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Notation N
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μ
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{displaystyle {mathcal {N}}(mu ,,sigma ^{2})} Parameters μ ∈ R — mean (location)
σ > 0 — variance (squared scale) PDF 1
2
σ
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π
e
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{displaystyle {rac {1}{sqrt {2sigma ^{2}pi }}},e^{-{rac {(x-mu )^{2}}{2sigma ^{2}}}}} CDF 1
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1
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erf
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{displaystyle {rac {1}{2}}left[1+operatorname {erf} left({rac {x-mu }{sigma {sqrt {2}}}}
ight)
ight]} Quantile μ
+
σ
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erf
−
1
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2
F
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{displaystyle mu +sigma {sqrt {2}}operatorname {erf} ^{-1}(2F-1)} |
In probability theory, the normal (or Gaussian) 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.
Contents
- Standard normal distribution
- General normal distribution
- Notation
- Alternative parameterizations
- Properties
- Symmetries and derivatives
- Differential equation
- Moments
- Fourier transform and characteristic function
- Moment and cumulant generating functions
- Cumulative distribution function
- Standard deviation and coverage
- Quantile function
- Zero variance limit
- Central limit theorem
- Maximum entropy
- Operations on normal deviates
- Infinite divisibility and Cramrs theorem
- Bernsteins theorem
- Operations on a single random variable
- Combination of two independent random variables
- Combination of two or more independent random variables
- Operations on the density function
- Extensions
- Normality tests
- Estimation of parameters
- Sample mean
- Sample variance
- Confidence intervals
- Bayesian analysis of the normal distribution
- Scalar form
- Vector form
- Sum of differences from the mean
- With known variance
- With known mean
- With unknown mean and unknown variance
- Occurrence and applications
- Exact normality
- Approximate normality
- Assumed normality
- Produced normality
- Generating values from normal distribution
- Numerical approximations for the normal CDF
- Development
- Naming
- References
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 random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of random variables 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. 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). Even the term Gaussian bell curve is ambiguous because it may be used to refer to a some function defined in terms of the Gaussian function which is not a probability distribution because it is not normalized in that it does not integrate to 1.
The probability density of the normal distribution is:
Where:
A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
Standard normal distribution
The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, and it is described by this probability density function:
The factor
Authors may differ also on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance
Stigler goes even further, defining the standard normal with variance
General normal distribution
Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ (the standard deviation) and then translated by μ (the mean value):
The probability density must be scaled by
If Z is a standard normal deviate, then X = Zσ + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a general normal deviate, then Z = (X − μ)/σ will have a standard normal distribution.
Every normal distribution is the exponential of a quadratic function:
where a is negative and c is
Notation
The standard Gaussian distribution (with zero mean and unit variance) is often denoted with the Greek letter ϕ (phi). The alternative form of the Greek phi letter, φ, is also used quite often.
The normal distribution is also often denoted by N(μ, σ2). Thus when a random variable X is distributed normally with mean μ and variance σ2, we write
Alternative parameterizations
Some authors advocate using the precision
This choice is claimed to have advantages in numerical computations when σ is very close to zero and simplify formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.
Also the reciprocal of the standard deviation
According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution.
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.
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 value of the normal distribution is practically zero when the value x lies more than a few standard deviations away from the mean. 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 heavy-tailed 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.
Symmetries and derivatives
The normal distribution f(x), with any mean μ and any positive deviation σ, has the following properties:
Furthermore, the density ϕ of the standard normal distribution (with μ = 0 and σ = 1) also has the following properties:
Differential equation
Moments
The plain and absolute moments of a variable X are the expected values of Xp and |X|p,respectively. If the expected value μ of X is zero, these parameters are called central moments. Usually we are interested only in moments with integer order p.
If X has a normal distribution, these moments exist and are finite for any p whose real part is greater than −1. For any non-negative integer p, the plain central moments are :
Here n!! denotes the double factorial, that is, the product of all numbers from n to 1 that have the same parity as n.
The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer p,
The last formula is valid also for any non-integer p > −1. When the mean μ is not zero, the plain and absolute moments can be expressed in terms of confluent hypergeometric functions 1F1 and U.
These expressions remain valid even if p is not integer. See also generalized Hermite polynomials.
The expectation of X conditioned on the event that X lies in an interval [a,b] is given by
where f(x) and F(x) respectively are the density and the cumulative distribution function of X. For b = ∞ this is known as the inverse Mills ratio. Note that above, density
Fourier transform and characteristic function
The Fourier transform of a normal distribution f with mean μ and deviation σ is
where i is the imaginary unit. If the mean μ is zero, the first factor is 1, and the Fourier transform is also a normal distribution on the frequency domain, with mean 0 and standard deviation 1/σ. In particular, the standard normal distribution ϕ (with μ = 0 and σ = 1) is an eigenfunction of the Fourier transform.
In probability theory, the Fourier transform of the probability distribution of a real-valued random variable X is called the characteristic function of that variable, and can be defined as the expected value of ei tX, as a function of the real variable t (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value parameter t.
Moment and cumulant generating functions
The moment generating function of a real random variable X is the expected value of etX, as a function of the real parameter t. For a normal distribution with mean μ and deviation σ, 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 t, only the first two cumulants are nonzero, namely the mean μ and the variance σ2.
Cumulative distribution function
The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter
In statistics one often uses the related error function, or erf(x), defined as the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range
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
For a generic normal distribution f with mean μ and deviation σ, the cumulative distribution function is
The complement of the standard normal CDF,
The graph of the standard normal CDF
where
Standard deviation and coverage
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.
More precisely, the probability that a normal deviate lies in the range μ − nσ and μ + nσ is given by
To 12 significant figures, the values for n = 1, 2, …, 6 are:
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:
For a normal random variable with mean μ and variance σ2, the quantile function is
The quantile
The following table gives the multiple n of σ such that X will lie in the range μ ± nσ with a specified probability p. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:
Zero-variance limit
In the limit when σ tends to zero, the probability density f(x) eventually tends to zero at any x ≠ μ, but grows without limit if x = μ, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when σ = 0.
However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function" δ translated by the mean μ, that is f(x) = δ(x−μ). Its CDF is then the Heaviside step function translated by the mean μ, namely
Central limit theorem
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 X1, …, Xn are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance σ2; and Z is their mean scaled by
Then, as n increases, the probability distribution of Z will tend to the normal distribution with zero mean and variance σ2.
The theorem can be extended to variables Xi 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.
Maximum entropy
Of all probability distributions over the reals with a specified mean μ and variance σ2, the normal distribution N(μ, σ2) is the one with maximum entropy. If X is a continuous random variable with probability density f(x), then the entropy of X is defined as
where f(x) log f(x) is understood to be zero whenever f(x) = 0. 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 f(x) is, for now, regarded as some function with mean μ and standard deviation
Since this must hold for any small δf(x), the term in brackets must be zero, and solving for f(x) yields:
Using the constraint equations to solve for λ0 and λ yields the normal distribution:
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.
Also if X1 and X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their sum X1 + X2 will also be normally distributed,[proof] with mean μ1 + μ2 and variance
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.
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.
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.
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.
Bernstein's theorem
Bernstein's theorem states that if X and Y are independent and X + Y and X − Y are also independent, then both X and Y must necessarily have normal distributions.
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 2
k = 0, where σ 2
k denotes the variance of Xk.
Operations on a single random variable
If X is distributed normally with mean μ and variance σ2, then
Combination of two independent random variables
If X1 and X2 are two independent standard normal random variables with mean 0 and variance 1, then
Combination of two or more independent random variables
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.
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.
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
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:
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:
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
Estimator
The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix
From the standpoint of the asymptotic theory,
Sample variance
The estimator
The difference between s2 and
The first of these expressions shows that the variance of s2 is equal to 2σ4/(n−1), which is slightly greater than the σσ-element of the inverse Fisher information matrix
Applying the asymptotic theory, both estimators s2 and
In particular, both estimators are asymptotically efficient for σ2.
Confidence intervals
By Cochran's theorem, for normal distributions the sample mean
This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ; similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:
where tk,p and χ 2
k,p are the pth quantiles of the t- and χ2-distributions respectively. These confidence intervals are of the confidence level 1 − α, meaning that the true values μ and σ2 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
Bayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
The formulas for the non-linear-regression cases are summarized in the conjugate prior article.
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
a y + b z a + b -
a b a + b = 1 1 a + 1 b = ( a − 1 + b − 1 ) − 1 . 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 thata b a + b
Vector form
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
where
Note that the form x′ A x 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
Sum of differences from the mean
Another useful formula is as follows:
where
With known variance
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows
This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ2. Then if
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
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/σ2) and mean of values
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
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows
The likelihood function from above, written in terms of the variance, is:
where
Then:
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
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows
- 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 respective numbers of pseudo-observations 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
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
Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:
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.
Assumed normality
I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.
There are statistical methods to empirically test that assumption, see the above Normality tests section.
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
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.
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.
Some more approximations can be found at: Error function#Approximation with elementary functions.
Development
Some authors attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 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
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: 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. 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:
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.
Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions. It was Laplace who first posed the problem of aggregating several observations in 1774, although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e−t ²dt = √π in 1782, providing the normalization constant for the normal distribution. Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.
It is of interest to note that in 1809 an American mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss. His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.
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: "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". However, by the end of the 19th century some authors 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." Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.
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".
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.