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From Wikipedia, the free encyclopedia

Probability density function
Probability density function for the normal distribution
The red line is the standard normal distribution
Cumulative distribution function
Cumulative distribution function for the normal distribution
Colors match the image above
notation: \mathcal{N}(\mu,\,\sigma^2)
parameters: μR — mean (location)
σ2 ≥ 0 — variance (squared scale)
support: xR   if σ2 > 0
x = μ   if σ2 = 0
pdf: \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2} \right)
cdf: \frac12\Big[1 + \operatorname{erf}\Big( \frac{x-\mu}{\sigma\sqrt2}\Big)\Big]
mean: μ
median: μ
mode: μ
variance: σ2
skewness: 0
kurtosis: 0
entropy: \ln\!\sqrt{2 \pi e \, \sigma^2}
mgf: \exp\!\Big(\mu t + \tfrac{1}{2}\sigma^2t^2\Big)
cf: \exp\!\Big(i\mu t - \tfrac{1}{2}\sigma^2 t^2\Big)
Fisher information: \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}

In probability theory and statistics, the normal distribution or Gaussian distribution is a continuous probability distribution that often gives a good description of data that cluster around the mean. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve.

The Gaussian distribution is one of many things named after Carl Friedrich Gauss, who used it to analyze astronomical data,[1] and determined the formula for its probability density function. However, Gauss was not the first to study this distribution or the formula for its density function—that had been done earlier by Abraham de Moivre.

The normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the heights of adult males in the United States are roughly normally distributed, with a mean of about 70 inches (1.8 m). Most men have a height close to the mean, though a small number of outliers have a height significantly above or below the mean. A histogram of male heights will appear similar to a bell curve, with the correspondence becoming closer if more data are used.

By the central limit theorem, under certain conditions the sum of a number of random variables with finite means and variances approaches a normal distribution as the number of variables increases. For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural science, and social science[2] as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption.

Contents

History

The bean machine is a device invented by Sir Francis Galton to demonstrate how the normal distribution appears in nature. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

The normal distribution was first introduced by de Moivre in an article in 1733,[3] which was 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 “Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum” (1809) by assuming the normal distribution of the errors. Gauss’s notation was quite different from the modern one, for the error Δ he writes

 \varphi\Delta = \tfrac{h}{\surd\pi}\, e^{-hh\Delta\Delta}

In the middle of the 19th century Maxwell demonstrated that the normal distribution is not only a convenient mathematical tool, but that it also appears in nature. He writes[4]: “The number of particles whose velocity, resolved in a certain direction, lies between x and x+dx is

 \mathrm{N}\frac{1}{\alpha\sqrt\pi}e^{-\frac{x^2}{\alpha^2}}dx

It was Pearson to first write the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher has added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:

 df = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-m)^2}{2\sigma^2}}dx

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. Curiously, it has never been known under the name of its inventor, de Moivre. The name “normal distribution” was coined independently by Peirce, Galton and Lexis around 1875; the term was derived from the fact that this distribution was seen as typical, common, normal. This name was popularized in statistical community by Pearson around the turn of the 20th century.[5]

The term “standard normal” which denotes the normal distribution with zero mean and unit variance came into general use around 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”.[6]

Definition

The simplest case of a normal distribution is known as the standard normal distribution, described by the probability density function

 \phi(x) = \tfrac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2},

The constant \scriptstyle\ 1/\sqrt{2\pi} in this expression ensures that the total area under the curve ϕ(x) is equal to one,[proof] and 12 in the exponent makes the “width” of the curve (measured as half of the distance between the inflection points of the curve) also equal to one. It is traditional[7] in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters ƒ or p. The alternative glyph φ is also used quite often, however within this article we reserve “φ” to denote characteristic functions.

More generally, a normal distribution results from exponentiating a quadratic function (just as an exponential distribution results from exponentiating a linear function):

 f(x) = e^{a x^2 + b x + c}. \,

This yields the classic “bell curve” shape (provided that a < 0 so that the quadratic function is concave). Notice that f(x) > 0 everywhere. One can adjust a to control the “width” of the bell, then adjust b to move the central peak of the bell along the x-axis, and finally adjust c to control the “height” of the bell. For f(x) to be a true probability density function over R, one must choose c such that \scriptstyle\int_{-\infty}^\infty f(x)\,dx\ =\ 1 (which is only possible when a < 0).

Rather than using a, b, and c, it is far more common to describe a normal distribution by its mean μ = −b/(2a) and variance σ2 = −1/(2a). Changing to these new parameters allows us to rewrite the probability density function in a convenient standard form,

 f(x) = \tfrac{1}{\sqrt{2\pi\sigma^2}}\, e^{\frac{-(x-\mu)^2}{2\sigma^2}} = \tfrac{1}{\sigma}\, \phi\!\left(\tfrac{x-\mu}{\sigma}\right).

Notice that for a standard normal distribution, μ = 0 and σ2 = 1. The last part of the equation above shows that any other normal distribution can be regarded as a version of the standard normal distribution that has been stretched horizontally by a factor σ and then translated rightward by a distance μ. Thus, μ specifies the position of the bell curve’s central peak, and σ specifies the “width” of the bell curve.

The parameter μ is at the same time the mean, the median and the mode of the normal distribution. The parameter σ2 is called the variance; as for any real-valued random variable, it describes how concentrated the distribution is around its mean. The square root of σ2 is called the standard deviation and is the width of the density function.

Some authors[8] instead of σ2 use its reciprocal τ = σ−2, which is called the precision. This parameterization has an advantage in numerical applications where σ2 is very close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution. Another advantage of using this parameterization is in the study of conditional distributions in multivariate normal case.

Normal distribution is denoted as N(μ, σ2). Commonly the letter N is written in calligraphic font (typed as \mathcal{N} in LaTeX). Thus when a random variable X is distributed normally with mean μ and variance σ2, we write

 X\ \sim\ \mathcal{N}(\mu,\,\sigma^2). \,

Characterization

In the previous section the normal distribution was defined by specifying its probability density function. However there are other ways to characterize a probability distribution. They include: the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, etc.

Probability density function

The continuous probability density function of the normal distribution exists only when the variance parameter σ2 is not equal to zero. Then it is given by the Gaussian function

 f(x;\,\mu,\sigma^2) = \tfrac{1}{\sqrt{2\pi\sigma^2}} \, e^{-(x- \mu)^2\!/(2\sigma^2)} = \tfrac{1}{\sigma} \,\phi\!\left(\tfrac{x - \mu}{\sigma}\right), \qquad x\in\mathbb{R}.

When σ2 = 0, the density can be represented as a Dirac delta function:

 f(x;\,\mu,0) = \delta(x-\mu)\ .

This isn’t a function in a usual sense, but rather a generalized function: it is equal to infinity at x = μ and zero elsewhere.

Properties:

  • Function ƒ(x) is symmetric around x = μ, which is at the same time the mode, the median and the mean of the distribution.
  • The inflection points of the curve occur one standard deviation away from the mean (i.e., at x = μσ and x = μ + σ).
  • The standard normal density ϕ(x) is an eigenfunction of the Fourier transform.
  • The function is supersmooth of order 2, implying that it is infinitely differentiable.
  • The derivative of ϕ(x) is ϕ′(x) = −x·ϕ(x), the second derivative is ϕ′′(x) = (x2 − 1)ϕ(x).

Cumulative distribution function

The cumulative distribution function (cdf) of a random variable X evaluated at a number x, is the probability of the event that X is less than or equal to x. The cdf of the standard normal distribution is denoted with the capital greek letter Φ (phi), and can be computed as an integral of the probability density function:

 \Phi(x) = \int_{-\infty}^x \phi(t) \, dt = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt.

This integral cannot be expressed in terms of standard functions, however with the use of a special function erf, called the error function, the standard normal cdf Φ(x) can be written as

 \Phi(x) = \frac12\Big[ 1 + \operatorname{erf}\Big(\frac{x}{\sqrt{2}}\Big)\Big],\quad x\in\mathbb{R}.

The complement of the standard normal cdf, 1 − Φ(x), is often denoted Q(x), and is referred to as the Q-function, especially in engineering texts.[9][10] This represents the tail probability of the Gaussian distribution, that is the probability that a standard normal random variable X is greater than the number x:

 Q(x) = \int_x^\infty \phi(t) \, dt = 1 - \Phi(x).

Other definitions of the Q-function, all of which are simple transformations of Φ, are also used occasionally.[11]

The inverse of the standard normal cdf, called the quantile function or probit function, can be expressed in terms of the inverse error function:

 \Phi^{-1}(z) = \sqrt2\,\operatorname{erf}^{-1}(2z - 1), \quad z\in(0,1).

It is recommended to use letter z to denote the quantiles of the standard normal cdf, unless that letter is already used for some other purpose.

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. For large values of x it is usually easier to work with the Q-function.

For a generic normal random variable with mean μ and variance σ2 > 0 the cdf will be equal to

 F(x;\,\mu,\sigma^2) = \int_{-\infty}^x f(t;\,\mu,\sigma^2)\,dt = \Phi\Big(\frac{x-\mu}{\sigma}\Big) = \frac12\Big[ 1 + \operatorname{erf}\Big(\frac{x-\mu}{\sigma\sqrt{2}}\Big)\Big],\quad x\in\mathbb{R}

and the corresponding quantile function is

 F^{-1}(p;\,\mu,\sigma^2) = \mu + \sigma\Phi^{-1}(p) = \mu + \sigma\sqrt2\,\operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).

For a normal distribution with zero variance, the cdf is the Heaviside function:

 F(x;\,\mu,0) = \mathbf{1}_{\{x\geq\mu\}}\,.

Properties:

  • The standard normal cdf is symmetric around point (0, ½):  Φ(−x) = 1 − Φ(x).
  • The derivative of Φ(x) is equal to the standard normal pdf ϕ(x):  Φ’(x) = ϕ(x).
  • The antiderivative of Φ(x) is: ∫ Φ(x) dx = xΦ(x) + ϕ(x).

Characteristic function

The characteristic function φX(t) of a random variable X is defined as the expected value of eitX, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function. Thus the characteristic function is the Fourier transform of the density ϕ(x).

For the standard normal random variable, the characteristic function is

 \varphi(t) = \int_{-\infty}^\infty e^{itx}\tfrac{1}{\sqrt{2\pi}}e^{-\frac12 x^2}dx = e^{-\frac12 t^2}.

For a generic normal distribution with mean μ and variance σ2, the characteristic function is [12]

 \varphi(t;\,\mu,\sigma^2) = \operatorname{E}[e^{it\mathcal{N}(\mu,\sigma^2)}] = e^{i\mu t - \frac12 \sigma^2t^2}.

Moment generating function

The moment generating function is defined as the expected value of etX. For a normal distribution, the moment generating function exists and is equal to

 M(t;\, \mu,\sigma^2) = \operatorname{E}[e^{tX}] = \varphi(-it;\, \mu,\sigma^2) = e^{ \mu t + \frac12 \sigma^2 t^2 }.

The cumulant generating function is the logarithm of the moment generating function:

 g(t;\,\mu,\sigma^2) = \ln M(t;\,\mu,\sigma^2) = \mu t + \tfrac{1}{2} \sigma^2 t^2

Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.

Properties

  1. The family of normal distributions is closed under linear transformations. That is, if X is normally distributed with mean μ and variance σ2, then a linear transform aX + b (for some real numbers a ≠ 0 and b) is also normally distributed:
     aX + b\ \sim\ \mathcal{N}(a\mu+b,\, a^2\sigma^2).
    Also if X1, X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their linear combination will also be normally distributed: [proof]
     aX_1 + bX_2 \ \sim\ \mathcal{N}(a\mu_1+b\mu_2,\, a^2\!\sigma_1^2 + b^2\sigma_2^2)
  2. The converse of (1) is also true: if X1 and X2 are independent and their sum X1 + X2 is distributed normally, then both X1 and X2 must also be normal. This is known as the Cramér’s theorem.
  3. Normal distribution is infinitely divisible: for a normally distributed X with mean μ and variance σ2 we can find n independent random variables {X1, …, Xn} each distributed normally with means μ/n and variances σ2/n such that
     X_1 + X_2 + \cdots + X_n \ \sim\ \mathcal{N}(\mu, \sigma^2)
  4. Normal distribution is stable (with exponent α = 2): if X1, X2 are two independent N(μ, σ2) random variables and a, b are arbitrary real numbers, then
     aX_1 + bX_2 \ \sim\ \sqrt{a^2+b^2}\cdot X_3\ +\ (a+b-\sqrt{a^2+b^2})\mu,
    where X3 is also N(μ, σ2). This relationship directly follows from property (1).
  5. The Kullback–Leibler divergence between two normal distributions X1N(μ1, σ21 )and X2N(μ2, σ22 )is given by:[13]
     D_\mathrm{KL}( X_1 \,\|\, X_2 ) = \frac{(\mu_1 - \mu_2)^2}{2\sigma_2^2} \,+\, \frac12\Bigg(\, \frac{\sigma_1^2}{\sigma_2^2} - 1 - \ln\frac{\sigma_1^2}{\sigma_2^2} \,\Bigg)\ .
    The Hellinger distance between the same distributions is equal to
     H^2(X_1,X_2) = 1 \,-\, \sqrt{\frac{2\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2}} \; e^{-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2+\sigma_2^2}}\ .
  6. The Fisher information matrix for normal distribution is diagonal and takes form
     \mathcal I = \begin{pmatrix} \frac{1}{\sigma^2} & 0 \\ 0 & \frac{1}{2\sigma^4} \end{pmatrix}
  7. Normal distributions belongs to an exponential family with natural parameters  \scriptstyle\theta_1=\frac{\mu}{\sigma^2} and \scriptstyle\theta_2=\frac{-1}{2\sigma^2}, and natural statistics x and x2. The dual, expectation parameters for normal distribution are η1 = μ and η2 = μ2 + σ2.
  8. Of all probability distributions over the reals with mean μ and variance σ2, the normal distribution N(μ, σ2) is the one with the maximum entropy.
  9. The family of normal distributions forms a manifold with constant curvature −1. The same family is flat with respect to the (±1)-connections ∇(e) and ∇(m).[14]

Standardizing normal random variables

As a consequence of property 1, it is possible to relate all normal random variables to the standard normal. For example if X is normal with mean μ and variance σ2, then

 Z = \frac{X - \mu}{\sigma}

has mean zero and unit variance, that is Z has the standard normal distribution. Conversely, having a standard normal random variable Z we can always construct another normal random variable with specific mean μ and variance σ2:

 X = \sigma Z + \mu. \,

This “standardizing” transformation is convenient as it allows one to compute the pdf and especially the cdf of a normal distribution having the table of pdf and cdf values for the standard normal. They will be related via

 F_X(x) = \Phi\bigg(\frac{x-\mu}{\sigma}\bigg), \quad f_X(x) = \frac{1}{\sigma}\,\phi\bigg(\frac{x-\mu}{\sigma}\bigg).

Moments

The normal distribution has moments of all orders. That is, for a normally distributed X with mean μ and variance σ2, the expectation E[|X|p] exists and is finite for all p such that Re[p] > −1. Usually we are interested only in moments of integer orders: p = 1, 2, 3, ….

  • Central moments are the moments of X around its mean μ. Thus, central moment of order p is the expected value of (X − μ)p. Using standardization of normal distribution, this expectation will be equal to σp·E[Zp], where Z is standard normal.
     \operatorname{E}\big[(X-\mu)^p\big] = \left.\begin{cases} 0 & \text{if }p\text{ is odd} \ \sigma^p(p-1)!! & \text{if }p\text{ is even} \end{cases}\right\} = \sigma^p \frac{p!}{2^{p/2}(p/2)!} \cdot \mathbf{1}_{\{p\text{ is even}\}}.
    Here n!! denotes the double factorial, that is the product of every other number from n to 1; and 1{…} is the indicator function.
  • Central absolute moments are the moments of |X − μ|. They coincide with regular moments for all even orders, but are nonzero for all odd p’s.
     \operatorname{E}\big[|X-\mu|^p\big] = \sigma^p(p-1)!! \cdot \begin{cases} \sqrt{2/\pi} & \text{if }p\text{ is odd}, \ 1 & \text{if }p\text{ is even}. \end{cases}
  • Raw moments and raw absolute moments are the moments of X and |X| respectively. The formulas for these moments are much more complicated, and are given in terms of confluent hypergeometric functions 1F1 and U.
    \begin{align} & \operatorname{E} \big[ X^p \big] = \sigma^p \cdot (-i\sqrt{2}\sgn\mu)^p \; U\Big( {-\tfrac{1}{2}p},\, \tfrac{1}{2},\, -\tfrac{1}{2}(\mu/\sigma)^2 \Big), \ & \operatorname{E} \big[ |X|^p \big] = \sigma^p \cdot 2^{\frac p 2} \frac {\Gamma\big(\frac{1+p}{2}\big)}{\sqrt\pi}\; _1F_1\Big( {-\tfrac{1}{2}p},\, \tfrac{1}{2},\, -\tfrac{1}{2}(\mu/\sigma)^2 \Big). \ \end{align}
    These expressions remain valid even if p is not integer.
  • First two cumulants are equal to μ and σ2 respectively, whereas all higher-order cumulants are equal to zero.
Order Raw moment Central moment Cumulant
1 \scriptstyle\mu 0 \scriptstyle\mu
2 \scriptstyle\mu^2 + \sigma^2 \scriptstyle\sigma^2 \scriptstyle\sigma^2
3 \scriptstyle\mu^3 + 3\mu\sigma^2 0 0
4 \scriptstyle\mu^4 + 6 \mu^2 \sigma^2 + 3 \sigma^4 \scriptstyle3 \sigma^4 0
5 \scriptstyle\mu^5 + 10 \mu^3 \sigma^2 + 15 \mu \sigma^4 0 0
6 \scriptstyle\mu^6 + 15 \mu^4 \sigma^2 + 45 \mu^2 \sigma^4 + 15 \sigma^6 \scriptstyle 15 \sigma^6 0
7 \scriptstyle\mu^7 + 21 \mu^5 \sigma^2 + 105 \mu^3 \sigma^4 + 105 \mu \sigma^6 0 0
8 \scriptstyle\mu^8 + 28 \mu^6 \sigma^2 + 210 \mu^4 \sigma^4 + 420 \mu^2 \sigma^6 + 105 \sigma^8 \scriptstyle 105 \sigma^8 0

Central limit theorem

The theorem states that under certain, fairly common conditions, the sum of a large number of random variables will have approximately normal distribution. For example if X1, …, Xn is a sequence of iid random variables, each having mean μ and variance σ2 but otherwise distributions of Xi’s can be arbitrary, then the central limit theorem states that

 \sqrt{n}\bigg( \frac{1}{n}\sum_{i=1}^n X_i - \mu \bigg)\ \xrightarrow{d}\ \mathcal{N}(0,\,\sigma^2).

The theorem will hold even if the summands Xi are not iid, although some constraints on the degree of dependence and the growth rate of moments still have to be imposed.

The importance of the central limit theorem cannot be overemphasized. A great number of test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, even more estimators can be represented as sums of random variables through the use of influence functions — all of these quantities are governed by the central limit theorem and will have asymptotically normal distribution as a result.

Plot of the pdf of a normal distribution with μ = 12 and σ = 3, approximating the pdf of a binomial distribution with n = 48 and p = 1/4

Another practical consequence of the central limit theorem is that certain other distributions can be approximated by the normal distribution, for example:

  • The binomial distribution B(n, p) is approximately normal N(np, np(1 − p)) for large n and for p not too close to zero or one.
  • The Poisson(λ) distribution is approximately normal N(λ, λ) for large values of λ.
  • The chi-squared distribution χ2(k) is approximately normal N(k, 2k) for large ks.
  • The Student’s t-distribution t(ν) is approximately normal N(0, 1) when ν is large.

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.

Standard deviation and confidence intervals

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue), while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.

About 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. This is known as the 68-95-99.7 rule, or the empirical rule, or the 3-sigma rule.

To be more precise, the area under the bell curve between μ −  and μ +  in terms of the cumulative normal distribution function is given by

\begin{align} & F(\mu+n\sigma;\,\mu,\sigma^2) - F(\mu-n\sigma;\,\mu,\sigma^2) \ &\quad = \Phi(n)-\Phi(-n) = 2\Phi(n)-1=\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right),\end{align}

where erf is the error function. To 12 decimal places, the values for the 1-, 2-, up to 6-sigma points are:

\scriptstyle\; n\; \;\scriptstyle\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right)\;
1 0.682689492137
2 0.954499736104
3 0.997300203937
4 0.999936657516
5 0.999999426697
6 0.999999998027

The next table gives the reverse relation of sigma multiples corresponding to a few often used values for the area under the bell curve. These values are useful to determine (asymptotic) confidence intervals of the specified levels based on normally distributed (or asymptotically normal) estimators:

\scriptstyle\;\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right)\; \scriptstyle\;n\;
0.80 1.281551565545
0.90 1.644853626951
0.95 1.959963984540
0.98 2.326347874041
0.99 2.575829303549
0.995 2.807033768344
0.998 3.090232306168
0.999 3.290526731492
0.9999 3.890591886413
0.99999 4.417173413469

where the value on the left of the table is the proportion of values that will fall within a given interval and n is a multiple of the standard deviation that specifies the width of the interval.

Related and derived distributions

Descriptive and inferential statistics

Scores

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

Normality tests

Normality tests assess the likelihood that the given data set {x1, …, xn} comes from a normal distribution. Typically the null hypothesis H0 is that the observations are distributed normally with unspecified mean μ and variance σ2, versus the alternative Ha that the distribution is arbitrary. A great number of tests (over 40) have been devised for this problem, the more prominent of them are outlined below:

  • “Visual” tests 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 — 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’s a plot of point of the form (Φ−1(pk), x(k)), where plotting points pk are equal to pk = (k−α)/(n+1−2α) and α is an adjustment constant which can be anything between 0 and 1. 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 (Φ(z(k)), pk), where \scriptstyle z_{(k)} = (x_{(k)}-\hat\mu)/\hat\sigma. For normally distributed data this plot should lie on a 45° line between (0,0) and (1,1).
    • Wilk–Shapiro test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
    • Normal probability plot (rankit plot)

Estimation of parameters

It is often the case that we don’t know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample X1, …, Xn from a normal N(μ,σ2) population we would like to learn the approximate values of parameters μ and σ2.

The standard approach to this problem is the maximum likelihood method, which gives as estimates the values that maximize the log-likelihood function:

\begin{align} \hat\ell(\mu,\sigma^2|\,X_1,\dots,X_n) &= \frac1n \sum_{i=1}^n \ln f(X_i;\,\mu,\sigma^2) \ &= -\frac{1}{2}\ln(2\pi) - \frac{1}{2}\ln\sigma^2 - \frac{1}{2n\sigma^2}\sum_{i=1}^n (X_i-\mu)^2. \end{align}

Maximizing this function with respect to μ and σ2 yields the maximum likelihood estimates

\begin{align} & \hat{\mu} = \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i, \ & \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \overline{X})^2. \end{align}

Estimator \scriptstyle\hat\mu is called the sample mean, as it is the arithmetic mean of the sample observations. The estimator \scriptstyle\hat\sigma^2 is similarly called the sample variance. Sometimes instead of \scriptstyle\hat\sigma^2 another estimator is considered, s2, which differs from the former by having (n − 1) instead of n in the denominator (so called Bessel’s correction):

 \textstyle s^2 = \frac{n}{n-1}\hat\sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2.

This quantity s2 is also called the sample variance, and its square root the sample standard deviation. The difference between s2 and \scriptstyle\hat\sigma^2 becomes negligibly small for large n’s.

These estimators have the following properties:

  • \hat\mu is the uniformly minimum variance unbiased (UMVU) estimator for μ, by the Lehmann–Scheffé theorem.
  • \hat\mu is a consistent estimator of μ, that is \scriptstyle\hat\mu converges in probability to μ as n → ∞.
  • \hat\mu has normal final sample distribution:
     \hat\mu \ \sim\ \mathcal{N}(\mu,\,\,\sigma^2/n),
    which implies that the standard error of \scriptstyle\hat\mu is equal to \scriptstyle\sigma/\sqrt{n}, that is if one wishes to decrease the standard error by a factor of 10, one must increase the number of samples by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and number of trials in Monte Carlo simulation.
  • \hat\sigma^2 is a biased estimator of σ2, whereas s2 is unbiased. On the other hand, \hat\sigma^2 is a superior estimator in terms of the mean squared error (MSE) criterion.
  • \hat\sigma^2 is a consistent and asymptotically normal estimator:
     \sqrt{n}(\hat\sigma^2 - \sigma^2)\ \xrightarrow{d}\ \mathcal{N}(0,\,2\sigma^4).
  • \hat\sigma^2 has a distribution proportional to chi-squared in finite sample:
     \hat\sigma^2 \ \sim\ \frac{\sigma^2}{n} \cdot \chi^2(n-1).
  • \hat\sigma^2 is independent from \hat\mu, by Cochran’s theorem. The normal distribution is the only distribution whose sample mean and sample variance are independent.
  • The ratio
     t = \frac{\hat\mu-\mu}{s/\sqrt{n}} = \frac{\overline{X}-\mu}{\sqrt{\frac{1}{n(n-1)}\sum(X_i-\overline{X})^2}}\ \sim\ t(n-1)
    has Student’s t-distribution. This t-statistic is ancillary, and is used for testing the hypothesis H0:μ = μ0 and in construction of confidence intervals.
  • The 1−α confidence intervals for μ and σ2 are:
    \begin{align} & \mu \in \Big[\, \hat\mu + q^{t(n-1)}_{\alpha/2}\,\tfrac{1}{\sqrt{n}}s,\ \ \hat\mu + q^{t(n-1)}_{1-\alpha/2}\,\tfrac{1}{\sqrt{n}}s \,\Big], \ & \sigma^2 \in \bigg[\, \frac{n\hat\sigma^2}{q^{\chi^2(n-1)}_{1-\alpha/2}},\ \ \frac{n\hat\sigma^2}{q^{\chi^2(n-1)}_{\alpha/2}} \,\bigg], \end{align}
    where q denotes the quantile function. For large n it is possible to replace the quantiles of t- and χ²-distributions with the normal quantiles. For example, the approximate 95% confidence intervals will be given by
    \begin{align} & \mu \in \big[\, \hat\mu - 1.96\tfrac{1}{\sqrt n}\hat\sigma,\ \ \hat\mu + 1.96\tfrac{1}{\sqrt n}\hat\sigma \,\big], \ & \sigma^2 \in \big[\, \hat\sigma^2 - 1.96\sqrt{\tfrac{2}{n}}\hat\sigma^2,\ \ \hat\sigma^2 + 1.96\sqrt{\tfrac{2}{n}}\hat\sigma^2 \,\big], \end{align}
    where 1.96 is the 97.5%-th quantile of the standard normal distribution.

Occurrence

The occurrence of normal distribution in practical problems can be loosely classified into three 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 one of the simplest and most convenient to use, frequently researchers are tempted to assume that certain quantity is distributed normally, without justifying such assumption rigorously. In fact, the maturity of a scientific field can be judged by the prevalence of the normality assumption in its methods.[citation needed]

Exact normality

Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:

  • Velocities of the molecules in the ideal gas. More generally, velocities of the particles in any system in thermodynamic equilibrium will have normal distribution, due to the maximum entropy principle.
  • Probability density function of a ground state in a quantum harmonic oscillator.
  • The density of an electron cloud in 1s state.
  • The position of a particle which experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is a dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation  \textstyle \frac{\partial}{\partial t} f(x) = \frac{1}{2} \frac{\partial^2}{\partial x^2} f(x). If the initial location is given by a certain density function g(x), then the density at time t is the convolution of g and the normal pdf.

Approximate normality

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by a large number of 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 which has a considerably larger magnitude than the rest of the effects.

Assumed normality

There are statistical methods to empirically test that assumption, see the #Normality tests section.

  • In biology:
    • The logarithm of measures of size of living tissue (length, height, skin area, weight)[16];
    • 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 (after separation on male/female subpopulations).
  • 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 Benoît Mandelbrot argue that log-Levy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes.
  • Measurement errors in physical experiments are often assumed to be normally distributed. This assumption allows for particularly simple practical rules for how to combine errors in measurements of different quantities. However, whether this assumption is valid or not in practice is debatable. A famous remark of Lippmann says: “Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics.” [17]
  • In standardized testing, results can be made to have a normal distribution. This is done by either selecting the number and difficulty of questions (as in the IQ test), or by 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.

Generating values from normal distribution

For computer simulations, especially in applications of Monte-Carlo method, it is often useful to generate values that have a normal distribution. All algorithms described here are concerned with generating the standard normal, since a N(μ, σ2) can be generated as X = μ + σZ, where Z is standard normal. The algorithms rely on the availability of a random number generator capable of producing random values distributed uniformly.

  • The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ−1(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ−1, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article.
  • A simple approximate approach that is easy to program is as follows: simply sum 12 uniform (0,1) deviates 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 (−6, 6).[18]
  • The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1]. Then two random variables X and Y
    \begin{align} & X = \sqrt{- 2 \ln U} \, \cos(2 \pi V) , \ & Y = \sqrt{- 2 \ln U} \, \sin(2 \pi V) . \end{align}
    will both have the standard normal distribution, and be independent. This formulation arises because for a bivariate normal random vector (X Y) the squared norm X2 + Y2 will have the chi-square distribution with two degrees of freedom, which is an easily-generated exponential random variable corresponding to the quantity −2ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.
  • Marsaglia polar method is a modification of the Box–Muller method algorithm, which does not require computation of functions sin() and cos(). In this method U and V are drawn from the uniform (−1,1) distribution, and then S = U2 + V2 is computed. If S is greater or equal to one then the method starts over, otherwise two quantities
     X = U\sqrt{\frac{-2\ln S}{S}}, \qquad Y = V\sqrt{\frac{-2\ln S}{S}}
    are returned. Again, X and Y here will be independent and standard normally distributed.
  • The ziggurat algorithm (Marsaglia & Tsang 2000) 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, exponentials and more uniform random numbers has to be employed.
  • 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 of the normal cdf

The standard normal cdf is widely used in scientific and statistical computing. Different approximations are used depending on the desired level of accuracy.

  • Abramowitz & Stegun (1964) give the approximation for Φ(x) with the absolute error |ε(x)| < 7.5·10−8 (algorithm 26.2.17):
     \Phi(x) = 1 - \phi(x)\Big(b_1t + b_2t^2 + b_3t^3 + b_4t^4 + b_5t^5\Big) + \varepsilon(x), \qquad t = \frac{1}{1+b_0x},
    where ϕ(x) is the standard normal pdf, and b0 = 0.2316419, b1 = 0.319381530, b2 = −0.356563782, b3 = 1.781477937, b4 = −1.821255978, b5 = 1.330274429.
  • Hart (1968) lists almost a hundred of rational function approximations for the erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart’s algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
  • Marsaglia (2004) suggested a simple algorithm[19] based on the Taylor series expansion
     \Phi(x) = \frac12 + \phi(x)\bigg( x + \frac{x^3}{3} + \frac{x^5}{3\cdot5} + \frac{x^7}{3\cdot5\cdot7} + \frac{x^9}{3\cdot5\cdot7\cdot9} + \cdots \bigg)
    for calculating Φ(x) 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 x = 10).
  • The GNU Scientific Library calculates values of the standard normal cdf using Hart’s algorithms and approximations with Chebyshev polynomials.

??? For a more detailed discussion of how to calculate the normal distribution, see Knuth’s The Art of Computer Programming, volume 2, section 3.4.1C.

Software for calculation of normal distribution

  • In Mathematica, function NormalDistribution[μ,σ] describes the normal distribution itself. For example a standard normal random variate can be generated as RandomReal[NormalDistribution[]]. Additional functions such as Erf[z], Erfc[z], InverseErf[z] can be helpful to calculate the cdf and quantiles of the distirbution.
  • In Stata, rnormal(μ,σ) generates a normal random variate, function normal(z) computes the standard normal cdf, and invnormal(p) the inverse cdf.
  • In Gauss, an r×c matrix of standard normal random variables can be generated using the command rndn(r,c). Function cdfn(x) computes the standard normal cdf, and cdfni(p) — its inverse (the quantile function).
  • In Excel, …

See also

Related distributions

others

Notes

  1. ^ Havil (2003)
  2. ^ Gale Encyclopedia of Psychology – Normal Distribution
  3. ^ Abraham de Moivre, “Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem expansi” (printed on 12 November 1733 in London for private circulation). This pamphlet has been reprinted in: (1) Richard C. Archibald (1926) “A rare pamphlet of Moivre and some of his discoveries,” Isis, vol. 8, pages 671–683; (2) Helen M. Walker, “De Moivre on the law of normal probability” in David Eugene Smith, A Source Book in Mathematics [New York, New York: McGraw–Hill, 1929; reprinted: New York, New York: Dover, 1959], vol. 2, pages 566–575.; (3) Abraham De Moivre, The Doctrine of Chances (2nd ed.) [London: H. Woodfall, 1738; reprinted: London: Cass, 1967], pages 235–243; (3rd ed.) [London: A Millar, 1756; reprinted: New York, New York: Chelsea, 1967], pages 243–254; (4) Florence N. David, Games, Gods and Gambling: A History of Probability and Statistical Ideas [London: Griffin, 1962], Appendix 5, pages 254–267.
  4. ^ Maxwell (1860)
  5. ^ "Earliest known uses of some of the words of mathematics (entry NORMAL)". http://jeff560.tripod.com/n.html. 
  6. ^ "Earliest known uses of some of the words in mathematics (entry STANDARD NORMAL CURVE)". http://jeff560.tripod.com/s.html. 
  7. ^ Halperin & et al. (1965, item 7)
  8. ^ Bernardo & Smith (2000)
  9. ^ Scott, Clayton; Robert Nowak (August 7, 2003). "The Q-function". Connexions. http://cnx.org/content/m11537/1.2/. 
  10. ^ Barak, Ohad (April 6, 2006). "Q function and error function". Tel Aviv University. http://www.eng.tau.ac.il/~jo/academic/Q.pdf. 
  11. ^ Weisstein, Eric W.. "Normal Distribution Function". MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/NormalDistributionFunction.html. 
  12. ^ Sanders, Mathijs A.. "Characteristic function of the univariate normal distribution". http://www.planetmathematics.com/CharNormal.pdf. Retrieved 2009-03-06. 
  13. ^ [1]
  14. ^ Amari & Nagaoka 2000
  15. ^ [2]
  16. ^ Huxley (1932)
  17. ^ Whittaker, E. T.; Robinson, G. (1967). The Calculus of Observations: A Treatise on Numerical Mathematics. New York: Dover. p. 179. 
  18. ^ Johnson & Kotz (1995, Equation (26.48))
  19. ^ see Bc programming language#A translated C function

References

External links

The normal distribution

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