Normal distribution
Normal distribution
![{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/187f33664b79492eedf4406c66d67f9fe5f524ea)
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[7][8] A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from the same distribution converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[9] Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
The normal distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
The probability density of the normal distribution is
where
is the mean or expectation of the distribution (and also its median and mode),
is the standard deviation, and
is the variance.
Definition
Standard normal distribution
General normal distribution
Every normal distribution is the exponential of a quadratic function:
Notation
Alternative parameterizations
According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution.
- σ
Properties
The normal distribution is the only absolutely continuous distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance.[14][15] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.[16][17]
The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the Cauchy distribution and the Lévy distribution.
Symmetries and derivatives
It is symmetric around the point which is at the same time the mode, the median and the mean of the distribution.[18]
It is unimodal: its first derivative is positive for negative for and zero only at
The area under the curve and over the -axis is unity (i.e. equal to one).
Its density has two inflection points (where the second derivative of is zero and changes sign), located one standard deviation away from the mean, namely at and [18]
Its density is log-concave.[18]
Its density is infinitely differentiable, indeed supersmooth of order 2.[19]
Its first derivative is
Its second derivative is
More generally, its nth derivative is where is the nth (probabilist) Hermite polynomial.[20]
The probability that a normally distributed variable with known and is in a particular set, can be calculated by using the fact that the fraction has a standard normal distribution.
Moments
![{\displaystyle \operatorname {E} \left[X^{p}\right]={\begin{cases}0&{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&{\text{if }}p{\text{ is even.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7dbebab41a0bdccc9ba057abd20e177396a3f5)
![{\displaystyle \operatorname {E} \left[|X|^{p}\right]=\sigma ^{p}(p-1)!!\cdot \left.{\begin{cases}{\sqrt {\frac {2}{\pi }}}&{\text{if }}p{\text{ is odd}}\\1&{\text{if }}p{\text{ is even}}\end{cases}}\right\}=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87925cf2a9fd2db8e4a3931532583b273c6c94d2)
![{\displaystyle \operatorname {E} \left[X^{p}\right]=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8e62ff69ee55b83475defdc6b46f842124c885)
![{\displaystyle \operatorname {E} \left[|X|^{p}\right]=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1d3b912eefbb43022fe2970e4c6885efb98e97)
| Order | Non-central moment | Central moment |
|---|---|---|
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| 8 |
![[a,b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle \operatorname {E} \left[X\mid a<X<b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82ec10bf31f0b63137699ae6e2b5a346770b097)
Fourier transform and characteristic function
Moment and cumulant generating functions
![{\displaystyle M(t)=\operatorname {E} [e^{tX}]={\hat {f}}(it)=e^{\mu t}e^{{\tfrac {1}{2}}\sigma ^{2}t^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bbd225c0fee5e58e9a8cd73b0f1b2bf535dc56)
The cumulant generating function is the logarithm of the moment generating function, namely
Cumulative distribution function
![[-x,x]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e23c41ff0bd6f01a0e27054c2b85819fcd08b762)
These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below.
The two functions are closely related, namely
![{\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7831a9a5f630df7170fa805c186f4c53219ca36)
![{\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75deccfbc473d782dacb783f1524abb09b8135c0)
The CDF of the standard normal distribution can be expanded by Integration by parts into a series:
![{\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54d12af9a3b12a7f859e4e7be105d172b53bcfb8)
An asymptotic expansion of the CDF for large x can also be derived using integration by parts; see Error function#Asymptotic expansion.[27]
Standard deviation and coverage
For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.
About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.
Quantile function
The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
| 0.80 | 1.281551565545 | 0.999 | 3.290526731492 | |
| 0.90 | 1.644853626951 | 0.9999 | 3.890591886413 | |
| 0.95 | 1.959963984540 | 0.99999 | 4.417173413469 | |
| 0.98 | 2.326347874041 | 0.999999 | 4.891638475699 | |
| 0.99 | 2.575829303549 | 0.9999999 | 5.326723886384 | |
| 0.995 | 2.807033768344 | 0.99999999 | 5.730728868236 | |
| 0.998 | 3.090232306168 | 0.999999999 | 6.109410204869 |
Zero-variance limit
Central limit theorem
As the number of discrete events increases, the function begins to resemble a normal distribution
Comparison of probability density functions, for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
The binomial distribution is approximately normal with mean and variance for large and for not too close to 0 or 1.
The Poisson distribution with parameter is approximately normal with mean and variance , for large values of .[31]
The chi-squared distribution is approximately normal with mean and variance , for large .
The Student's t-distribution is approximately normal with mean 0 and variance 1 when 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.
Maximum entropy
Operations on normal deviates
The family of normal distributions is closed under linear transformations: if X is normally distributed with mean μ and standard deviation σ, then the variable Y = aX + b, for any real numbers a and b, is also normally distributed, with mean aμ + b and standard deviation |a|σ.
- X
- μ
In particular, if X and Y are independent normal deviates with zero mean and variance σ2, then X + Y and X − Y are also independent and normally distributed, with zero mean and variance 2σ2. This is a special case of the polarization identity.[36]
Also, if X1, X2 are two independent normal deviates with mean μ and deviation σ, and a, b are arbitrary real numbers, then the variable
is also normally distributed with mean μ and deviation σ. It follows that the normal distribution is stable (with exponent α = 2).
More generally, any linear combination of independent normal deviates is a normal deviate.
Infinite divisibility and Cramér's theorem
For any positive integer n, any normal distribution with mean μ and variance σ2 is the distribution of the sum of n independent normal deviates, each with mean μ/n and variance σ2/n. This property is called infinite divisibility.[37]
Conversely, if X1 and X2 are independent random variables and their sum X1 + X2 has a normal distribution, then both X1 and X2 must be normal deviates.[38]
This result is known as Cramér’s decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.[39]
Bernstein's theorem
More generally, if X1, ..., Xn are independent random variables, then two distinct linear combinations ∑akXk and ∑bkXk will be independent if and only if all Xk's are normal and ∑akbkσ 2k = 0, where σ 2k denotes the variance of Xk.[40]
Other properties
Related distributions
Operations on a single random variable
If X is distributed normally with mean μ and variance σ2, then
The exponential of X is distributed log-normally: eX ~ ln(N (μ, σ2)).
The absolute value of X has folded normal distribution: |X| ~ Nf (μ, σ2). If μ = 0 this is known as the half-normal distribution.
The absolute value of normalized residuals, |X − μ|/σ, has chi distribution with one degree of freedom: |X − μ|/σ ~ χ1(|X − μ|/σ).
The square of X/σ has the noncentral chi-squared distribution with one degree of freedom: X2/σ2 ~ χ21(μ2/σ2). If μ = 0, the distribution is called simply chi-squared.
The distribution of the 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.
Combination of two independent random variables
If X1 and X2 are two independent standard normal random variables with mean 0 and variance 1, 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[47] with density function fZ(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 φZ(t) = (1 + t 2)−1/2.
Their ratio follows the standard Cauchy distribution: X1 / X2 ∼ Cauchy(0, 1).
Their Euclidean norm has the Rayleigh distribution.
Combination of two or more independent random variables
If X1, X2, ..., Xn are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with n degrees of freedom
If X1, X2, ..., Xn are independent normally distributed random variables with means μ and variances σ2, then their sample mean is independent from the sample standard deviation,[48] which can be demonstrated using Basu's theorem or Cochran's theorem.[49] The ratio of these two quantities will have the Student's t-distribution with n − 1 degrees of freedom:
![{\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\sim t_{n-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ff0d3c79a0504e8f259ef99192b825357914d7)
If X1, ..., Xn, Y1, ..., Ym 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:[50]
Operations on the density function
The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.
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 (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ Rk is multivariate-normally distributed if any linear combination of its components ∑k**j=1aj Xj has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ Ck is said to be normal if both its 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 that represents a "q-analogue" of the normal distribution.
the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. Note that this distribution is different from the Gaussian q-distribution above.
A random variable X has a two-piece normal distribution if it has a distribution
where μ is the mean and σ1 and σ2 are the standard deviations of the distribution to the left and right of the mean respectively.
The mean, variance and third central moment of this distribution have been determined[51]
![{\displaystyle \operatorname {T} (X)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9959f2c5186e2ed76884054edaf837a602ac6fac)
where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.
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-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.
Normality tests
Normality tests assess the likelihood that the given data set {x1, ..., xn} 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 Ha that the distribution is arbitrary. Many 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(pk), x(k)), where plotting points pk are equal to pk = (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)), pk), where . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1). Shapiro-Wilk 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: Lilliefors test (an adaptation of the Kolmogorov–Smirnov 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, ..., xn) 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:
Taking derivatives with respect to μ and σ2 and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Sample mean
Sample variance
In particular, both estimators are asymptotically efficient for σ2.
Confidence intervals
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;[53] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[54]
![{\displaystyle \mu \in \left[{\hat {\mu }}-t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s\right]\approx \left[{\hat {\mu }}-|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0f0c9f6d6cc43a7443c61181d37e2636797770)
![{\displaystyle \sigma ^{2}\in \left[{\frac {(n-1)s^{2}}{\chi _{n-1,1-\alpha /2}^{2}}},{\frac {(n-1)s^{2}}{\chi _{n-1,\alpha /2}^{2}}}\right]\approx \left[s^{2}-|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2},s^{2}+|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27cb861f2528b26c460e44132a762091bfba3f42)
- and
k,pare the pthquantilesof the t- and χ2-distributions respectively. These confidence intervals are of the *confidence level
Bayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
Either the mean, or the variance, or neither, may be considered a fixed quantity.
When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
Both univariate and multivariate cases need to be considered.
Either conjugate or improper prior distributions may be placed on the unknown variables.
An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data, but more complex.
The formulas for the non-linear-regression cases are summarized in the conjugate prior article.
Sum of two quadratics
Scalar form
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.
This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:
The factor has the form of a weighted average of y and z.
This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that is one-half the harmonic mean of a and b.
Vector form
where
Note that the form x′ A x is called a quadratic form and is a scalar:
Sum of differences from the mean
Another useful formula is as follows:
With known variance
First, the likelihood function is (using the formula above for the sum of differences from the mean):
![{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\tau )&=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bcd1c34520a24e29b758a0f7427e79e9d8a414)
Then, we proceed as follows:
![{\displaystyle {\begin{aligned}p(\mu \mid \mathbf {X} )&\propto p(\mathbf {X} \mid \mu )p(\mu )\\&=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96e309ead00fbc8603eced5342aa5df534522d6a)
This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:
The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
With known mean
![{\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2528fe4774a93087d4adae570ef9ab84707f52)
The likelihood function from above, written in terms of the variance, is:
![{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc06aa31588bba03e4748f8f345f0638a75dc156)
where
Then:
![{\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/381c1b93f6dc76e2cdca9f3f1f77132dd51dc55f)
The above is also a scaled inverse chi-squared distribution where
or equivalently
Reparameterizing in terms of an inverse gamma distribution, the result is:
With unknown mean and unknown variance
From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Note that each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.
The priors are normally defined as follows:
The update equations can be derived, and look as follows:
The prior distributions are
![{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\Gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7afb8e3b63fb1526171840344b32458e55cf8b)
Therefore, the joint prior is
![{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f808161077baef3854dbfd90b870698d721090)
The likelihood function from the section above with known variance is:
![{\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d77342aadcb34c5d84418cecaefdb52842b6b7)
Writing it in terms of variance rather than precision, we get:
![{\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b915f070b522a1e9f419be05624c86c854ca14)
Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
![{\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {X} )&\propto p(\mu ,\sigma ^{2})\,p(\mathbf {X} \mid \mu ,\sigma ^{2})\\&\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&={\mathcal {N}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {IG}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad9489034d77d53c12c7ee6044f712cfdb77831)
In other words, the posterior distribution has the form of a product of a normal distribution over p(μ | σ2) times an inverse gamma distribution over p(σ2), with parameters that are the same as the update equations above.
Occurrence and applications
The occurrence of normal distribution in practical problems can be loosely classified into four categories:
Exactly normal distributions;
Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.
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:
Probability density function of a ground state in a quantum harmonic oscillator.
The position of a particle that experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is the dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation . If the initial location is given by a certain density function , 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 many 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 that 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 radiation 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
Histogram of sepal widths for Iris versicolor from Fisher's Iris flower data set, with superimposed best-fitting normal distribution.
Fitted cumulative normal distribution to October rainfalls, see distribution fitting
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 above Normality tests section.
In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including: Measures of size of living tissue (length, height, skin area, weight);[55] 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.
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 Benoit Mandelbrot have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.
Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.[56]
In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or 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, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem.[57] The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Produced normality
In regression analysis, lack of normality in residuals simply indicates that the model postulated is inadequate in accounting for the tendency in the data and needs to be augmented; in other words, normality in residuals can always be achieved given a properly constructed model.
Generating values from normal distribution
The bean machine, a device invented by 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.
In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ2) can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.
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. Wichura gives a fast algorithm for computing this function to 16 decimal places,[58] which is used by R to compute random variates of the normal distribution.
An easy to program approximate approach, that relies on the central limit theorem, is as follows: generate 12 uniform U(0,1) deviates, add them all up, 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).[59]
The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1). Then the two random variables X and Y
- Y
The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. 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 1, then the method starts over, otherwise the two quantities
The Ratio method[60] is a rejection method. The algorithm proceeds as follows: Generate two independent uniform deviates U and V; Compute X = √8/e (V − 0.5)/U; Optional: if X2 ≤ 5 − 4e1/4U then accept X and terminate algorithm; Optional: 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 two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved[61] so that the logarithm is rarely evaluated.
The ziggurat algorithm[62] 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), do exponentials and more uniform random numbers have to be employed.
Integer arithmetic can be used to sample from the standard normal distribution.[63] This method is exact in the sense that it satisfies the conditions of ideal approximation;[64] i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
There is also some investigation[65] 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.
Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting p=Φ(z), the simplest approximation for the quantile function is:
![{\displaystyle z=\Phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2df7f1427d0c90d075faef38f4f5ab7acce5c9)
This approximation delivers for z a maximum absolute error of 0.026 (for 0.5 ≤ p ≤ 0.9999, corresponding to 0 ≤ z ≤ 3.719). For p < 1/2 replace p by 1 − p and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
![{\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1edea9f990058f741db6735799c8b40999b833b)
The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
![{\displaystyle {\begin{aligned}L(z)&=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\Phi (u)]\,du\\[5pt]L(z)&\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&p<1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\L(z)&\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&p<1/2,\\\\0.4115{\dfrac {1-p}{p}},&p\geq 1/2.\end{cases}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b69fa586cffdfbbd40a94c65629726e4ae78bf)
This approximation is particularly accurate for the right far-tail (maximum error of 10−3 for z≥1.4). Highly accurate approximations for the CDF, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).
History
Development
Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.
Marquis de Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.
In 1809 Gauss published his monograph "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, … to denote the measurements of some unknown quantity V, and sought the "most probable" estimator of that quantity: the one that 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.[4] Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:[70]
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.[71]
Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.[5] It was Laplace who first posed the problem of aggregating several observations in 1774,[72] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e−t2 dt = √π in 1782, providing the normalization constant for the normal distribution.[73] Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.[74]
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:[77] "The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is
Naming
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. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual".[78] However, by the end of the 19th century some authors[6] had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus "normal". Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."[79] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.[80]
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)
Also, 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:
The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 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".[81]
When the name is used, the "Gaussian distribution" was named after Carl Friedrich Gauss, who introduced the distribution in 1809 as a way of rationalizing the method of least squares as outlined above. Among English speakers, both "normal distribution" and "Gaussian distribution" are in common use, with different terms preferred by different communities.
See also
Wrapped normal distribution — the Normal distribution applied to a circular domain
Bates distribution — similar to the Irwin–Hall distribution, but rescaled back into the 0 to 1 range
Behrens–Fisher problem — the long-standing problem of testing whether two normal samples with different variances have same means;
Bhattacharyya distance – method used to separate mixtures of normal distributions
Erdős–Kac theorem—on the occurrence of the normal distribution in number theory
Gaussian blur—convolution, which uses the normal distribution as a kernel
Normally distributed and uncorrelated does not imply independent
Standard normal table
Sub-Gaussian distribution
Sum of normally distributed random variables
Tweedie distribution — The normal distribution is a member of the family of Tweedie exponential dispersion models
Z-test— using the normal distribution
Stein's lemma