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Wikipedia / Normal distribution

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued 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 is the variance. The standard deviation of the distribution is the positive value (sigma). 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 real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that the average of many statistically independent 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.

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of independent 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.

However, normal distributions are frequently misused in contexts where the assumption that the data are normally distributed is not met and the normal distribution is a poor model.

A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). (For other names, see Naming.)

The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.

Definitions

Standard normal distribution

The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution, and it is described by this probability density function (or density): The variable and inflection points and .

Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss which has a variance of , and Stephen Stigler

General normal distribution

If is a standard normal deviate will have a normal distribution with expected value and standard deviation . This is equivalent to saying that the standard normal distribution can be scaled/stretched by a factor of and shifted by to yield a different normal distribution, called .

Conversely, if is a normal deviate with parameters , then this to convert it to the standard normal distribution. This variate is also called the standardized form of .

In particular, the probability density function for can be written in terms of the standard normal distribution so that the integral is still1.

Notation

The probability density function of the standard normal distribution is commonly denoted by the Greek letter phi, . The variant form is also used.

The cumulative distribution function of the standard normal distribution is commonly denoted by the capital Greek letter phi, .

or . When a random variable is normally distributed with mean and standard deviation , one may write

Alternative parameterizations

Some authors advocate using the precision as the parameter defining the width of the distribution, instead of the standard deviation or the variance . The precision is normally defined as the reciprocal of the variance, .

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.

might be defined as the precision

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.

Normal distributions form an exponential family with natural parameters, and natural statistics and . The dual expectation parameters for normal distribution are and .

Cumulative distribution function

The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter

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:

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.

For a generic normal distribution with density , mean

The probability that lies between and with is therefore

, is often called the Q-function, especially in engineering texts. 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.

The graph of the standard normal cumulative distribution function has 2-fold rotational symmetry around the point (0,1/2); that is, . Its antiderivative

An asymptotic expansion of the cumulative distribution function for large can be derived using integration by parts denotes the double factorial. For more, see .

Taylor series representation

The Taylor series for the normal distribution can be derived by substituting into the Taylor series for the exponential function:

This series can be integrated term by term to obtain the Taylor series for the cumulative distribution function:

However, this series is ineffective for calculation due to slow convergence, except when is small.

Both of these series describe entire functions, which converge for all real and complex values of .

Recursive computation with Taylor series

The recurrence relation for Hermite polynomials may be used to efficiently construct the Taylor series expansion about any point

Standard deviation and coverage

About 68% of values drawn from a normal distribution are within one standard deviation from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.

are:For large

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 The quantile of the standard normal distribution is commonly denoted as . These values are used in hypothesis testing, construction of confidence intervals and Q–Q plots. A normal random variable with probability is 1.96 in only 5% of cases.

such that 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. as defined above.For small , the quantile function has the useful asymptotic expansion

Using root finding to compute the quantile function

can be used with Newton's method (or another root-finding algorithm such as Halley's method) to find the value of for which for some desired quantile . For example, starting with an initial, approximately correct guess , increasingly better approximations ,

Properties

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. 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.

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 of stock. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.

The value of the normal density 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 heavy-tailed distribution should be assumed and 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

(mean ) has the following properties:

Furthermore, the density ) also has the following properties:

  • More generally, its is the th (probabilist) Hermite polynomial.

  • The probability that a normally distributed variable with known has a standard normal distribution.

Moments

The plain and absolute moments of a variable , respectively. If the expected value of is zero, these parameters are called central moments; otherwise, these parameters are called non-central moments. Usually we are interested only in moments with integer order .

If has a normal distribution, the non-central moments exist and are finite for any whose real part is greater than−1. For any non-negative integer , the plain central moments are: denotes the double factorial, that is, the product of all numbers from to

the plain and absolute moments can be expressed in terms of confluent hypergeometric functions

These expressions remain valid even when is not an integer. See also generalized Hermite polynomials.The expectation of conditioned on the event that where and respectively are the density and the cumulative distribution function of this is known as the inverse Mills ratio. Note that above, density of instead of .

Fourier transform and characteristic function

The Fourier transform of a normal density with mean is

where is the imaginary unit, 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 variance . 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 real-valued random variable is closely connected to the characteristic function of that variable, which is defined as the expected value, as a function of the real variable (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable .

, which also could be seen as consequences of the Isserlis's theorem.

Moment- and cumulant-generating functions

The moment generating function of a real random variable , as a function of the real parameter . For a normal distribution with density , mean , the moment generating function exists and is equal to

For any , the coefficient of in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value .

The cumulant generating function

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in , only the first two cumulants are nonzero, namely the mean and the variance.

Some authors prefer to instead work with the characteristic function and .

Stein operator and class

Within Stein's method the class of all absolutely continuous functions such that .

Zero-variance limit

In the limit, while its integral remains equal to 1. An extension of the normal distribution to the case with zero variance can be defined using the Dirac delta measure, although the resulting random variables are not absolutely continuous and thus do not have probability density functions. The cumulative distribution function of such a random variable is then the Heaviside step function

Maximum entropy

Of all probability distributions over the reals with a specified finite mean and finite variance is the one with maximum entropy. To see this, let be a continuous random variable with probability density . The entropy of is defined as 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 mean and variance, by using variational calculus. A function with three Lagrange multipliers

about

Since this must hold for any small , the factor multiplying must be zero, and solving for

The Lagrange constraints that is properly normalized and has the specified mean and variance are satisfied if and only if , , and which is independent of the mean .

Other properties

Related distributions

Central limit theorem

are independent and identically distributed and Then, as increases, the probability distribution of will tend to the normal distribution with zero mean and variance .

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.

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

Operations and functions of normal variables

Operations on a single normal variable

If is distributed normally with mean , then

Operations on two independent normal variables

  • are two independent will also be normally distributed,[proof].

  • In particular, if and . This is a special case of the polarization identity.

  • are two independent normal deviates with mean , and , is also normally distributed with mean . It follows that the normal distribution is stable).

  • are normal distributions, then their normalized geometric mean.

Operations on two independent standard normal variables

are two independent standard normal random variables with mean 0 and variance 1, then

Operations on multiple independent normal variables

Operations on multiple correlated normal 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.

Infinite divisibility and Cramér's theorem

For any positive integer , any normal distribution with mean is the distribution of the sum of . This property is called infinite divisibility.

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.

The Kac–Bernstein theorem

The Kac–Bernstein theorem and are also independent, then both and must necessarily have normal distributions.

.

Extensions

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.

  • The multivariate normal distribution describes the Gaussian law in the -dimensional Euclidean space. A vector is multivariate-normally distributed if any linear combination of its components has a (univariate) normal distribution. The variance of is a symmetric positive-definite matrix . The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the case are ellipses and in the case of arbitrary are ellipsoids.

  • Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0.

  • Complex normal distribution deals with the complex normal vectors. A complex vector is said to be normal if both its real and imaginary components jointly possess a -dimensional multivariate normal distribution. The variance-covariance structure of is described by two matrices: the matrix , and the matrix .

  • Matrix normal distribution describes the case of normally distributed matrices.

Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space , and thus are the analogues of multivariate normal vectors for the case . A random element is said to be normal if for any constant the scalar product has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator . Several Gaussian processes became popular enough to have their own names:

A random variable where is the mean and and are the variances 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

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

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

  • The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inference

Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate population we would like to learn the approximate values of parameters . The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function: Taking derivatives with respect to and solving the resulting system of first order conditions yields the maximum likelihood estimates

Sample mean

is called the sample mean is complete and sufficient for , and therefore by the Lehmann–Scheffé theorem is the uniformly minimum variance unbiased (UMVU) estimator. The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix. This implies that the estimator is finite-sample efficient. Of practical importance is the standard error, 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 . The estimator is also asymptotically normal

Sample variance

is called the sample variance, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root is called the sample standard deviation by having instead of in the denominator (the so-called Bessel's correction becomes negligibly small for large is that it is an unbiased estimator is uniformly minimum variance unbiased (UMVU), in terms of the mean squared error have scaled chi-squared distribution with , which is slightly greater than the σσ does not exist.

.

Confidence intervals

By Cochran's theorem and the sample variance are independent, which means there can be no gain in considering their joint distribution and can be employed to construct the so-called t-statistic This quantity has the Student's t-distribution with degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this -statistics will allow us to construct the confidence interval for ; similarly, inverting the distribution of the statistic will give us the confidence interval for : where and are the th quantiles of the - and -distributions respectively. These confidence intervals are of the confidence level , meaning that the true values and fall outside of these intervals with probability (or significance level) . In practice people usually take , resulting in the 95% confidence intervals. The confidence interval for can be found by taking the square root of the interval bounds for .

and The approximate formulas become valid for large values of , and are more convenient for the manual calculation since the standard normal quantiles do not depend on . In particular, the most popular value of , results in .

Normality tests

Normality tests assess the likelihood that the given data set comes from a normal distribution. Typically the null hypothesis is that the observations are distributed normally with unspecified mean and variance , versus the alternative that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

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.

  • Q–Q plot, also known as normal probability plot or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form , where plotting points are equal to and is an adjustment constant, which can be anything between 0 and1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.

  • P–P plot – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points . For normally distributed data this plot should lie on a straight line between and.

Goodness-of-fit tests:

Moment-based tests:

Tests based on the empirical distribution function:

Bayesian analysis of the normal distribution

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:

  • Either the mean, or the variance, or neither, may be considered a fixed quantity.

  • When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.

  • Both univariate and multivariate cases need to be considered.

  • Either conjugate or improper prior distributions may be placed on the unknown variables.

  • An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior article.

Sum of two quadratics

Scalar form

The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

This equation rewrites the sum of two quadratics in by expanding the squares, grouping the terms in , and completing the square. Note the following about the complex constant factors attached to some of the terms:

  1. has the form of a weighted average of and .

  2. This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities and add directly, so to combine and themselves, it is 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 is one-half the harmonic mean of and .

Vector form

A similar formula can be written for the sum of two vector quadratics: If , , are vectors of length , and and are symmetric, invertible matrices, then

The form 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 matters for any off-diagonal elements of , and there is no loss of generality in assuming that is symmetric. Furthermore, if

Sum of differences from the mean

With known variance

For a set of i.i.d. normally distributed data points of size where each individual point with known variance , the conjugate prior distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using we proceed as follows.

First, the likelihood function

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving. The result is the kernel

That is, to combine data points with total precision of (or equivalently, total variance of , 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 precision-weighted 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

With known mean

For a set of i.i.d. normally distributed data points of size where each individual point with known mean , the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for

The likelihood function

Reparameterizing in terms of an inverse gamma distribution

With unknown mean and unknown variance

For a set of i.i.d. normally distributed data points of size where each individual point with unknown mean and unknown variance , a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution. Logically, this originates as follows:

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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. 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.

  6. 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.

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.

Occurrence and applications

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

  1. Exactly normal distributions;

  2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and

  3. Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.

  4. Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normality

A normal distribution occurs in some physical theories:

Approximate normality

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

In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as

Assumed normality

There are statistical methods to empirically test that assumption; see the above Normality tests section.

In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including:

  • Measures of size of living tissue (length, height, skin area, weight);

  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;

  • Certain physiological measurements, such as blood pressure of adult humans.

  • In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot have argued that log-Levy distributions, which possess heavy tails, would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.

  • Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.

  • In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

  • Many scores are derived from the normal distribution, including percentile ranks (percentiles or quantiles), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.

  • In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem. The plot on the right illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer review

John Ioannidis argued 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 falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if they were 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.

Computational methods

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 can be generated as , where is standard normal. All these algorithms rely on the availability of a random number generator capable of producing uniform random variates.

  • The most straightforward method is based on the probability integral transform property: if is distributed uniformly on (0,1), then will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ, which cannot be done analytically. Some approximate methods are described in and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places, which is used by R to compute random variates of the normal distribution.

  • An easy-to-program approximate approach that relies on the central limit theorem is as follows: generate 12 uniform deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of . Note that in a true normal distribution, only 0.00034% of all samples will fall outside.

  • The Box–Muller method uses two independent random numbers and distributed uniformly on (0,1). Then the two random variables and will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector the squared norm will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable .

  • The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, and are drawn from the uniform (−1,1) distribution, and then is computed. If are returned. Again, and are independent, standard normal random variables.

The Ratio method is a rejection method. The algorithm proceeds as follows:

  • Generate two independent uniform deviates and ;

  • Compute ;

  • Optional: if then accept and terminate algorithm;

  • Optional: if then reject and start over from step 1;

  • If then accept , otherwise start over the algorithm.

  • The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved so that the logarithm is rarely evaluated.

  • The ziggurat algorithm is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.

  • Integer arithmetic can be used to sample from the standard normal distribution. This method is exact in the sense that it satisfies the conditions of ideal approximation; i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.

  • There is also some investigation into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile function

The standard normal cumulative distribution function is widely used in scientific and statistical computing.

The values 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.

  • give the approximation for for with the absolute error (algorithm 26.2.17 where is the standard normal probability density function, and , , , , , .

  • lists dozens of approximations by means of rational functions, with or without exponentials, for the function, where erfc(x) = 1 - erf(x). His algorithms vary in the degree of complexity and the resulting precision, with a maximum absolute precision of 24 digits. An algorithm by combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with 16-digit precision.

  • , after recalling the Hart68 solution is not suited for erf, gave a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.

  • suggested a simple algorithm for calculating with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when ).

  • The GNU Scientific Library calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomials.

  • ,

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

This approximation delivers for a maximum absolute error of 0.026 (for , corresponding to ). For replace by

This approximation is particularly accurate for the right far-tail (maximum error of 10 for ). Highly accurate approximations for the cumulative distribution function, 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 cumulative distribution function as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

History

Development

Some authors attribute 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 , and that "If or be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ." 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.

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 , , to denote the measurements of some unknown quantity, and sought the most probable estimator of that quantity: the one that maximizes the probability 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 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 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 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−t2 dt in 1782, providing the normalization constant for the normal distribution. For this accomplishment, Gauss acknowledged the priority of Laplace. 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 Irish-American mathematician Robert Adrain published two insightful but flawed 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 exhumed 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 and

Naming

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

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 this distribution being 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.

Also, it was Pearson who first wrote the distribution in terms of the standard deviation as in modern notation. Soon after this, in year 1915, Fisher

The term standard normal distribution, 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 Alexander M. Mood (1950) Introduction to the Theory of Statistics.

See also

Notes

References

Citations

Sources