In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a realvalued random variable. The general form of its probability density function is
The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation.^{ [1] } The variance of the distribution is .^{ [2] } A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Normal distributions are important in statistics and are often used in the natural and social sciences to represent realvalued random variables whose distributions are not known.^{ [3] }^{ [4] } Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.^{ [5] }
Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. 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.
A normal distribution is sometimes informally called a bell curve.^{ [6] } However, many other distributions are bellshaped (such as the Cauchy, Student's t, and logistic distributions).
The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when and , and it is described by this probability density function:^{ [1] }
Here, the factor ensures that the total area under the curve is equal to one.^{ [note 1] } The factor in the exponent ensures that the distribution has unit variance (i.e., variance being equal to one), and therefore also unit standard deviation. This function is symmetric around , where it attains its maximum value and has inflection points at and .
Authors differ on which normal distribution should be called the "standard" one. Carl Friedrich Gauss, for example, defined the standard normal as having a variance of . That is:
On the other hand, Stephen Stigler ^{ [7] } goes even further, defining the standard normal as having a variance of :
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor (the standard deviation) and then translated by (the mean value):
The probability density must be scaled by so that the integral is still 1.
If is a standard normal deviate, then will have a normal distribution with expected value and standard deviation . Conversely, if is a normal deviate with parameters and , then the distribution will have a standard normal distribution. This variate is also called the standardized form of .
The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter (phi).^{ [8] } The alternative form of the Greek letter phi, , is also used quite often.^{ [1] }
The normal distribution is often referred to as or .^{ [1] }^{ [9] } Thus when a random variable is normally distributed with mean and variance , one may write
Some authors advocate using the precision as the parameter defining the width of the distribution, instead of the deviation or the variance . The precision is normally defined as the reciprocal of the variance, .^{ [10] } The formula for the distribution then becomes
This choice is claimed to have advantages in numerical computations when is very close to zero, and simplifies formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.
Alternatively, the reciprocal of the standard deviation might be defined as the precision, in which case the expression of the normal distribution becomes
According to Stigler, this formulation is advantageous because of a much simpler and easiertoremember formula, and simple approximate formulas for the quantiles of the distribution.
Normal distributions form an exponential family with natural parameters and , and natural statistics x and x^{2}. The dual expectation parameters for normal distribution are η_{1} = μ and η_{2} = μ^{2} + σ^{2}.
The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter (phi),^{ [1] } is the integral
The related error function gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range . That is:^{ [1] }
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 for more.
The two functions are closely related, namely
For a generic normal distribution with density , mean and deviation , the cumulative distribution function is
The complement of the standard normal CDF, , is often called the Qfunction, especially in engineering texts.^{ [11] }^{ [12] } It gives the probability that the value of a standard normal random variable will exceed : . Other definitions of the function, all of which are simple transformations of , are also used occasionally.^{ [13] }
The graph of the standard normal CDF has 2fold rotational symmetry around the point (0,1/2); that is, . Its antiderivative (indefinite integral) can be expressed as follows:
The CDF of the standard normal distribution can be expanded by Integration by parts into a series:
where denotes the double factorial.
An asymptotic expansion of the CDF for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion.^{ [14] }
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.^{ [6] } This fact is known as the 689599.7 (empirical) rule, or the 3sigma rule.
More precisely, the probability that a normal deviate lies in the range between and is given by
To 12 significant figures, the values for are:^{ [15] }
OEIS  

1  0.682689492137  0.317310507863 
 OEIS: A178647  
2  0.954499736104  0.045500263896 
 OEIS: A110894  
3  0.997300203937  0.002699796063 
 OEIS: A270712  
4  0.999936657516  0.000063342484 
 
5  0.999999426697  0.000000573303 
 
6  0.999999998027  0.000000001973 

For large , one can use the approximation .
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:
For a normal random variable with mean and variance , the quantile function is
The quantile of the standard normal distribution is commonly denoted as . These values are used in hypothesis testing, construction of confidence intervals and QQ plots. A normal random variable will exceed with probability , and will lie outside the interval with probability . In particular, the quantile is 1.96; therefore a normal random variable will lie outside the interval in only 5% of cases.
The following table gives the quantile such that will lie in the range with a specified probability . These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:.^{ [16] }^{ [17] } NOTE: the following table shows , not as defined above.
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 
For small , the quantile function has the useful asymptotic expansion
The normal distribution is the only 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.^{ [18] }^{ [19] } 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.^{ [20] }^{ [21] }
The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is nonzero 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 lognormal distribution or the Pareto distribution.
The value of the normal distribution is practically zero when the value lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavytailed distribution should be assumed and the appropriate robust statistical inference methods applied.
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.
The normal distribution with density (mean and standard deviation ) has the following properties:
Furthermore, the density of the standard normal distribution (i.e. and ) also has the following properties:
The plain and absolute moments of a variable are the expected values of and , respectively. If the expected value of is zero, these parameters are called central moments. Usually we are interested only in moments with integer order .
If has a normal distribution, these moments exist and are finite for any whose real part is greater than −1. For any nonnegative integer , the plain central moments are:^{ [25] }
Here denotes the double factorial, that is, the product of all numbers from to 1 that have the same parity as
The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any nonnegative integer
The last formula is valid also for any noninteger When the mean the plain and absolute moments can be expressed in terms of confluent hypergeometric functions and ^{[ citation needed ]}
These expressions remain valid even if is not an integer. See also generalized Hermite polynomials.
Order  Noncentral moment  Central moment 

1  
2  
3  
4  
5  
6  
7  
8 
The expectation of conditioned on the event that lies in an interval is given by
where and respectively are the density and the cumulative distribution function of . For this is known as the inverse Mills ratio. Note that above, density of is used instead of standard normal density as in inverse Mills ratio, so here we have instead of .
The Fourier transform of a normal density with mean and standard deviation is^{ [26] }
where is the imaginary unit. If the mean , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and standard deviation . In particular, the standard normal distribution is an eigenfunction of the Fourier transform.
In probability theory, the Fourier transform of the probability distribution of a realvalued random variable is closely connected to the characteristic function of that variable, which is defined as the expected value of , as a function of the real variable (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complexvalue variable .^{ [27] } The relation between both is:
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density , mean and deviation , the moment generating function exists and is equal to
The cumulant generating function is the logarithm of the moment generating function, namely
Since this is a quadratic polynomial in , only the first two cumulants are nonzero, namely the mean and the variance .
Within Stein's method the Stein operator and class of a random variable are and the class of all absolutely continuous functions .
In the limit when tends to zero, the probability density eventually tends to zero at any , but grows without limit if , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when .
However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function" translated by the mean , that is Its CDF is then the Heaviside step function translated by the mean , namely
Of all probability distributions over the reals with a specified mean and variance , the normal distribution is the one with maximum entropy.^{ [28] } If is a continuous random variable with probability density , then the entropy of is defined as^{ [29] }^{ [30] }^{ [31] }
where is understood to be zero whenever . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified variance, by using variational calculus. A function with two Lagrange multipliers is defined:
where is, for now, regarded as some density function with mean and standard deviation .
At maximum entropy, a small variation about will produce a variation about which is equal to 0:
Since this must hold for any small , the term in brackets must be zero, and solving for yields:
Using the constraint equations to solve for and yields the density of the normal distribution:
The entropy of a normal distribution is equal to
The family of normal distributions is closed under linear transformations: if is normally distributed with mean and standard deviation , then the variable , for any real numbers and , is also normally distributed, with mean and standard deviation .
Also if and are two independent normal random variables, with means , and standard deviations , , then their sum will also be normally distributed,^{ [proof] } with mean and variance .
In particular, if and are independent normal deviates with zero mean and variance , then and are also independent and normally distributed, with zero mean and variance . This is a special case of the polarization identity.^{ [32] }
Also, if , are two independent normal deviates with mean and deviation , and , are arbitrary real numbers, then the variable
is also normally distributed with mean and deviation . It follows that the normal distribution is stable (with exponent ).
More generally, any linear combination of independent normal deviates is a normal deviate.
For any positive integer , any normal distribution with mean and variance is the distribution of the sum of independent normal deviates, each with mean and variance . This property is called infinite divisibility.^{ [33] }
Conversely, if and are independent random variables and their sum has a normal distribution, then both and must be normal deviates.^{ [34] }
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 nonGaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.^{ [35] }
Bernstein's theorem states that if and are independent and and are also independent, then both X and Y must necessarily have normal distributions.^{ [36] }^{ [37] }
More generally, if are independent random variables, then two distinct linear combinations and will be independent if and only if all are normal and , where denotes the variance of .^{ [36] }
The Hellinger distance between the same distributions is equal to
The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance and is their mean scaled by
Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .
The theorem can be extended to variables that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.
Many 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:
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.
If X is distributed normally with mean μ and variance σ^{2}, then
If and are two independent standard normal random variables with mean 0 and variance 1, then
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.
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 onedimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
A random variable X has a twopiece normal distribution if it has a distribution
where μ is the mean and σ_{1} and σ_{2} 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^{ [48] }
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:
It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample from a normal population we would like to learn the approximate values of parameters and . The standard approach to this problem is the maximum likelihood method, which requires maximization of the loglikelihood function:
Taking derivatives with respect to and and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Estimator is called the sample mean , since it is the arithmetic mean of all observations. The statistic is complete and sufficient for , and therefore by the Lehmann–Scheffé theorem, is the uniformly minimum variance unbiased (UMVU) estimator.^{ [49] } In finite samples it is distributed normally:
The variance of this estimator is equal to the μμelement of the inverse Fisher information matrix . This implies that the estimator is finitesample efficient. Of practical importance is the fact that the standard error of is proportional to , 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, is consistent, that is, it converges in probability to as . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
The estimator is called the sample variance , since it is the variance of the sample (). In practice, another estimator is often used instead of the . This other estimator is denoted , and is also called the sample variance, which represents a certain ambiguity in terminology; its square root is called the sample standard deviation. The estimator differs from by having (n − 1) instead of n in the denominator (the socalled Bessel's correction):
The difference between and becomes negligibly small for large n's. In finite samples however, the motivation behind the use of is that it is an unbiased estimator of the underlying parameter , whereas is biased. Also, by the Lehmann–Scheffé theorem the estimator is uniformly minimum variance unbiased (UMVU),^{ [49] } which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator is "better" than the in terms of the mean squared error (MSE) criterion. In finite samples both and have scaled chisquared distribution with (n − 1) degrees of freedom:
The first of these expressions shows that the variance of is equal to , which is slightly greater than the σσelement of the inverse Fisher information matrix . Thus, is not an efficient estimator for , and moreover, since is UMVU, we can conclude that the finitesample efficient estimator for does not exist.
Applying the asymptotic theory, both estimators and are consistent, that is they converge in probability to as the sample size . The two estimators are also both asymptotically normal:
In particular, both estimators are asymptotically efficient for .
By Cochran's theorem, for normal distributions the sample mean and the sample variance s^{2} are independent, which means there can be no gain in considering their joint distribution. There is also a converse 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 and s can be employed to construct the socalled tstatistic:
This quantity t has the Student's tdistribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this tstatistics will allow us to construct the confidence interval for μ;^{ [50] } similarly, inverting the χ^{2} distribution of the statistic s^{2} will give us the confidence interval for σ^{2}:^{ [51] }
where t_{k,p} and χ 2
k,p are the pth quantiles of the t and χ^{2}distributions respectively. These confidence intervals are of the confidence level 1 − α, meaning that the true values μ and σ^{2} fall outside of these intervals with probability (or significance level) 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 and s^{2}. 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 z_{0.025} = 1.96 .
Normality tests assess the likelihood that the given data set {x_{1}, ..., x_{n}} comes from a normal distribution. Typically the null hypothesis H_{0} is that the observations are distributed normally with unspecified mean μ and variance σ^{2}, versus the alternative H_{a} that the distribution is arbitrary. Many tests (over 40) have been devised for this problem, the more prominent of them are outlined below:
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
The formulas for the nonlinearregression cases are summarized in the conjugate prior article.
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:
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size , then
where
Note that the form x′ Ax is called a quadratic form and is a scalar:
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since , only the sum matters for any offdiagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form
Another useful formula is as follows:
where
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with known variance σ^{2}, the conjugate prior distribution is also normally distributed.
This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ^{2}. Then if and we proceed as follows.
First, the likelihood function is (using the formula above for the sum of differences from the mean):
Then, we proceed as follows:
In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean and precision , i.e.
This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:
That is, to combine n data points with total precision of nτ (or equivalently, total variance of n/σ^{2}) and mean of values , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precisionweighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)
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 precisionweighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chisquared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chisquared for the sake of convenience. The prior for σ^{2} is as follows:
The likelihood function from above, written in terms of the variance, is:
where
Then:
The above is also a scaled inverse chisquared distribution where
or equivalently
Reparameterizing in terms of an inverse gamma distribution, the result is:
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with unknown mean μ and unknown variance σ^{2}, a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normalinversegamma distribution. Logically, this originates as follows:
The priors are normally defined as follows:
The update equations can be derived, and look as follows:
The respective numbers of pseudoobservations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new "interaction term" needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.
The prior distributions are
Therefore, the joint prior is
The likelihood function from the section above with known variance is:
Writing it in terms of variance rather than precision, we get:
where
Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
In other words, the posterior distribution has the form of a product of a normal distribution over p(μ  σ^{2}) times an inverse gamma distribution over p(σ^{2}), with parameters that are the same as the update equations above.
The occurrence of normal distribution in practical problems can be loosely classified into four categories:
Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:
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.
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.
There are statistical methods to empirically test that assumption, see the above Normality tests section.
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.^{[ citation needed ]}
In computer simulations, especially in applications of the MonteCarlo 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, σ^{2}
_{}) 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 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:
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 singleparameter approximation:
The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
This approximation is particularly accurate for the right fartail (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).
Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small relative error on the whole domain for the CDF and the quantile function as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.
Some authors^{ [63] }^{ [64] } attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738^{ [note 3] } published in the second edition of his " The Doctrine of Chances " the study of the coefficients in the binomial expansion of (a + b)^{n}. De Moivre proved that the middle term in this expansion has the approximate magnitude of , 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 ."^{ [65] } 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.^{ [66] }
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 wellknown answer: the arithmetic mean of the measured values.^{ [note 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:^{ [67] }
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 nonlinear weighted least squares (NWLS) method.^{ [68] }
Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.^{ [note 5] } It was Laplace who first posed the problem of aggregating several observations in 1774,^{ [69] } 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.^{ [70] } 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.^{ [71] }
It is of interest to note that in 1809 an Irish mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss.^{ [72] } His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.^{ [73] }
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:^{ [74] } "The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is
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".^{ [75] } However, by the end of the 19th century some authors^{ [note 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."^{ [76] } Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.^{ [77] }
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'.
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".^{ [78] }
In probability theory, a lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is lognormally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a lognormal distribution. A random variable which is lognormally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics.
In probability and statistics, Student's tdistribution is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym Student.
The scaled inverse chisquared distribution is the distribution for x = 1/s^{2}, where s^{2} is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ^{2} = τ^{2}. The distribution is therefore parametrised by the two quantities ν and τ^{2}, referred to as the number of chisquared degrees of freedom and the scaling parameter, respectively.
In probability theory, the Rice distribution or Rician distribution is the probability distribution of the magnitude of a circularlysymmetric bivariate normal random variable, possibly with nonzero mean (noncentral). It was named after Stephen O. Rice.
In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships.
The noncentral tdistribution generalizes Student's tdistribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral tdistribution is also known as the singly noncentral tdistribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.
In geometry, the elliptic coordinate system is a twodimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the axis of the Cartesian coordinate system.
Toroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the axis is the axis of rotation. The focal ring is also known as the reference circle.
Elliptic cylindrical coordinates are a threedimensional orthogonal coordinate system that results from projecting the twodimensional elliptic coordinate system in the perpendicular direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the axis of the Cartesian coordinate system.
Prolate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.
Oblate spheroidal coordinates are a threedimensional orthogonal coordinate system that results from rotating the twodimensional elliptic coordinate system about the nonfocal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the xy plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semiaxes are equal in length.
In probability theory, the inverse Gaussian distribution is a twoparameter family of continuous probability distributions with support on (0,∞).
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ^{2}, the random variable Y = X has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin.
In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the LeviCivita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above. The truncated normal distribution has wide applications in statistics and econometrics. For example, it is used to model the probabilities of the binary outcomes in the probit model and to model censored data in the Tobit model.
In probability theory and statistics, the halfnormal distribution is a special case of the folded normal distribution.
In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.
In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0. It is essentially a mixture of a discrete distribution and a continuous distribution as a result of censoring.
In probability theory, an exponentially modified Gaussian distribution describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ^{2}, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.
In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in nonparametric regression, semiparametric regression and functional data analysis. In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity. In a nonparametric setting, the variance function is assumed to be a smooth function.