n)^2 \over 2\sigma^2}, \quad\sigma>0,$

and then

$\begin{align} {\partial \over \partial\sigma}\ell(\overline{X}_n,\sigma) &=-{n \over \sigma} +{\sum_{i=1}^n (X_i-\overline{X}_n)^2 \over \sigma^3}\\ &=-{n \over \sigma^3}\biggl(\sigma^2-{1 \over n}\sum_{i=1}^n (X_i-\overline{X}_n)^2 \biggr), \quad\sigma>0. \end{align}$

This derivative is positive, zero, or negative according as σ² is between 0 and

$\hat\sigma_n^2:={1 \over n}\sum_{i=1}^n(X_i-\overline{X}_n)^2,$

or equal to that quantity, or greater than that quantity. (If there is just one observation, meaning that n = 1, or if X₁ = ... = X_n, which only happens with probability zero, then $\hat\sigma{}_n^2=0$ by this formula, reflecting the fact that in these cases the likelihood function is unbounded as σ decreases to zero.)

Consequently this average of squares of residuals is the maximum-likelihood estimate of σ², and its square root is the maximum-likelihood estimate of σ based on the n observations. This estimator $\hat\sigma{}_n^2$ is biased, but has a smaller mean squared error than the usual unbiased estimator, which is n/(n − 1) times this estimator.

Surprising generalization

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is subtle. It involves the spectral theorem and the reason it can be better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices.

Occurrence

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption, for example the Kolmogorov-Smirnov test.

Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.

Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting marginal distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the theory of errors (see below).

To summarize, here is a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.

In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as
- Binomial random variables, associated with yes/no questions;
- Poisson random variables, associated with rare events;
In physiological measurements of biological specimens:
- The logarithm of measures of size of living tissue (length, height, skin area, weight);
- The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
- Other physiological measures may be normally distributed, but there is no reason to expect that a priori;
Measurement errors are often assumed to be normally distributed, and any deviation from normality is considered something which should be explained;
Financial variables, in the Black–Scholes model
- Changes in the logarithm of exchange rates, price indices, and stock market indices; these variables behave like compound interest, not like simple interest, and so are multiplicative;
- While the Black–Scholes model assumes normality, in reality these variables exhibit heavy tails, as seen in stock market crashes;
- Other financial variables may be normally distributed, but there is no reason to expect that a priori;
Light intensity
- The intensity of laser light is normally distributed;
- Thermal light has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.

Photon counting

Light intensity from a single source varies with time, as thermal fluctuations can be observed if the light is analyzed at sufficiently high time resolution. Quantum mechanics interprets measurements of light intensity as photon counting, where the natural assumption is to use the Poisson distribution. When light intensity is integrated over large times longer than the coherence time, the Poisson-to-normal approximation is appropriate.

Measurement errors

Normality is the central assumption of the mathematical theory of errors. Similarly, in statistical model-fitting, an indicator of goodness of fit is that the residuals (as the errors are called in that setting) be independent and normally distributed. The assumption is that any deviation from normality needs to be explained. In that sense, both in model-fitting and in the theory of errors, normality is the only observation that need not be explained, being expected. However, if the original data are not normally distributed (for instance if they follow a Cauchy distribution), then the residuals will also not be normally distributed. This fact is usually ignored in practice.

Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Whether this assumption is valid is debatable.

A famous and oft-quoted remark attributed to Gabriel Lippmann says: "Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics." ^[10]

Physical characteristics of biological specimens

The sizes of full-grown animals is approximately lognormal. The evidence and an explanation based on models of growth was first published in the 1932 book Problems of Relative Growth by Julian Huxley.

Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further make the distribution of sizes deviate from lognormality.

The assumption that linear size of biological specimens is normal (rather than lognormal) leads to a non-normal distribution of weight (since weight or volume is roughly proportional to the 2nd or 3rd power of length, and Gaussian distributions are only preserved by linear transformations), and conversely assuming that weight is normal leads to non-normal lengths. This is a problem, because there is no a priori reason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.

On the other hand, there are some biological measures where normality is assumed, such as blood pressure of adult humans. This is supposed to be normally distributed, but only after separating males and females into different populations (each of which is normally distributed).

Financial variables

The normal model of asset price movements does not capture extreme movements such as stock market crashes.

Already in 1900 Louis Bachelier proposed representing price changes of stocks using the normal distribution. This approach has since been modified slightly. Because of the multiplicative nature of compounding of returns, financial indicators such as stock values and commodity prices exhibit "multiplicative behavior". As such, their periodic changes (e.g., yearly changes) are not normal, but rather lognormal - i.e. logarithmic returns as opposed to values are normally distributed. This is still the most commonly used hypothesis in finance, in particular in option pricing in the Black–Scholes model.

However, in reality financial variables exhibit heavy tails, and thus the assumption of normality understates the probability of extreme events such as stock market crashes. Corrections to this model have been suggested by mathematicians such as Benoît Mandelbrot, who observed that the changes in logarithm over short periods (such as a day) are approximated well by distributions that do not have a finite variance, and therefore the central limit theorem does not apply. Rather, the sum of many such changes gives log-Levy distributions.

Distribution in standardized testing and intelligence

In standardized testing, results can be scaled to have a normal distribution; for example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100. As the entire population is known, this normalization can be done, and allows the use of the Z test in standardized testing.

Sometimes, the difficulty and number of questions on an IQ test is selected in order to yield normal distributed results. Or else, the raw test scores are converted to IQ values by fitting them to the normal distribution. In either case, it is the deliberate result of test construction or score interpretation that leads to IQ scores being normally distributed for the majority of the population. However, the question whether intelligence itself is normally distributed is more involved, because intelligence is a latent variable, therefore its distribution cannot be observed directly.

Diffusion equation

The probability density function of the normal distribution is closely related to the (homogeneous and isotropic) diffusion equation and therefore also to the heat equation. This partial differential equation describes the time evolution of a mass-density function under diffusion. In particular, the probability density function

$\varphi_{0,t}(x) = \frac{1}{\sqrt{2\pi t\,}}\exp\left(-\frac{x^2}{2t}\right),$

for the normal distribution with expected value 0 and variance t satisfies the diffusion equation:

$\frac{\partial}{\partial t} \varphi_{0,t}(x) = \frac{1}{2} \frac{\partial^2}{\partial x^2} \varphi_{0,t}(x).$

If the mass-density at time t = 0 is given by a Dirac delta, which essentially means that all mass is initially concentrated in a single point, then the mass-density function at time t will have the form of the normal probability density function with variance linearly growing with t. This connection is no coincidence: diffusion is due to Brownian motion which is mathematically described by a Wiener process, and such a process at time t will also result in a normal distribution with variance linearly growing with t.

More generally, if the initial mass-density is given by a function φ(x), then the mass-density at time t will be given by the convolution of φ and a normal probability density function.

Use in computational statistics

The normal distribution arises in many areas of statistics. For example, for a random variable with finite variance, the sampling distribution of the sample mean is approximately normal, even if the distribution of the population from which the sample is taken is not normal. However, for distributions with infinite or undefined variance, such as the Cauchy distribution, the sampling distribution of the sample mean need not be approximately normal.

In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are based on the assumption of normality.

Generating values for normal random variables

For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods and the most basic is to invert the standard normal cdf. More efficient methods are also known, one such method being the Box-Muller transform. An even faster algorithm is the ziggurat algorithm. These are discussed below. A simple approach that is easy to program is as follows. Simply sum 12 uniform (0,1) deviates and subtract 6 (half of 12). This is quite usable in many applications. The sum over these 12 values has an Irwin–Hall distribution; 12 is chosen to give the sum a variance of exactly one. The resulting random deviates are limited to the range (−6, 6) and have a density which is a 12-section eleventh-order polynomial approximation to the normal distribution.^[11]

The Box-Muller method says that, if you have two independent random numbers U and V uniformly distributed on (0, 1], (e.g. the output from a random number generator), then two independent standard normally distributed random variables are X and Y, where:

$X = \sqrt{- 2 \ln U} \, \cos(2 \pi V) ,$

$Y = \sqrt{- 2 \ln U} \, \sin(2 \pi V) .$

This formulation arises because the chi-square distribution with two degrees of freedom (see property 4 above) is an easily-generated exponential random variable (which corresponds to the quantity lnU in these equations). Thus an angle is chosen uniformly around the circle via the random variable V, a radius is chosen to be exponential and then transformed to (normally distributed) x and y coordinates.

George Marsaglia developed the Ziggurat algorithm, which is faster than the Box-Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases where the combination of those two falls outside the "core of the ziggurat" a kind of rejection sampling using logarithms, exponentials and more uniform random numbers has to be employed.

There is also some investigation into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally-distributed data.

Numerical approximations of the normal distribution and its cdf

The normal distribution function is widely used in scientific and statistical computing. Therefore, it has been implemented in various ways.

The GNU Scientific Library calculates values of the standard normal cdf using piecewise approximations by rational functions. Another approximation method uses third-degree polynomials on intervals.^[12] The article on the bc programming language gives an example of how to compute the cdf in Gnu bc.

For a more detailed discussion of how to calculate the normal distribution, see Knuth's The Art of Computer Programming, section 3.4.1C.

Notes

^ Gale Encyclopedia of Psychology - Normal Distribution
^ Havil, 2003
^ Abraham de Moivre, "Approximatio ad Summam Terminorum Binomii (a + b)ⁿ in Seriem expansi" (printed on 12 November 1733 in London for private circulation). This pamphlet has been reprinted in: (1) Richard C. Archibald (1926) “A rare pamphlet of Moivre and some of his discoveries,” Isis, vol. 8, pages 671-683; (2) Helen M. Walker, “De Moivre on the law of normal probability” in David Eugene Smith, A Source Book in Mathematics [New York, New York: McGraw-Hill, 1929; reprinted: New York, New York: Dover, 1959], vol. 2, pages 566-575.; (3) Abraham De Moivre, The Doctrine of Chances (2nd ed.) [London: H. Woodfall, 1738; reprinted: London: Cass, 1967], pages 235-243; (3rd ed.) [London: A Millar, 1756; reprinted: New York, New York: Chelsea, 1967], pages 243-254; (4) Florence N. David, Games, Gods and Gambling: A History of Probability and Statistical Ideas [London: Griffin, 1962], Appendix 5, pages 254-267.
^ Scott, Clayton; Robert Nowak (August 7, 2003). "The Q-function". Connexions. http://cnx.org/content/m11537/1.2/.
^ Barak, Ohad (April 6, 2006). "Q function and error function". Tel Aviv University. http://www.eng.tau.ac.il/~jo/academic/Q.pdf.
^ Weisstein, Eric W.. "Normal Distribution Function". MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NormalDistributionFunction.html.
^ Sanders, Mathijs A.. "Characteristic function of the univariate normal distribution". http://www.planetmathematics.com/CharNormal.pdf. Retrieved on 2009-03-06.
^ Duncan, Acheson J. (1974). Quality Control and Industrial Statistics (4th ed.). Homewood, Ill.: R.D. Irwin. p. 139. ISBN 0-256-01558-9.
^ Johnson, N.L.; S. Kotz, and N. Balakrishnan (1994). Continuous Univariate Distributions. 1 (2nd ed.). Wiley and Sons. ISBN 0-471-58495-9. "Chapter 13, Section 8.2"
^ Whittaker, E. T.; Robinson, G. (1967). The Calculus of Observations: A Treatise on Numerical Mathematics. New York: Dover. p. 179.
^ Johnson, NL; Kotz S, Balakrishnan N. (1995). Continuous Univariate Distributions. 2. Wiley. "Equation(26.48)"
^ Salter, Andy. "B-Spline curves". Spline Curves and Surfaces. http://www.doc.ic.ac.uk/~dfg/AndysSplineTutorial/BSplines.html. Retrieved on 2008-12-05.

References

John Aldrich. Earliest Uses of Symbols in Probability and Statistics. Electronic document, retrieved March 20, 2005. (See "Symbols associated with the Normal Distribution".)
Abraham de Moivre (1738). The Doctrine of Chances.
Stephen Jay Gould (1981). The Mismeasure of Man. First edition. W. W. Norton. ISBN 0-393-01489-4 .
Havil, 2003. Gamma, Exploring Euler's Constant, Princeton, NJ: Princeton University Press, p. 157.
R. J. Herrnstein and Charles Murray (1994). The Bell Curve: Intelligence and Class Structure in American Life. Free Press. ISBN 0-02-914673-9 .
Pierre-Simon Laplace (1812). Analytical Theory of Probabilities.
Jeff Miller, John Aldrich, et al. Earliest Known Uses of Some of the Words of Mathematics. In particular, the entries for "bell-shaped and bell curve", "normal" (distribution), "Gaussian", and "Error, law of error, theory of errors, etc.". Electronic documents, retrieved December 13, 2005.
S. M. Stigler (1999). Statistics on the Table, chapter 22. Harvard University Press. (History of the term "normal distribution".)
Eric W. Weisstein et al. Normal Distribution at MathWorld. Electronic document, retrieved March 20, 2005.
Marvin Zelen and Norman C. Severo (1964). Probability Functions. Chapter 26 of Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ed, by Milton Abramowitz and Irene A. Stegun. National Bureau of Standards.

External links

The normal distribution

Online results and applications

Normal distribution table
Public Domain Normal Distribution Table
Distribution Calculator – Calculates probabilities and critical values for normal, t, chi-square and F-distribution.
Java Applet on Normal Distributions
Interactive Distribution Modeler (incl. Normal Distribution).
Free Area Under the Normal Curve Calculator from Daniel Soper's Free Statistics Calculators website.
Interactive Graph of the Standard Normal Curve Quickly Visualize the one and two-tailed area of the Standard Normal Curve
Javascript calculator which calculates the probability that a value randomly chosen from a Normal Distribution is greater than, less than or between chosen values
Standard Normal Distribution Table for the iPhone
Random Normal Distribution numbers generator. Quantitative Variable Statistical Analyzer

Algorithms and approximations

GNU Scientific Library – Reference Manual – The Gaussian Distribution
Calculating the Cumulative Normal distribution, C++, VBA, sitmo.com
An algorithm for computing the inverse normal cumulative distribution function by Peter J. Acklam – has examples for several programming languages
An Approximation to the Inverse Normal(0, 1) Distribution, gatech.edu
Handbook of Mathematical Functions: Polynomial and Rational Approximations for P(x) and Z(x), Abramowitz and Stegun

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · stable · Tweedie

v • d • e Common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) · Median · Mode

Dispersion	Range · Standard deviation · Coefficient of variation · Percentile

Moments	Variance · Semivariance · Skewness · Kurtosis

Categorical data

Frequency · Contingency table

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and
hypothesis testing

Inference	Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis

Design of experiments	Population · Sampling · Stratified sampling · Replication · Blocking · Sensitivity and specificity

Sample size estimation	Statistical power · Effect size · Standard error

General estimation	Bayesian estimator · Maximum likelihood · Method of moments · Minimum distance · Maximum spacing

Specific tests	Z-test (normal) · Student's t-test · F-test · Chi-square test · Pearson's chi-square test · Wald test · Mann–Whitney U · Wilcoxon signed-rank test

Survival analysis	Survival function · Kaplan-Meier · Logrank test · Failure rate · Proportional hazards models

Correlation and
regression

Correlation	Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Confounding variable

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