Normal distribution

This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution.

Probability density function The red line is the standard normal distribution
Cumulative distribution function Colors match the image above
Notation	${\mathcal {N}}(\mu ,\,\sigma ^{2})$
Parameters	μ ∈ R — mean (location) σ2 ≥ 0 — variance (squared scale)
Support	x ∈ R   if σ2 > 0 x = μ   if σ2 = 0
PDF	${\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$
CDF	${\frac {1}{2}}{\Big [}1+\operatorname {erf} {\Big (}{\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}{\Big )}{\Big ]}$
Mean	μ
Median	μ
Mode	μ
Variance	σ2
Skewness	0
Excess kurtosis	0
Entropy	${\tfrac {1}{2}}\ln(2\pi e\,\sigma ^{2})$
MGF	$e^{\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}}$
CF	$e^{i\mu t-{\tfrac {1}{2}}\sigma ^{2}t^{2}}$
Fisher information	${\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$

In probability theory and statistics, the normal distribution or Gaussian distribution is an absolutely continuous probability distribution with zero cumulants of all orders higher than two. 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:

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

where μ and σ² are the mean and the variance of the distribution. The Gaussian distribution with μ = 0 and σ² = 1 is called the standard normal distribution.

The Gaussian distribution is one of many things named after Carl Friedrich Gauss, who used it to analyze astronomical data, and determined the formula for its probability density function. However, Gauss was not the first to study this distribution or the formula for its density function—that had been done earlier by Abraham de Moivre.

The normal distribution is often 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 (1.8 m). 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 are used.

By the central limit theorem, under certain conditions the sum of a number of random variables with finite means and variances approaches a normal distribution as the number of variables increases. For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural science, and social science 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.

Definition

The simplest case of a normal distribution is known as the standard normal distribution, described by the probability density function

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

The constant $\scriptstyle \ 1/{\sqrt {2\pi }}$ in this expression ensures that the total area under the curve ϕ(x) is equal to one,^[proof] and 1⁄2 in the exponent makes the “width” of the curve (measured as half of the distance between the inflection points of the curve) also equal to one. It is traditional in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters ƒ or p. The alternative glyph φ is also used quite often, however within this article we reserve “φ” to denote characteristic functions.

More generally, a normal distribution results from exponentiating a quadratic function (just as an exponential distribution results from exponentiating a linear function):

$f(x)=e^{ax^{2}+bx+c}.\,$

This yields the classic “bell curve” shape (provided that a < 0 so that the quadratic function is concave). Notice that f(x) > 0 everywhere. One can adjust a to control the “width” of the bell, then adjust b to move the central peak of the bell along the x-axis, and finally adjust c to control the “height” of the bell. For f(x) to be a true probability density function over R, one must choose c such that $\scriptstyle \int _{-\infty }^{\infty }f(x)\,dx\ =\ 1$ (which is only possible when a < 0).

Rather than using a, b, and c, it is far more common to describe a normal distribution by its mean μ = −b/(2a) and variance σ² = −1/(2a). Changing to these new parameters allows us to rewrite the probability density function in a convenient standard form,

$f(x)={\tfrac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}={\tfrac {1}{\sigma }}\,\phi \!\left({\tfrac {x-\mu }{\sigma }}\right).$

Notice that for a standard normal distribution, μ = 0 and σ² = 1. The last part of the equation above shows that any other normal distribution can be regarded as a version of the standard normal distribution that has been stretched horizontally by a factor σ and then translated rightward by a distance μ. Thus, μ specifies the position of the bell curve’s central peak, and σ specifies the “width” of the bell curve.

The parameter μ is at the same time the mean, the median and the mode of the normal distribution. The parameter σ² is called the variance; as for any real-valued random variable, it describes how concentrated the distribution is around its mean. The square root of σ² is called the standard deviation and is the width of the density function.

The normal distribution is usually denoted by N(μ, σ²). Commonly the letter N is written in calligraphic font (typed as \mathcal{N} in LaTeX). Thus when a random variable X is distributed normally with mean μ and variance σ², we write

$X\ \sim \ {\mathcal {N}}(\mu ,\,\sigma ^{2}).\,$

Alternative formulations

Some authors instead of σ² use its reciprocal τ = σ⁻², which is called the precision. 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. Another advantage of using this parameterization is in the study of conditional distributions in multivariate normal case.

The question which normal distribution should be called the “standard” one is also answered differently by various authors. Starting from the works of Gauss the standard normal was considered to be the one with variance σ² = 1/2:

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

Stigler (1982) goes even further and suggests the standard normal with variance σ² = 1/(2π):

$f(x)=e^{-\pi x^{2}}$

According to the author, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution.

Characterization

In the previous section the normal distribution was defined by specifying its probability density function. However there are other ways to characterize a probability distribution. They include: the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, etc.

Probability density function

The probability density function (pdf) of a random variable describes the relative frequencies of different values for that random variables. The pdf of the normal distribution is given by the formula explained in detail in the previous section:

$f(x;\,\mu ,\sigma ^{2})={\tfrac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-(x-\mu )^{2}\!/(2\sigma ^{2})}={\tfrac {1}{\sigma }}\,\phi \!\left({\tfrac {x-\mu }{\sigma }}\right),\qquad x\in \mathbb {R} .$

This is a proper function only when the variance σ² is not equal to zero. In that case this is a continuous smooth function, defined on the entire real line, and which is called the “Gaussian function”.

When σ² = 0, the density function doesn’t exist. However we can consider a generalized function that would behave in a manner similar to the regular density function (in the sense that it defines a measure on the real line, and it can be plugged in into an integral in order to calculate expected values of different quantities):

$f(x;\,\mu ,0)=\delta (x-\mu ).$

This is the Dirac delta function, it is equal to infinity at x = μ and is zero elsewhere.

Properties:

• Function ƒ(x) is symmetric around the point x = μ, which is at the same time the mode, the median and the mean of the distribution.
• The inflection points of the curve occur one standard deviation away from the mean (i.e., at x = μ − σ and x = μ + σ).
• The standard normal density ϕ(x) is an eigenfunction of the Fourier transform.
• The function is supersmooth of order 2, implying that it is infinitely differentiable.
• The derivative of ϕ(x) is ϕ′(x) = −x·ϕ(x), and the second derivative is ϕ′′(x) = (x² − 1)ϕ(x).

Cumulative distribution function

The cumulative distribution function (cdf) describes probabilities for a random variable to fall in the intervals of the form (−∞, x]. The cdf of the standard normal distribution is denoted with the capital Greek letter Φ (phi), and can be computed as an integral of the probability density function:

$\Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt={\frac {1}{2}}{\Big [}\,1+\operatorname {erf} {\Big (}{\frac {x}{\sqrt {2}}}{\Big )}\,{\Big ]},\quad x\in \mathbb {R} .$

This integral can only be expressed in terms of a special function erf, called the error function. The numerical methods for calculation of the standard normal cdf are discussed below. For a generic normal random variable with mean μ and variance σ² > 0 the cdf will be equal to

$F(x;\,\mu ,\sigma ^{2})=\Phi {\Big (}{\frac {x-\mu }{\sigma }}{\Big )}={\frac {1}{2}}{\Big [}\,1+\operatorname {erf} {\Big (}{\frac {x-\mu }{\sigma {\sqrt {2}}}}{\Big )}\,{\Big ]},\quad x\in \mathbb {R} .$

For a normal distribution with zero variance, the cdf is the Heaviside step function:

$F(x;\,\mu ,0)=\mathbf {1} \{x\geq \mu \}\,.$

The complement of the standard normal cdf, Q(x) = 1 − Φ(x), is referred to as the Q-function, especially in engineering texts. This represents the tail probability of the Gaussian distribution, that is the probability that a standard normal random variable X is greater than the number x. Other definitions of the Q-function, all of which are simple transformations of Φ, are also used occasionally.

Properties:

• The standard normal cdf is 2-fold rotationally symmetric around point (0, ½):  Φ(−x) = 1 − Φ(x).
• The derivative of Φ(x) is equal to the standard normal pdf ϕ(x):  Φ′(x) = ϕ(x).
• The antiderivative of Φ(x) is:  ∫ Φ(x) dx = x Φ(x) + ϕ(x).

Quantile function

The inverse of the standard normal cdf, called the quantile function or probit function, is expressed in terms of the inverse error function:

$\Phi ^{-1}(p)\equiv z_{p}={\sqrt {2}}\;\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).$

Quantiles of the standard normal distribution are commonly denoted as z_p. The quantile z_p represents such a value that a standard normal random variable X has the probability of exactly p to fall inside the (−∞, z_p] interval. The quantiles are used in hypothesis testing, construction of confidence intervals and Q-Q plots. The most “famous” normal quantile is 1.96 = z_0.975. A standard normal random variable is greater than 1.96 in absolute value in only 5% of cases.

For a normal random variable with mean μ and variance σ², the quantile function is

$F^{-1}(p;\,\mu ,\sigma ^{2})=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).$

Characteristic function

The characteristic function φ_X(t) of a random variable X is defined as the expected value of e^itX, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function. Thus the characteristic function is the Fourier transform of the density ϕ(x). For a normally distributed X with mean μ and variance σ², the characteristic function is

$\varphi (t;\,\mu ,\sigma ^{2})=\int _{-\infty }^{\infty }\!e^{itx}{\tfrac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {1}{2}}(x-\mu )^{2}}dx=e^{i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}}.$

Moment generating function

The moment generating function is defined as the expected value of e^tX. For a normal distribution, the moment generating function exists and is equal to

$M(t;\,\mu ,\sigma ^{2})=\operatorname {E} [e^{tX}]=\varphi (-it;\,\mu ,\sigma ^{2})=e^{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}}.$

The cumulant generating function is the logarithm of the moment generating function:

$g(t;\,\mu ,\sigma ^{2})=\ln M(t;\,\mu ,\sigma ^{2})=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}.$

Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.

Properties

1. The family of normal distributions is closed under linear transformations. That is, if X is normally distributed with mean μ and variance σ², then a linear transform aX + b (for some real numbers a ≠ 0 and b) is also normally distributed:

   $aX+b\ \sim \ {\mathcal {N}}(a\mu +b,\,a^{2}\sigma ^{2}).$

   Also if X₁, X₂ are two independent normal random variables, with means μ₁, μ₂ and standard deviations σ₁, σ₂, then their linear combination will also be normally distributed: ^[proof]

   $aX_{1}+bX_{2}\ \sim \ {\mathcal {N}}(a\mu _{1}+b\mu _{2},\,a^{2}\!\sigma _{1}^{2}+b^{2}\sigma _{2}^{2})$

2. The converse of (1) is also true: if X₁ and X₂ are independent and their sum X₁ + X₂ is distributed normally, then both X₁ and X₂ must also be normal. This is known as Cramér’s theorem.
3. It is sometimes mistakenly believed that if two normal random variables are uncorrelated then they are also independent. Such “property” is false^[proof]. The correct statement is that if the two random variables are jointly normal and uncorrelated, only then they are independent.
4. Normal distribution is infinitely divisible: for a normally distributed X with mean μ and variance σ² we can find n independent random variables {X₁, …, X_n} each distributed normally with means μ/n and variances σ²/n such that

   $X_{1}+X_{2}+\cdots +X_{n}\ \sim \ {\mathcal {N}}(\mu ,\sigma ^{2})$

5. Normal distribution is stable (with exponent α = 2): if X₁, X₂ are two independent N(μ, σ²) random variables and a, b are arbitrary real numbers, then

   $aX_{1}+bX_{2}\ \sim \ {\sqrt {a^{2}+b^{2}}}\cdot X_{3}\ +\ (a+b-{\sqrt {a^{2}+b^{2}}})\mu ,$

   where X₃ is also N(μ, σ²). This relationship directly follows from property (1).
6. The Kullback–Leibler divergence between two normal distributions X₁ ∼ N(μ₁, σ²₁ )and X₂ ∼ N(μ₂, σ²₂ )is given by:

   $D_{\mathrm {KL} }(X_{1}\,\|\,X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}\,+\,{\frac {1}{2}}{\Bigg (}\,{\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\,{\Bigg )}\ .$

   The Hellinger distance between the same distributions is equal to

   $H^{2}(X_{1},X_{2})=1\,-\,{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\;e^{-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\ .$

7. The Fisher information matrix for normal distribution is diagonal and takes form

   ${\mathcal {I}}={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&0\\0&{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}$

8. Normal distributions belongs to an exponential family with natural parameters $\scriptstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}$ and $\scriptstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}$, and natural statistics x and x². The dual, expectation parameters for normal distribution are η₁ = μ and η₂ = μ² + σ².
9. Of all probability distributions over the reals with mean μ and variance σ², the normal distribution N(μ, σ²) is the one with the maximum entropy.
10. The family of normal distributions forms a manifold with constant curvature −1. The same family is flat with respect to the (±1)-connections ∇^(e) and ∇^(m).

Standardizing normal random variables

As a consequence of property 1, it is possible to relate all normal random variables to the standard normal. For example if X is normal with mean μ and variance σ², then

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

has mean zero and unit variance, that is Z has the standard normal distribution. Conversely, having a standard normal random variable Z we can always construct another normal random variable with specific mean μ and variance σ²:

$X=\sigma Z+\mu .\,$

This “standardizing” transformation is convenient as it allows one to compute the pdf and especially the cdf of a normal distribution having the table of pdf and cdf values for the standard normal. They will be related via

$F_{X}(x)=\Phi {\bigg (}{\frac {x-\mu }{\sigma }}{\bigg )},\quad f_{X}(x)={\frac {1}{\sigma }}\,\phi {\bigg (}{\frac {x-\mu }{\sigma }}{\bigg )}.$

Moments

The normal distribution has moments of all orders. That is, for a normally distributed X with mean μ and variance σ², the expectation E[|X|^p] exists and is finite for all p such that Re[p] > −1. Usually we are interested only in moments of integer orders: p = 1, 2, 3, ….

• Central moments are the moments of X around its mean μ. Thus, central moment of order p is the expected value of (X − μ)^p. Using standardization of normal distribution, this expectation will be equal to σ^p·E[Z^p], where Z is standard normal.

  $\operatorname {E} {\big [}(X-\mu )^{p}{\big ]}=\left.{\begin{cases}0&{\text{if }}p{\text{ is odd}}\\\sigma ^{p}(p-1)!!&{\text{if }}p{\text{ is even}}\end{cases}}\right\}=\sigma ^{p}{\frac {p!}{2^{p/2}(p/2)!}}\cdot \mathbf {1} _{\{p{\text{ is even}}\}}.$

  Here n!! denotes the double factorial, that is the product of every other odd number from 1 to n; and 1_{…} is the indicator function.
• Central absolute moments are the moments of |X − μ|. They coincide with regular moments for all even orders, but are nonzero for all odd p’s.

  $\operatorname {E} {\big [}|X-\mu |^{p}{\big ]}=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {2/\pi }}&{\text{if }}p{\text{ is odd}},\\1&{\text{if }}p{\text{ is even}}.\end{cases}}$

• Raw moments and raw absolute moments are the moments of X and |X| respectively. The formulas for these moments are much more complicated, and are given in terms of confluent hypergeometric functions ₁F₁ and U.

  ${\begin{aligned}&\operatorname {E} {\big [}X^{p}{\big ]}=\sigma ^{p}\cdot (-i{\sqrt {2}}\operatorname {sgn} \mu )^{p}\;U{\Big (}{-{\tfrac {1}{2}}p},\,{\tfrac {1}{2}},\,-{\tfrac {1}{2}}(\mu /\sigma )^{2}{\Big )},\\&\operatorname {E} {\big [}|X|^{p}{\big ]}=\sigma ^{p}\cdot 2^{\frac {p}{2}}{\frac {\Gamma {\big (}{\frac {1+p}{2}}{\big )}}{\sqrt {\pi }}}\;_{1}F_{1}{\Big (}{-{\tfrac {1}{2}}p},\,{\tfrac {1}{2}},\,-{\tfrac {1}{2}}(\mu /\sigma )^{2}{\Big )}.\\\end{aligned}}$

  These expressions remain valid even if p is not integer. See also generalized Hermite polynomials.
• First two cumulants are equal to μ and σ² respectively, whereas all higher-order cumulants are equal to zero.

Order	Raw moment	Central moment	Cumulant
1	$\scriptstyle \mu$	0	$\scriptstyle \mu$
2	$\scriptstyle \mu ^{2}+\sigma ^{2}$	$\scriptstyle \sigma ^{2}$	$\scriptstyle \sigma ^{2}$
3	$\scriptstyle \mu ^{3}+3\mu \sigma ^{2}$	0	0
4	$\scriptstyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}$	$\scriptstyle 3\sigma ^{4}$	0
5	$\scriptstyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}$	0	0
6	$\scriptstyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}$	$\scriptstyle 15\sigma ^{6}$	0
7	$\scriptstyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}$	0	0
8	$\scriptstyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}$	$\scriptstyle 105\sigma ^{8}$	0

Central limit theorem

Main article: Central limit theorem

The theorem states that under certain, fairly common conditions, the sum of a large number of random variables will have approximately normal distribution. For example if (x₁, …, x_n) is a sequence of iid random variables, each having mean μ and variance σ² but otherwise distributions of x_i’s can be arbitrary, then the central limit theorem states that

${\sqrt {n}}{\bigg (}{\frac {1}{n}}\sum _{i=1}^{n}x_{i}-\mu {\bigg )}\ {\xrightarrow {d}}\ {\mathcal {N}}(0,\,\sigma ^{2}).$

The theorem will hold even if the summands x_i are not iid, although some constraints on the degree of dependence and the growth rate of moments still have to be imposed.

The importance of the central limit theorem cannot be overemphasized. A great number of test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, even more estimators can be represented as sums of random variables through the use of influence functions — all of these quantities are governed by the central limit theorem and will have asymptotically normal distribution as a result.

As the number of discrete events increases, the function begins to resemble a normal distribution

Another practical consequence of the central limit theorem is that certain other distributions can be approximated by the normal distribution, for example:

• The binomial distribution B(n, p) is approximately normal N(np, np(1 − p)) for large n and for p not too close to zero or one.
• The Poisson(λ) distribution is approximately normal N(λ, λ) for large values of λ.
• The chi-squared distribution χ²(k) is approximately normal N(k, 2k) for large ks.
• The Student’s t-distribution t(ν) is approximately normal N(0, 1) when ν 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 Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

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

${\begin{aligned}&F(\mu +n\sigma ;\,\mu ,\sigma ^{2})-F(\mu -n\sigma ;\,\mu ,\sigma ^{2})\\&\quad =\Phi (n)-\Phi (-n)=2\Phi (n)-1=\mathrm {erf} \left({\frac {n}{\sqrt {2}}}\right),\end{aligned}}$

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

$\scriptstyle \;n\;$	$\;\scriptstyle \mathrm {erf} \left({\frac {n}{\sqrt {2}}}\right)\;$	i.e. 1 minus ...	or 1 in ...
1	0.682689492137	0.317310507863	3.15148718753
2	0.954499736104	0.045500263896	21.9778945081
3	0.997300203937	0.002699796063	370.398347380
4	0.999936657516	0.000063342484	15,787.192684
5	0.999999426697	0.000000573303	1,744,278.331
6	0.999999998027	0.000000001973	506,842,375.7

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:

$\scriptstyle \;\mathrm {erf} \left({\frac {n}{\sqrt {2}}}\right)\;$	$\scriptstyle \;n\;$
0.80	1.281551565545
0.90	1.644853626951
0.95	1.959963984540
0.98	2.326347874041
0.99	2.575829303549
0.995	2.807033768344
0.998	3.090232306168
0.999	3.290526731492
0.9999	3.890591886413
0.99999	4.417173413469

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.

Related distributions

• If X is distributed normally with mean μ and variance σ², then
  • The exponent of X is distributed log-normally: e^X ~ lnN (μ, σ²).
  • The absolute value of X has folded normal distribution: IXI ~ N_f (μ, σ²). If μ = 0 this is known as the half-normal distribution.
  • The square of X/σ has the non-central chi-square distribution with one degree of freedom: X²/σ² ~ χ²₁(μ²/σ²). If μ = 0, the distribution is called simply chi-square.
  • Variable X restricted to an interval [a, b] is called the truncated normal distribution.
  • (X − μ)⁻² has a Lévy distribution with location 0 and scale σ⁻².
• If X₁ and X₂ are two independent standard normal random variables, then
  • Their sum and difference is distributed normally with mean zero and variance two: X₁ ± X₂ ∼ N(0, 2).
  • Their product Z = X₁·X₂ follows the “product-normal” distribution with density function f_Z(z) = π⁻¹K₀(|z|), where K₀ is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at z = 0, and has the characteristic function φ_Z(t) = (1 + t²)^−1/2.
  • Their ratio follows the standard Cauchy distribution: X₁ ÷ X₂ ∼ Cauchy(0, 1).
  • Their Euclidean norm $\scriptstyle {\sqrt {X_{1}^{2}\,+\,X_{2}^{2}}}$ has the Rayleigh distribution, also known as the chi distribution with 2 degrees of freedom.
• If X₁, X₂, …, X_n are independent standard normal random variables, then the sum of their squares has the chi-square distribution with n degrees of freedom: $\scriptstyle X_{1}^{2}+\cdots +X_{n}^{2}\ \sim \ \chi _{n}^{2}$.
• If X₁, X₂, …, X_n are independent normally distributed random variables with means μ and variances σ², then their sample mean is independent from the sample standard deviation, which can be demonstrated using the Basu’s theorem or Cochran’s theorem. The ratio of these two quantities will have the Student’s t-distribution with n − 1 degrees of freedom:

  $t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\tfrac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\tfrac {1}{n(n-1)}}{\big [}(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}{\big ]}}}}\ \sim \ t_{n-1}.$

• If X₁, …, X_n, Y₁, …, Y_m are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom:

  $F={\frac {{\big (}X_{1}^{2}+X_{2}^{2}+\ldots +X_{n}^{2}{\big )}/n}{{\big (}Y_{1}^{2}+Y_{2}^{2}+\ldots +Y_{m}^{2}{\big )}/m}}\ \sim \ F_{n,\,m}.$

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case. All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.

• Multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R^k is multivariate-normally distributed if any linear combination of its components  $\scriptstyle \sum _{j=1}^{k}a_{j}X_{j}$  has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V.
• Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ C^k is said to be normal if both his real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
• Matrix normal distribution describes the case of normally distributed matrices.
• Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element h ∈ H is said to be normal if for any constant a ∈ H the scalar product (a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
  • Brownian motion,
  • Brownian bridge,
  • Ornstein-Uhlenbeck process.
• Gaussian q-distribution is an abstract mathematical construction which represents a “q-analogue” of the normal distribution.

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:

• Pearson distribution — a four-parametric family of probability distributions that extend the normal law to include different skewness and kurtosis values.

Normality tests

Main article: Normality tests

Normality tests assess the likelihood that the given data set {x₁, …, x_n} comes from a normal distribution. Typically the null hypothesis H₀ is that the observations are distributed normally with unspecified mean μ and variance σ², versus the alternative H_a that the distribution is arbitrary. A great number of tests (over 40) have been devised for this problem, the more prominent of them are outlined below:

• “Visual” tests are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
  • Q-Q plot — is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it’s a plot of point of the form (Φ⁻¹(p_k), x_(k)), where plotting points p_k are equal to p_k = (k−α)/(n+1−2α) and α is an adjustment constant which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
  • P-P plot — similar to the Q-Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_(k)), p_k), where $\scriptstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}$. For normally distributed data this plot should lie on a 45° line between (0,0) and (1,1).
  • Wilk–Shapiro test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
  • Normal probability plot (rankit plot)

• Moment tests:
  • D’Agostino’s K-squared test
  • Jarque–Bera test

• Empirical distribution function tests:
  • Kolmogorov–Smirnov test
  • Lilliefors test
  • Anderson–Darling test

Estimation of parameters

It is often the case that we don’t know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x₁, …, x_n) from a normal N(μ, σ²) population we would like to learn the approximate values of parameters μ and σ². The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:

$\ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i};\,\mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.$

Taking derivatives with respect to μ and σ² and solving the resulting system of first order conditions yields the maximum likelihood estimates:

${\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.$

Estimator $\scriptstyle {\hat {\mu }}$ is called the sample mean, since it is the arithmetic mean of all observations. The statistic $\scriptstyle {\overline {x}}$ is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, $\scriptstyle {\hat {\mu }}$ is the uniformly minimum variance unbiased (UMVU) estimator. In finite samples it is distributed normally:

${\hat {\mu }}\ \sim \ {\mathcal {N}}(\mu ,\,\,\sigma ^{2}\!\!\;/n).$

The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix $\scriptstyle {\mathcal {I}}^{-1}$. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of $\scriptstyle {\hat {\mu }}$ is proportional to $\scriptstyle 1/{\sqrt {n}}$, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.

From the standpoint of the asymptotic theory, $\scriptstyle {\hat {\mu }}$ is consistent, that is, it converges in probability to μ as n → ∞. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:

${\sqrt {n}}({\hat {\mu }}-\mu )\ {\xrightarrow {d}}\ {\mathcal {N}}(0,\,\sigma ^{2}).$

The estimator $\scriptstyle {\hat {\sigma }}^{2}$ is called the sample variance, since it is the variance of the sample (x₁, …, x_n). In practice, another estimator is often used instead of the $\scriptstyle {\hat {\sigma }}^{2}$. This other estimator is denoted s², and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. The estimator s² differs from $\scriptstyle {\hat {\sigma }}^{2}$ by having (n − 1) instead of n in the denominator (the so called Bessel’s correction):

$s^{2}={\frac {n}{n-1}}\,{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.$

The difference between s² and $\scriptstyle {\hat {\sigma }}^{2}$ becomes negligibly small for large n’s. In finite samples however, the motivation behind the use of s² is that it is an unbiased estimator of the underlying parameter σ², whereas $\scriptstyle {\hat {\sigma }}^{2}$ is biased. Also, by the Lehmann–Scheffé theorem the estimator s² is uniformly minimum variance unbiased (UMVU), which makes it the “best” estimator among all unbiased ones. However it can be shown that the biased estimator $\scriptstyle {\hat {\sigma }}^{2}$ is “better” than the s² in terms of the mean squared error (MSE) criterion. In finite samples both s² and $\scriptstyle {\hat {\sigma }}^{2}$ have scaled chi-squared distribution with (n − 1) degrees of freedom:

$s^{2}\ \sim \ {\tfrac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\ \sim \ {\tfrac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}\ .$

The first of these expressions shows that the variance of s² is equal to 2σ⁴/(n−1), which is slightly greater than the σσ-element of the inverse Fisher information matrix $\scriptstyle {\mathcal {I}}^{-1}$. Thus, s² is not an efficient estimator for σ², and moreover, since s² is UMVU, we can conclude that the finite-sample efficient estimator for σ² does not exist.

Applying the asymptotic theory, both estimators s² and $\scriptstyle {\hat {\sigma }}^{2}$ are consistent, that is they converge in probability to σ² as the sample size n → ∞. The two estimators are also both asymptotically normal:

${\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\ {\xrightarrow {d}}\ {\mathcal {N}}(0,\,2\sigma ^{4}).$

In particular, both estimators are asymptotically efficient for σ².

By Cochran’s theorem, for normal distribution the sample mean $\scriptstyle {\hat {\mu }}$ and the sample variance s² are independent, which means there can be no gain in considering their joint distribution. There is also a reverse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between $\scriptstyle {\hat {\mu }}$ and s can be employed to construct the so-called t-statistic:

$t={\frac {{\hat {\mu }}-\mu }{s/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\ \sim \ t_{n-1}$

This quantity t has the Student’s t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ; similarly, inverting the χ² distribution of the statistic s² will give us the confidence interval for σ²:

${\begin{aligned}&\mu \in {\Big [}\,{\hat {\mu }}+t_{n-1,\alpha /2}\,{\tfrac {1}{\sqrt {n}}}s,\ \ {\hat {\mu }}+t_{n-1,1-\alpha /2}\,{\tfrac {1}{\sqrt {n}}}s\,{\Big ]}\approx {\Big [}\,{\hat {\mu }}-|z_{\alpha /2}|{\tfrac {1}{\sqrt {n}}}s,\ \ {\hat {\mu }}+|z_{\alpha /2}|{\tfrac {1}{\sqrt {n}}}s\,{\Big ]},\\&\sigma ^{2}\in {\bigg [}\,{\frac {(n-1)s^{2}}{\chi _{n-1,1-\alpha /2}^{2}}},\ \ {\frac {(n-1)s^{2}}{\chi _{n-1,\alpha /2}^{2}}}\,{\bigg ]}\approx {\Big [}\,s^{2}-|z_{\alpha /2}|{\tfrac {\sqrt {2}}{\sqrt {n}}}s^{2},\ \ s^{2}+|z_{\alpha /2}|{\tfrac {\sqrt {2}}{\sqrt {n}}}s^{2}\,{\Big ]},\end{aligned}}$

where t_k,p and χ_k,p² are the p^th quantiles of the t- and χ²-distributions respectively. These confidence intervals are of the level 1 − α, meaning that the true values μ and σ² fall outside of these intervals with probability α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The approximate formulas in the display above were derived from the asymptotic distributions of $\scriptstyle {\hat {\mu }}$ and s². The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_{α/2 do not depend on n. In particular, the most popular value of α = 5%, results in |z0.025| = 1.96.}

Occurrence

The occurrence of normal distribution in practical problems can be loosely classified into three categories:

1. Exactly normal distributions;
2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
3. Distributions modeled as normal — the normal distribution being one of the simplest and most convenient to use, frequently researchers are tempted to assume that certain quantity is distributed normally, without justifying such assumption rigorously. In fact, the maturity of a scientific field can be judged by the prevalence of the normality assumption in its methods.

Exact normality

The ground state of a quantum harmonic oscillator has the Gaussian distribution.

Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:

• Velocities of the molecules in the ideal gas. More generally, velocities of the particles in any system in thermodynamic equilibrium will have normal distribution, due to the maximum entropy principle.
• Probability density function of a ground state in a quantum harmonic oscillator.
• The density of an electron cloud in 1s state.
• The position of a particle which experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is a dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation  $\scriptstyle {\frac {\partial }{\partial t}}f(x)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x)$. If the initial location is given by a certain density function g(x), then the density at time t is the convolution of g and the normal pdf.

Approximate normality

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by a large number of small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence which has a considerably larger magnitude than the rest of the effects.

• 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 binary response variables;
  • Poisson random variables, associated with rare events;
• 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.

Assumed normality

I can only recognize the occurrence of the normal curve — the Laplacian curve of errors — as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations. — Pearson (1901)

There are statistical methods to empirically test that assumption, see the #Normality tests section.

• In biology:
  • 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;
  • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).
• In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoît Mandelbrot argue that log-Levy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes.
• Measurement errors in physical experiments are often assumed to be normally distributed. This assumption allows for particularly simple practical rules for how to combine errors in measurements of different quantities. However, whether this assumption is valid or not in practice is debatable. A famous remark of 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.”
• In standardized testing, results can be made to have a normal distribution. This is done by either selecting the number and difficulty of questions (as in the IQ test), or by transforming the raw test scores into “output” scores by fitting them to the 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.
• 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. Bell curve grading assigns relative grades based on a normal distribution of scores.

Generating values from normal distribution

The bean machine, a device invented by sir Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

For computer simulations, especially in applications of Monte-Carlo method, it is often useful to generate values that have a normal distribution. All algorithms described here are concerned with generating the standard normal, since a N(μ, σ²) can be generated as X = μ + σZ, where Z is standard normal. The algorithms rely on the availability of a random number generator capable of producing random values distributed uniformly.

• The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ⁻¹(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article.
• A simple approximate approach that is easy to program is as follows: simply sum 12 uniform (0,1) deviates and subtract 6 — the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).
• The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1]. Then two random variables X and Y

  ${\begin{aligned}&X={\sqrt {-2\ln U}}\,\cos(2\pi V),\\&Y={\sqrt {-2\ln U}}\,\sin(2\pi V).\end{aligned}}$

  will both have the standard normal distribution, and be independent. This formulation arises because for a bivariate normal random vector (X Y) the squared norm X² + Y² will have the chi-square distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.
• Marsaglia polar method is a modification of the Box–Muller method algorithm, which does not require computation of functions sin() and cos(). In this method U and V are drawn from the uniform (−1,1) distribution, and then S = U² + V² is computed. If S is greater or equal to one then the method starts over, otherwise two quantities

  $X=U{\sqrt {\frac {-2\ln S}{S}}},\qquad Y=V{\sqrt {\frac {-2\ln S}{S}}}$

  are returned. Again, X and Y here will be independent and standard normally distributed.
• Ratio method starts with generating two independent uniform deviates U and V. The algorithm proceeds as follows:
  • Compute X = √(8/e) (V − 0.5)/U;
  • If X² ≤ 5 − 4e^1/4U then accept X and terminate algorithm;
  • If X² ≥ 4e^−1.35/U + 1.4 then reject X and start over from step 1;
  • If X² ≤ −4 / lnU then accept X, otherwise start over the algorithm.
• The ziggurat algorithm (Marsaglia & Tsang 2000) 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 for the normal cdf

The standard normal cdf is widely used in scientific and statistical computing. The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.

• Abramowitz & Stegun (1964) give the approximation for Φ(x) with the absolute error |ε(x)| < 7.5·10⁻⁸ (algorithm 26.2.17):

  $\Phi (x)=1-\phi (x){\Big (}b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}{\Big )}+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},$

  where ϕ(x) is the standard normal pdf, and b₀ = 0.2316419, b₁ = 0.319381530, b₂ = −0.356563782, b₃ = 1.781477937, b₄ = −1.821255978, b₅ = 1.330274429.
• Hart (1968) lists almost a hundred of rational function approximations for the erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart’s algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
• Marsaglia (2004) suggested a simple algorithm based on the Taylor series expansion

  $\Phi (x)={\frac {1}{2}}+\phi (x){\bigg (}x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+{\frac {x^{7}}{3\cdot 5\cdot 7}}+{\frac {x^{9}}{3\cdot 5\cdot 7\cdot 9}}+\cdots {\bigg )}$

  for calculating Φ(x) with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when x = 10).
• The GNU Scientific Library calculates values of the standard normal cdf using Hart’s algorithms and approximations with Chebyshev polynomials.

Implementation

Most of statistical packages have built-in procedures for dealing with the normal distributions. The typically required tasks are: to generate standard normal random variates for Monte-Carlo simulations; to calculate the standard normal cdf so that the p-values for a given z-score may be obtained; and to compute the inverse standard normal cdf to find the p-value for a given z-score. Examples of functions performing these tasks in various applications are given below:

	Generating standard normal r.v’s	Standard normal cdf Φ(x)	Quantile function Φ−1(p)
Excel	— a	NORMSDIST(x)	NORMSINV(p)
Matlab	normrnd(0,1)	normcdf(x,0,1)	norminv(p,0,1)
Mathematicab	RandomReal[NormalDistribution[]]	CDF[NormalDistribution[],x]	Quantile[NormalDistribution[],p]
Stata	rnormal(0,1)	normal(x)	invnormal(p)
Gauss	rndn(1,1)	cdfn(x)	cdfni(p)
R	rnorm(1)	pnorm(x)	qnorm(p)
online resources	[3]	[4] [5]	[6]

• ^a You can generate normal random variates using any of the methods given above, for example by inverting the standard normal cdf: NORMSINV(RAND()).
• ^b In Mathematica there is a special object NormalDistribution[μ,σ] to represent the normal distribution. This object can be used to compute a variety of properties of the distribution.

History

Some authors attribute at least a partial credit for the discovery of the normal distribution to de Moivre, who was studying the magnitudes of the terms in the binomial expansion of (1 + 1)ⁿ, and published his results in the second edition of his “The Doctrine of Chances” (1738). De Moivre proved that the middle term in this expansion has the approximate magnitude of $\scriptstyle 2/{\sqrt {2\pi n}}$, and that “If m or ½n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term diſtant from the middle by the Interval ℓ, has to the middle Term, is $\scriptstyle -{\frac {2\ell \ell }{n}}$.” Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.

Carl Friedrich Gauss invented the normal distribution in 1809 as a way to rationalize the method of least squares.

In 1809 Gauss publishes his tract “Theoria motus corporum coelestium in sectionibus conicis solem ambientium” where among other things he introduces and describes several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Denoting M, M′, M′′, … the measurements of some unknown quantity V, Gauss seeks the “most probable” estimator: the one which maximizes the probability φ(M−V) · φ(M′−V) · φ(M′′−V) · … of obtaining the observed experimental results. In his notation φΔ is the probability law of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values. Starting from these principles, Gauss demonstrates that the only law which rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:

$\varphi \Delta ={\tfrac {h}{\surd \pi }}\,e^{-hh\Delta \Delta }\ ,$

where h is “the measure of the precision of the observations”. Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares (NWLS) method.

Marquis de Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.

Although Gauss was the first to suggest the normal distribution law, the merit of the contributions of Laplace cannot be underestimated. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e^−t ²dt = √π in 1782, providing the normalization constant for the normal distribution. Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental central limit theorem, which underlined the theoretical importance of the normal distribution.

It is of interest to notice that in 1809 an american mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss. His works remained unnoticed until 1871 when they were rediscovered by Abbe, mainly because the scientific community was virtually non-existent in United States at that time.

In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena. He writes: “The number of particles whose velocity, resolved in a certain direction, lies between x and x+dx is

$\mathrm {N} {\frac {1}{\alpha {\sqrt {\pi }}}}e^{-{\frac {x^{2}}{\alpha ^{2}}}}dx$

It was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:

$df={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-m)^{2}}{2\sigma ^{2}}}}dx$

Since its introduction, the normal distribution has been known by many different names: the law of error, the law of facility of errors, Laplace’s second law, Gaussian law, etc. The name “normal distribution” was coined independently by Peirce, Galton and Lexis around 1875; the term was derived from the fact that this distribution was seen as typical, common, normal. This name was popularized in statistical community by Pearson around the turn of the 20th century.

Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another ‘abnormal.’

— Pearson (1920)

The term “standard normal” which denotes the normal distribution with zero mean and unit variance came into general use around 1950s, appearing in the popular textbooks by P.G. Hoel (1947) “Introduction to mathematical statistics” and A.M. Mood (1950) “Introduction to the theory of statistics”.

See also

• Behrens–Fisher problem — the long-standing problem of testing whether two normal samples with different variances have same means;
• Erdős-Kac theorem — on the occurrence of the normal distribution in number theory
• Gaussian blur — convolution which uses the normal distribution as a kernel

Notes

Bibliography

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