Probability density function of Gaussian distribution (bell curve).

The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. It is actually a family of distributions of the same general form, differing only in their location and scale parameters: the mean and standard deviation. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. Because the graph of its probability density resembles a bell, it is often called the bell curve.

History

The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the Theorem of de Moivre-Laplace.

Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors.

The name "bell curve" goes back to Jouffret who used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875. This terminology is unfortunate, since it reflects and encourages the fallacy that many or all probability distributions are "normal". (See the discussion of "occurrence" below.)

That the distribution is called the normal or Gaussian distribution is an instance of Stigler's law of eponymy: "No scientific discovery is named after its original discoverer."

Specification of the normal distribution

There are various ways to specify a random variable. The most visual is the probability density function (plot at the top), which represents how likely each value of the random variable is. The cumulative density function is a conceptually cleaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the moments, the cumulants, the characteristic function, the moment-generating function, and the cumulant-generating function. Some of these are very useful for theoretical work, but not intuitive. See probability distribution for a discussion.

All of the cumulants of the normal distribution are zero, except the first two.

Probability density function

The probability density function of the normal distribution with mean μ and standard deviation σ (equivalently, variance σ²) is an example of a Gaussian function,

<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu )^2 / 2\sigma^2}}.<math>

(See also exponential function and pi.) If a random variable X has this distribution, we write X ~ N(μ, σ²). If μ = 0 and σ = 1, the distribution is called the standard normal distribution, with formula

<math>f(x) = {1 \over \sqrt{2\pi} }\,e^{-{x^2 / 2}}.<math>

The picture at the top of this article gives the graph of the probability density function of the normal distribution with μ = 0 and several values of σ.

For all normal distributions, the density function is symmetric about its mean value. About 68% of the area under the curve is within one standard deviation of the mean, 95.5% within two standard deviations, and 99.7% within three standard deviations. The inflection points of the curve occur at one standard deviation away from the mean.

Cumulative distribution function

Plot of the cumulative distribution for values of z from −4 to +4. One can see that the probability of a standard normal variable taking on a value less than 0.25 is approximately equal to 0.60.

The cumulative distribution function (hereafter cdf) is defined as the probability that a variable X has a value less than or equal to x, and it is expressed in terms of the density function as

<math>\Pr(X \le x) = \int_{-\infty}^x \frac{1}{\sigma\sqrt{2\pi}} e^{-(u-\mu)^2/(2\sigma^2)}\,du.<math>

The standard normal cdf, conventionally denoted <math>\Phi<math>, is just the general cdf evaluated with <math>\mu=0<math> and <math>\sigma=1<math>,

<math>\Phi(z) = {1 \over \sqrt{2\pi} } \int_{-\infty}^z \,e^{-{x^2 / 2}}\,dx.<math>

The standard normal cdf can be expressed in terms of a special function called the error function, as

<math>\Phi(z) = \frac{1}{2} \left(1+\operatorname{erf}\,\frac{z}{\sqrt{2}}\right).<math>

Generating functions

Moment generating function

The moment generating function is defined as the expected value of e^tX. For a normal distribution, it can be shown that the moment generating function is

<math>M_X(t)=E\left[e^{tX}\right]=\int_{-\infty}^{\infty} \frac{1} {\sigma\sqrt{2\pi}}\,e^{-{(x-\mu )^2 / 2\sigma^2}}\,e^{tx}\,dx = e^{\mu t+\sigma^2 t^2/2}<math>

as can be seen by completing the square in the exponent.

Characteristic function

The characteristic function is defined as the expected value of e^itX. For a normal distribution, the characteristic function is

<math>\phi_X(t)=E\left[e^{itX}\right]=\int_{-\infty}^{\infty} \frac{1} {\sigma\sqrt{2\pi}}\,e^{-{(x-\mu )^2 / 2\sigma^2}}\,e^{itx}\,dx = e^{i\mu t-\sigma^2 t^2/2}\,.<math>

Obviously, the characteristic function is obtained by replacing t with it in the moment-generating function.

Properties

1. If X ~ N(μ, σ²) and a and b are real numbers, then aX + b ~ N(aμ + b, (aσ)²) (see expected value and variance).
2. If X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²), and X₁ and X₂ are independent, then X₁ + X₂ ~ N(μ₁ + μ₂, σ₁² + σ₂²).
3. If X₁, ..., X_n are independent standard normal variables, then X₁² + ... + X_n² has a chi-squared distribution with n degrees of freedom.

Standardizing normal random variables

As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.

If X is a normal random variable with mean μ and variance σ², then

<math> Z = \frac{X - \mu}{\sigma} <math>

is a standard normal random variable: Z~N(0,1). An important consequence is that the cdf of a general normal distribution is therefore

<math>\Pr(X \le x) = \Phi\left(\frac{x-\mu}{\sigma}\right) = \frac{1}{2} \left(1+\mbox{erf}\,\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right).<math>

Conversely, if Z is a standard normal random variable, then

<math>X=\sigma Z+\mu \,<math>

is a normal random variable with mean μ and variance σ².

The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.

Generating normal random variables

For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods; the most basic is to invert the standard normal cdf. More efficient methods are also known. One such method is the Box-Muller transform. The Box-Muller transform takes two uniformly distributed values as input and maps them to two normally distributed values. This requires generating values from a uniform distribution, for which many methods are known. See also random number generators.

The Box-Muller transform is a consequence of Property 3 and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which is easy to generate).

The central limit theorem

The normal distribution has the very important property that under certain conditions, the distribution of a sum of a large number of independent variables is approximately normal. This is the so-called central limit theorem.

The practical importance of the central limit theorem is that the normal distribution can be used as an approximation to some other distributions.

• A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 (some books recommend using this approximation only if np and n(1 − p) are both at least 5; in this case, a continuity correction should be applied). The approximating normal distribution has mean μ = np and standard deviation σ = (n p (1 - p))^1/2.

• A Poisson distribution with parameter λ is approximately normal for large λ. The approximating normal distribution has mean μ = λ and standard deviation σ = √λ.

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.

Infinite divisibility

The normal distributions are infinitely divisible probability distributions.

Occurrence

Approximately normal distributions occur in many situations, as a result of the central limit theorem. When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption.

Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.

Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting marginal distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the theory of errors (see below).

To summarize, here's a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.

• In counting problems (so the central limit theorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as
  • Binomial random variables, associated to yes/no questions;
  • Poisson random variables, associated to rare events;
• In physiological measurements of biological specimens:
  • The logarithm of 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;
  • Other physiological measures may be normally distributed, but there is no reason to expect that a priori;
• Measurement errors are assumed to be normally distributed, and any deviation from normality must be explained;
• Financial variables
  • The logarithm of interest rates, exchange rates, and inflation; these variables behave like compound interest, not like simple interest, and so are multiplicative;
  • Stock-market indices are supposed to be multiplicative too, but some researchers claim that they are log-L�vy variables instead of lognormal;
  • Other financial variables may be normally distributed, but there is no reason to expect that a priori;
• Light intensity
  • The intensity of laser light is normally distributed;
  • Thermal light has a Bose-Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.

Photon counts

Light intensity from a single source varies with time, and is usually assumed to be normally distributed. However, quantum mechanics interprets measurements of light intensity as photon counting. Ordinary light sources which produce light by thermal emission, should follow a Poisson distribution or Bose-Einstein distribution on very short time scales. On longer time scales (longer than the coherence time), the addition of independent variables yields an approximately normal distribution. The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution and long coherence times.

Measurement errors

Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Note that this is the central assumption of the mathematical theory of errors.

Physical characteristics of biological specimens

The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore measures of body size should at most follow a lognormal rather than normal distribution. Despite common claims of normality, the sizes of plants and animals is approximately lognormal. The evidence and an explanation based on models of growth was first published in the classic book

Huxley, Julian: Problems of Relative Growth (1932)

Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further make the joint distribution of sizes deviate from lognormality.

The assumption that linear size of biological specimens is normal leads to a non-normal distribution of weight (since weight/volume is roughly the 3rd power of length, and Gaussian distributions are only preserved by linear transformations), and conversely assuming that weight is normal leads to non-normal lengths. This is a problem, because there is no a priori reason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.

• blood pressure of adult humans is supposed to be normally distributed, but only after separating males and females into different populations (each of which is normally distributed)
• The length of inert appendages such as hair, nails, teeth, claws and shells is expected to be normally distributed if measured in the direction of growth. This is because the growth of inert appendages depends on the size of the root, and not on the length of the appendage, and so proceeds by additive increments. Hence, we have an example of a sum of very many small lognormal increments approaching a normal distribution. Another plausible example is the width of tree trunks, where a new thin ring is produced every year whose width is affected by a large number of factors.

Financial variables

Because of the exponential nature of interest and inflation, financial indicators such as interest rates, stock values, or commodity prices make good examples of multiplicative behavior. As such, they should not be expected to be normal, but lognormal.

Beno�t Mandelbrot, the popularizer of fractals, has claimed that even the assumption of lognormality is flawed, and advocates the use of log-Levy distributions.

It is accepted that financial indicators deviate from lognormality. The distribution of price changes on short time scales is observed to have "heavy tails", so that very small or very large price changes are more likely to occur than a lognormal model would predict. Deviation from lognormality indicates that the assumption of independence of the multiplicative influences is flawed.

Lifetime

Other examples of variables that are not normally distributed include the lifetimes of humans or mechanical devices. Examples of distributions used in this connection are the exponential distribution (memoryless) and the Weibull distribution. In general, there is no reason that waiting times should be normal, since they are not directly related to any kind of additive influence.

Test scores

A great deal of confusion exists over whether or not IQ test scores and intelligence are normally distributed. While for most practical purposes the distributions of IQ and intelligence (or at least psychometric g) can be seen as the same thing, it is important to distinguish between the two terms when discussing whether they are normally distributed.

As a deliberate result of test construction, IQ scores are always and obviously normally distributed for the majority of the population. The fact that intelligence is normally distributed is less clear. The difficulty and number of questions on an IQ test is decided based on which combinations will yield a normal distribution. This does not mean, however, that the information is in any way being misrepresented, or that there is any kind of "true" distribution that is being artificially forced into the shape of a normal curve. Intelligence tests can be constructed to yield any kind of score distribution desired. All true IQ tests have a normal distribution of scores as a result of test design; otherwise IQ scores would be meaningless without knowing what test produced them. Intelligence tests in general, however, can produce any kind of distribution.

For an example of how arbitrary the distribution of intelligence test scores really is, imagine a 20-item multiple-choice test entirely composed of problems that consist mostly of finding the areas of circles. Such a test, if given to a population of high-school students, would likely yield a U-shaped distribution, with the bulk of the scores being very high or very low, instead of a normal curve. If a student understands how to find the area of a circle, he can likely do so repeatedly and with few errors, and thus would get a perfect or high score on the test, whereas a student who has never had geometry lessons would likely get every question wrong, possibly with a few right due to guessing luck. If a test is composed mostly of easy questions, then most of the test-takers will have high scores and very few will have low scores. If a test is composed entirely of questions so easy or so hard that every person gets either a perfect score or a zero, it fails to make any kind of statistical discrimination at all and yields a rectangular distribution. These are just a few examples of the many varieties of distributions that could theoretically be produced by carefully designing intelligence tests.

Whether intelligence itself is normally distributed has been at times a matter of some debate. Some critics maintain that the choice of a normal distribution is entirely arbitrary. Brian Simon once claimed that the normal distribution was specifically chosen by psychometricians to falsely support the idea that superior intelligence is only held by a small minority, thus legitimizing the rule of a privileged elite over the masses of society. Historically, though, intelligence tests were designed without any concern for producing a normal distribution, and scores came out approximately normally distributed anyway. American educational psychologist Arthur Jensen claims that any test that contains "a large number of items," "a wide range of item difficulties," "a variety of content or forms," and "items that have a significant correlation with the sum of all other scores" will inevitably produce a normal distribution. Furthermore, there exists a number of correlations between IQ scores and other human characteristics that are more provably normally distributed, such as nerve conduction velocity and the glucose metabolism rate of a person's brain, supporting the idea that intelligence is normally distributed.

Some critics, such as Stephen Jay Gould in his book The Mismeasure of Man, question the validity of intelligence tests in general, not just the fact that intelligence is normally distributed. For further discussion see the article IQ.

The Bell Curve is a controversial book on the topic of the heritability of intelligence. However, despite its title, the book does not primarily address whether IQ is normally distributed.

Maximum likelihood estimation of parameters

Suppose

<math>X_1,\dots,X_n<math>

are independent and identically distributed, and are normally distributed with expectation μ and variance σ². In the language of statisticians, the observed values of these random variables make up a "sample from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on observed values of this sample. The joint probability density function of these random variables is

<math>f(x_1,\dots,x_n) \propto \sigma^{-n} \prod_{i=1}^n \exp\left({-1 \over 2} \left({x_i-\mu \over \sigma}\right)^2\right).<math>

(Nota bene: Here the proportionality symbol <math>\propto<math> means proportional as a function of <math>\mu<math> and <math>\sigma<math>, not proportional as a function of <math>x_1,\dots,x_n<math>. That may be considered one of the differences between the statistician's point of view and the probabilist's point of view. The reason why this is important will appear below.)

As a function of μ and σ this is the likelihood function

<math>L(\mu,\sigma) \propto \sigma^{-n} \exp\left({-\sum_{i=1}^n (x_i-\mu)^2 \over 2\sigma^2}\right).<math>

In the method of maximum likelihood, the values of μ and σ that maximize the likelihood function are taken to be estimates of the population parameters μ and σ.

Usually in maximizing a function of two variables one might consider partial derivatives. But here we will exploit the fact that the value of μ that maximizes the likelihood function with σ fixed does not depend on σ. Therefore, we can find that value of μ, then substitute it from μ in the likelihood function, and finally find the value of σ that maximizes the resulting expression.

It is evident that the likelihood function is a decreasing function of the sum

<math>\sum_{i=1}^n (x_i-\mu)^2.<math>

So we want the value of μ that minimizes this sum. Let

<math>\overline{x}=(x_1+\cdots+x_n)/n<math>

be the "sample mean". Observe that

<math>\sum_{i=1}^n (x_i-\mu)^2=\sum_{i=1}^n((x_i-\overline{x})+(\overline{x}-\mu))^2<math>

<math>=\sum_{i=1}^n(x_i-\overline{x})^2 + 2\sum_{i=1}^n (x_i-\overline{x})(\overline{x}-\mu) + \sum_{i=1}^n (\overline{x}-\mu)^2

<math>

<math>

=\sum_{i=1}^n(x_i-\overline{x})^2 + 0 + n(\overline{x}-\mu)^2. <math>

Only the last term depends on μ and it is minimized by

<math>\hat{\mu}=\overline{x}.<math>

That is the maximum-likelihood estimate of μ. Substituting that for μ in the sum above makes the last term vanish. Consequently, when we substitute that estimate for μ in the likelihood function, we get

<math>L(\overline{x},\sigma) \propto \sigma^{-n} \exp\left({-\sum_{i=1}^n (x_i-\overline{x})^2 \over 2\sigma^2}\right).<math>

It is conventional to denote the "loglikelihood function", i.e., the logarithm of the likelihood function, by a lower-case <math>\ell<math>, and we have

<math>\ell(\hat{\mu},\sigma)=[\mathrm{constant}]-n\log(\sigma)-{\sum_{i=1}^n(x_i-\overline{x})^2 \over 2\sigma^2}<math>

and then

<math>{\partial \over \partial\sigma}\ell(\hat{\mu},\sigma)

={-n \over \sigma} +{\sum_{i=1}^n (x_i-\overline{x})^2 \over \sigma^3} ={-n \over \sigma^3}\left(\sigma^2-{1 \over n}\sum_{i=1}^n (x_i-\overline{x})^2 \right).<math>

This derivative is positive, zero, or negative according as σ² is between 0 and

<math>{1 \over n}\sum_{i=1}^n(x_i-\overline{x})^2,<math>

or equal to that quantity, or greater than that quantity.

Consequently this average of squares of residuals is maximum-likelihood estimate of σ², and its square root is the maximum-likelihood estimate of σ.

Surprising generalization

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices.

See also

• Multivariate normal distribution

External links and references

• A. Kropinski's normal distribution tutorial (http://ce597n.www.ecn.purdue.edu/CE597N/1997F/students/michael.a.kropinski.1/project/tutorialMichael)
• S. M.Stigler: Statistics on the Table, Harvard University Press 1999, chapter 22. History of the term "normal distribution".
• Earliest Known uses of some of the Words of Mathematics (http://members.aol.com/jeff570/mathword.html). See: [1] (http://members.aol.com/jeff570/n.html) for "normal", [2] (http://members.aol.com/jeff570/g.html) for "Gaussian", and [3] (http://members.aol.com/jeff570/e.html) for "error".
• Earliest Uses of Symbols in Probability and Statistics (http://members.aol.com/jeff570/stat.html). See Symbols associated with the Normal Distribution.
• Distribution Calculator (http://www.vias.org/simulations/simusoft_distcalc.html) Calculates probabilities and critical values for normal, t-, chi2- and F-distribution

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