Normal distribution

In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

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.[2][3] 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.[4]

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.[5] However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

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.

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 CDF of the standard normal distribution can be expanded by Integration by parts into a series:

An asymptotic expansion of the CDF for large x can also be derived using integration by parts. For more, see Error function#Asymptotic expansion.[13]

A quick approximation to the standard normal distribution's CDF can be found by using a taylor series approximation:

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.[5] This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.

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:

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.[17][18] 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.[19][20]

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.

The cumulant generating function is the logarithm of the moment generating function, namely

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

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.

This theorem can also be used to justify modeling the sum of many uniform noise sources as gaussian noise. See AWGN.

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing[39] (). In the following sections we look at some special cases.

Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.

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.

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.[32]

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.

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[49]

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:

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;[51] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[52]

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.

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.

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:

First, the likelihood function is (using the formula above for the sum of differences from the mean):

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

The likelihood function from above, written in terms of the variance, is:

Reparameterizing in terms of an inverse gamma distribution, the result is:

The likelihood function from the section above with known variance is:

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 | σ2) times an inverse gamma distribution over p2), 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.

Fitted cumulative normal distribution to October rainfalls, see distribution fitting

John Ioannidis argues that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of faslsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as "validated" by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.[56]

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 and normal quantile function

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 single-parameter 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 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).

In 1823 Gauss published his monograph "Theoria combinationis observationum erroribus minimis obnoxiae" 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 density function 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.[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:[69]

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.[70]

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,[71] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral et2 dt = π in 1782, providing the normalization constant for the normal distribution.[72] 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.[73]

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.[74] His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.[75]

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:[76] "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".[77] 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."[78] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.[79]

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".[80]