Normal distribution

Probability density function

The red line is the standard normal distribution
Cumulative distribution function

Colors match the image above
Notation \mathcal{N}(\mu,\,\sigma^2)
Parameters μ ∈ R — mean (location)
σ2 > 0 — variance (squared scale)
Support x ∈ R
PDF \frac{1}{\sqrt{2\pi\sigma^2}}\,e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
CDF \frac12\left[1 %2B \operatorname{erf}\left( \frac{x-\mu}{\sqrt{2\sigma^2}}\right)\right]
Mean μ
Median μ
Mode μ
Variance σ2
Skewness 0
Ex. kurtosis 0
Entropy \frac12 \ln(2 \pi e \, \sigma^2)
MGF \exp\{ \mu t %2B \frac{1}{2}\sigma^2t^2 \}
CF \exp \{ i\mu t - \frac{1}{2}\sigma^2 t^2 \}
Fisher information \begin{pmatrix}1/\sigma^2&0\\0&1/(2\sigma^4)\end{pmatrix}

In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function or informally the bell curve:[nb 1]

f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }

where parameter μ is the mean or expectation (location of the peak) and σ 2 is the variance, the mean of the squared deviation, (a "measure" of the width of the distribution). σ is the standard deviation. The distribution with μ = 0 and σ 2 = 1 is called the standard normal. A normal distribution is often used as a first approximation to describe real-valued random variables that cluster around a single mean value.

The normal distribution is considered the most prominent probability distribution in statistics. There are several reasons for this:[1] First, the normal distribution is very tractable analytically, that is, a large number of results involving this distribution can be derived in explicit form. Second, the normal distribution arises as the outcome of the central limit theorem, which states that under mild conditions the sum of a large number of random variables is distributed approximately normally. Finally, the "bell" shape of the normal distribution makes it a convenient choice for modelling a large variety of random variables encountered in practice.

For this reason, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural sciences, and social sciences[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. Note that a normally-distributed variable has a symmetric distribution about its mean. Quantities that grow exponentially, such as prices, incomes or populations, are often skewed to the right, and hence may be better described by other distributions, such as the log-normal distribution or Pareto distribution. In addition, the probability of seeing a normally-distributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavy-tailed distribution such as the Student's t-distribution.

From a technical perspective, alternative characterizations are possible, for example:

Contents

Definition

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

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

The factor \scriptstyle\ 1/\sqrt{2\pi} in this expression ensures that the total area under the curve ϕ(x) is equal to one,[proof] and 1/2 in the exponent makes the "width" of the curve (measured as half the distance between the inflection points) also equal to one. It is traditional in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters f or p.[5] The alternative glyph φ is also used quite often, however within this article "φ" is reserved 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 %2B b x %2B c}. \,

This yields the classic "bell curve" shape, provided that a < 0 so that the quadratic function is concave. 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 one to rewrite the probability density function in a convenient standard form,

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

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 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.

The normal distribution is usually denoted by N(μ, σ2).[6] 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). \,

Alternative formulations

Some authors advocate using the precision instead of the variance, and variously define it as τ = σ−2 or τ = σ−1.[7] This parametrization 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 parametrization is in the study of conditional distributions in multivariate normal case.

The question which normal distribution should be called the "standard" one is also answered differently by various authors. Starting from the works of Gauss the standard normal was considered to be the one with variance σ2 = 1/2 :

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

Stigler (1982) goes even further and insists the standard normal to be with the variance σ2 = 1/2π :

f(x) = e^{-\pi x^2}

According to the author, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution. In Stigler's formulation the density of a normal N(μ, τ) with mean μ and precision τ will be equal to

f(x;\,\mu,\tau) = \tau\,e^{-\pi\tau^2(x-\mu)^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 probability density function (pdf) of a random variable describes the relative frequencies of different values for that random variable. The pdf of the normal distribution is given by the formula explained in detail in the previous section:

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

This is a proper function only when the variance σ2 is not equal to zero. In that case this is a continuous smooth function, defined on the entire real line, and which is called the "Gaussian function".

Properties:

When σ2 = 0, the density function doesn't exist. However a generalized function that defines a measure on the real line, and it can be used to calculate, for example, expected value is

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

where δ(x) is the Dirac delta function which is equal to infinity at x = μ and is zero elsewhere.

Cumulative distribution function

The cumulative distribution function (CDF) describes probability of a random variable falling in the interval (−∞, 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) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt = \frac12\left[\, 1 %2B \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\,\right],\quad x\in\mathbb{R}.

This integral cannot be expressed in terms of elementary functions, so is simply called the error function, or erf, a special function. Numerical methods for calculation of the standard normal CDF are discussed below. For a generic normal random variable with mean μ and variance σ2 > 0 the CDF will be equal to

F(x;\,\mu,\sigma^2) = \Phi\left(\frac{x-\mu}{\sigma}\right) = \frac12\left[\, 1 %2B \operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\,\right],\quad x\in\mathbb{R}.

The complement of the standard normal CDF, Q(x) = 1 − Φ(x), is referred to as the Q-function, especially in engineering texts.[11][12] 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. Other definitions of the Q-function, all of which are simple transformations of Φ, are also used occasionally.[13]

Properties:

For a normal distribution with zero variance, the CDF is the Heaviside step function (with H(0) = 1 convention):

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

Quantile function

The inverse of the standard normal CDF, called the quantile function or probit function, is expressed in terms of the inverse error function:

\Phi^{-1}(p) \equiv z_p = \sqrt2\;\operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1).

Quantiles of the standard normal distribution are commonly denoted as zp. The quantile zp represents such a value that a standard normal random variable X has the probability of exactly p to fall inside the (−∞, zp] interval. The quantiles are used in hypothesis testing, construction of confidence intervals and Q-Q plots. The most "famous" normal quantile is 1.96 = z0.975. A standard normal random variable is greater than 1.96 in absolute value in 5% of cases.

For a normal random variable with mean μ and variance σ2, the quantile function is

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

Characteristic function and moment generating 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 a normally distributed X with mean μ and variance σ2, the characteristic function is [14]

\varphi(t;\,\mu,\sigma^2) = \int_{-\infty}^\infty\! e^{itx}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12 (x-\mu)^2/\sigma^2} dx = e^{i\mu t - \frac12 \sigma^2t^2}.

The characteristic function can be analytically extended to the entire complex plane: one defines φ(z) = eiμz − 1/2σ2z2 for all z ∈ C.[15]

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 %2B \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 %2B \frac{1}{2} \sigma^2 t^2.

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

Moments

See also: List of integrals of Gaussian functions

The normal distribution has moments of all orders. That is, for a normally distributed X with mean μ and variance σ 2, 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, ….

Order Raw moment Central moment Cumulant
1 μ 0 μ
2 μ2 + σ2 σ 2 σ 2
3 μ3 + 3μσ2 0 0
4 μ4 + 6μ2σ2 + 3σ4 3σ 4 0
5 μ5 + 10μ3σ2 + 15μσ4 0 0
6 μ6 + 15μ4σ2 + 45μ2σ4 + 15σ6 15σ 6 0
7 μ7 + 21μ5σ2 + 105μ3σ4 + 105μσ6 0 0
8 μ8 + 28μ6σ2 + 210μ4σ4 + 420μ2σ6 + 105σ8 105σ 8 0

Properties

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 %2B \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\left(\frac{x-\mu}{\sigma}\right), \quad f_X(x) = \frac{1}{\sigma}\,\phi\left(\frac{x-\mu}{\sigma}\right).

Standard deviation and confidence intervals

About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 rule, or the empirical rule, or the 3-sigma rule. To be more precise, the area under the bell curve between μ − nσ and μ + nσ is given by

F(\mu%2Bn\sigma;\,\mu,\sigma^2) - F(\mu-n\sigma;\,\mu,\sigma^2) = \Phi(n)-\Phi(-n) = \mathrm{erf}\left(\frac{n}{\sqrt{2}}\right),

where erf is the error function. To 12 decimal places, the values for the 1-, 2-, up to 6-sigma points are:[16]

\scriptstyle\; n\; \;\scriptstyle\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right)\; i.e. 1 minus ... or 1 in ...
1 0.682689492137 0.317310507863 3.15148718753
2 0.954499736104 0.045500263896 21.9778945080
3 0.997300203937 0.002699796063 370.398347345
4 0.999936657516 0.000063342484 15,787.1927673
5 0.999999426697 0.000000573303 1,744,277.89362
6 0.999999998027 0.000000001973 506,797,345.897

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:[17]

\scriptstyle\;\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right)\; n     \scriptstyle\;\mathrm{erf}\left(\frac{n}{\sqrt{2}}\right)\; n
0.80 1.281551565545 0.999 3.290526731492
0.90 1.644853626951 0.9999 3.890591886413
0.95 1.959963984540 0.99999 4.417173413469
0.98 2.326347874041 0.999999 4.891638475699
0.99 2.575829303549 0.9999999 5.326723886384
0.995 2.807033768344 0.99999999 5.730728868236
0.998 3.090232306168 0.999999999 6.109410204869

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.

Central limit theorem

Main article: Central limit theorem

The theorem states that under certain (fairly common) conditions, the sum of a large number of random variables will have an approximately normal distribution. For example if (x1, …, xn) is a sequence of iid random variables, each having mean μ and variance σ2, then the central limit theorem states that

\sqrt{n}\left( \frac{1}{n}\sum_{i=1}^n x_i - \mu \right)\ \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.

Another practical consequence of the central limit theorem is that certain other 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.

Miscellaneous

  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 and b) is also normally distributed:
    aX %2B b\ \sim\ \mathcal{N}(a\mu%2Bb,\, 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 %2B bX_2 \ \sim\ \mathcal{N}(a\mu_1%2Bb\mu_2,\, a^2\!\sigma_1^2 %2B 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.[18] This is known as Cramér's decomposition theorem. The interpretation of this property is that a normal distribution is only divisible by other normal distributions. Another application of this property is in connection with the central limit theorem: although the CLT asserts that the distribution of a sum of arbitrary non-normal iid random variables is approximately normal, the Cramér's theorem shows that it can never become exactly normal.[19]
  3. If the characteristic function φX of some random variable X is of the form φX(t) = eQ(t), where Q(t) is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that Q can be at most a quadratic polynomial, and therefore X a normal random variable.[20] The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero cumulants.
  4. If X and Y are jointly normal and uncorrelated, then they are independent. The requirement that X and Y should be jointly normal is essential, without it the property does not hold.[proof] For non-normal random variables uncorrelatedness does not imply independence.
  5. If X and Y are independent N(μ, σ 2) random variables, then X + Y and X − Y are also independent and identically distributed (this follows from the polarization identity).[21] This property uniquely characterizes normal distribution, as can be seen from the Bernstein's theorem: if X and Y are independent and such that X + Y and X − Y are also independent, then both X and Y must necessarily have normal distributions.
    More generally, if X1, ..., Xn are independent random variables, then two linear combinations ∑akXk and ∑bkXk will be independent if and only if all Xk's are normal and ∑akbkσ 2
    k          = 0    , where σ 2
    k          denotes the variance of Xk.[22]
  6. Normal distribution is infinitely divisible:[23] 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 %2B X_2 %2B \cdots %2B X_n \ \sim\ \mathcal{N}(\mu, \sigma^2)
  7. 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 %2B bX_2 \ \sim\ \sqrt{a^2%2Bb^2}\cdot X_3\ %2B\ (a%2Bb-\sqrt{a^2%2Bb^2})\mu,
    where X3 is also N(μ, σ2). This relationship directly follows from property (1).
  8. The Kullback–Leibler divergence between two normal distributions X1 ∼ N(μ1, σ21 )and X2 ∼ N(μ2, σ22 )is given by:[24]
    D_\mathrm{KL}( X_1 \,\|\, X_2 ) = \frac{(\mu_1 - \mu_2)^2}{2\sigma_2^2} \,%2B\, \frac12\left(\, \frac{\sigma_1^2}{\sigma_2^2} - 1 - \ln\frac{\sigma_1^2}{\sigma_2^2} \,\right)\ .
    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%2B\sigma_2^2}} \; e^{-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2%2B\sigma_2^2}}\ .
  9. 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}
  10. 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.
  11. The conjugate prior of the mean of a normal distribution is another normal distribution.[25] Specifically, if x1, …, xn are iid N(μ, σ2) and the prior is μ ~ N(μ0, σ2
    0    )    , then the posterior distribution for the estimator of μ will be
    \mu | x_1,\ldots,x_n\ \sim\ \mathcal{N}\left( \frac{\frac{\sigma^2}{n}\mu_0 %2B \sigma_0^2\bar{x}}{\frac{\sigma^2}{n}%2B\sigma_0^2},\ \left( \frac{n}{\sigma^2} %2B \frac{1}{\sigma_0^2} \right)^{\!-1} \right)
  12. 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.[26]
  13. 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).[27]

Related distributions

Operations on a single random variable

If X is distributed normally with mean μ and variance σ2, then

Combination of two independent random variables

If X1 and X2 are two independent standard normal random variables, then

Combination of two or more independent random variables

t = \frac{\overline X - \mu}{S/\sqrt{n}} = \frac{\frac{1}{n}(X_1%2B\cdots%2BX_n) - \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_1-\overline X)^2%2B\cdots%2B(X_n-\overline X)^2\right]}} \ \sim\ t_{n-1}.
F = \frac{\left(X_1^2%2BX_2^2%2B\cdots%2BX_n^2\right)/n}{\left(Y_1^2%2BY_2^2%2B\cdots%2BY_m^2\right)/m}\ \sim\ F_{n,\,m}.

Operations on the density function

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.

Extensions

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

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

Normality tests

Main article: 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:

Estimation of parameters

It is often the case that we don't know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x1, …, xn) from a normal N(μ, σ2) population we would like to learn the approximate values of parameters μ and σ2. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:

\ln\mathcal{L}(\mu,\sigma^2) = \sum_{i=1}^n \ln f(x_i;\,\mu,\sigma^2) = -\frac{n}{2}\ln(2\pi) - \frac{n}{2}\ln\sigma^2 - \frac{1}{2\sigma^2}\sum_{i=1}^n (x_i-\mu)^2.

Taking derivatives with respect to μ and σ2 and solving the resulting system of first order conditions yields the maximum likelihood estimates:

\hat{\mu} = \overline{x} \equiv \frac{1}{n}\sum_{i=1}^n x_i, \qquad \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2.

Estimator \scriptstyle\hat\mu is called the sample mean, since it is the arithmetic mean of all observations. The statistic \scriptstyle\overline{x} is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, \scriptstyle\hat\mu is the uniformly minimum variance unbiased (UMVU) estimator.[29] In finite samples it is distributed normally:

\hat\mu \ \sim\ \mathcal{N}(\mu,\,\,\sigma^2\!\!\;/n).

The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix \scriptstyle\mathcal{I}^{-1}. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of \scriptstyle\hat\mu is proportional to \scriptstyle1/\sqrt{n}, 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, \scriptstyle\hat\mu is consistent, that is, it converges in probability to μ as n → ∞. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:

\sqrt{n}(\hat\mu-\mu) \ \xrightarrow{d}\ \mathcal{N}(0,\,\sigma^2).

The estimator \scriptstyle\hat\sigma^2 is called the sample variance, since it is the variance of the sample (x1, …, xn). In practice, another estimator is often used instead of the \scriptstyle\hat\sigma^2. This other estimator is denoted s2, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. The estimator s2 differs from \scriptstyle\hat\sigma^2 by having (n − 1) instead of n in the denominator (the so called Bessel's correction):

s^2 = \frac{n}{n-1}\,\hat\sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2.

The difference between s2 and \scriptstyle\hat\sigma^2 becomes negligibly small for large n's. In finite samples however, the motivation behind the use of s2 is that it is an unbiased estimator of the underlying parameter σ2, whereas \scriptstyle\hat\sigma^2 is biased. Also, by the Lehmann–Scheffé theorem the estimator s2 is uniformly minimum variance unbiased (UMVU),[29] which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator \scriptstyle\hat\sigma^2 is "better" than the s2 in terms of the mean squared error (MSE) criterion. In finite samples both s2 and \scriptstyle\hat\sigma^2 have scaled chi-squared distribution with (n − 1) degrees of freedom:

s^2 \ \sim\ \frac{\sigma^2}{n-1} \cdot \chi^2_{n-1}, \qquad \hat\sigma^2 \ \sim\ \frac{\sigma^2}{n} \cdot \chi^2_{n-1}\ .

The first of these expressions shows that the variance of s2 is equal to 2σ4/(n−1), which is slightly greater than the σσ-element of the inverse Fisher information matrix \scriptstyle\mathcal{I}^{-1}. Thus, s2 is not an efficient estimator for σ2, and moreover, since s2 is UMVU, we can conclude that the finite-sample efficient estimator for σ2 does not exist.

Applying the asymptotic theory, both estimators s2 and \scriptstyle\hat\sigma^2 are consistent, that is they converge in probability to σ2 as the sample size n → ∞. The two estimators are also both asymptotically normal:

\sqrt{n}(\hat\sigma^2 - \sigma^2) \simeq \sqrt{n}(s^2-\sigma^2)\ \xrightarrow{d}\ \mathcal{N}(0,\,2\sigma^4).

In particular, both estimators are asymptotically efficient for σ2.

By Cochran's theorem, for normal distribution the sample mean \scriptstyle\hat\mu and the sample variance s2 are independent, which means there can be no gain in considering their joint distribution. There is also a reverse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between \scriptstyle\hat\mu and s can be employed to construct the so-called t-statistic:

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}

This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;[30] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[31]

\begin{align} & \mu \in \left[\, \hat\mu %2B t_{n-1,\alpha/2}\, \frac{1}{\sqrt{n}}s,\ \ \hat\mu %2B t_{n-1,1-\alpha/2}\,\frac{1}{\sqrt{n}}s \,\right] \approx \left[\, \hat\mu - |z_{\alpha/2}|\frac{1}{\sqrt n}s,\ \ \hat\mu %2B |z_{\alpha/2}|\frac{1}{\sqrt n}s \,\right], \\ & \sigma^2 \in \left[\, \frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}},\ \ \frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}} \,\right] \approx \left[\, s^2 - |z_{\alpha/2}|\frac{\sqrt{2}}{\sqrt{n}}s^2,\ \ s^2 %2B |z_{\alpha/2}|\frac{\sqrt{2}}{\sqrt{n}}s^2 \,\right], \end{align}

where tk,p and χ 2
k,p  are the pth quantiles of the t- and χ2-distributions respectively. These confidence intervals are of the level 1 − α, meaning that the true values μ and σ2 fall outside of these intervals with probability α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The approximate formulas in the display above were derived from the asymptotic distributions of \scriptstyle\hat\mu and s2. The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles zα/2 do not depend on n. In particular, the most popular value of α = 5%, results in |z0.025| = 1.96.

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 the distribution with maximum entropy for a given mean and variance.

Exact normality

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

Approximate normality

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by 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

“ I can only recognize the occurrence of the normal curve — the Laplacian curve of errors — as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations. — Pearson (1901) ”

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

Generating values from normal distribution

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ2
) can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.

Numerical approximations for the normal CDF

The standard normal CDF is widely used in scientific and statistical computing. The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.

History

Development

Some authors[37][38] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 [nb 3] published in the second edition of his "The Doctrine of Chances" the study of the coefficients in the binomial expansion of (a + b)n. De Moivre proved that the middle term in this expansion has the approximate magnitude of \scriptstyle 2/\sqrt{2\pi n}, and that "If m or ½n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is \scriptstyle -\frac{2\ell\ell}{n}."[39] 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.[40]

In 1809 Gauss published his monograph "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, … to denote the measurements of some unknown quantity V, and sought the "most probable" estimator: the one which maximizes the probability φ(M−V) · φ(M′−V) · φ(M′′−V) · … of obtaining the observed experimental results. In his notation φΔ is the probability law of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.[nb 4] Starting from these principles, Gauss demonstrates that the only law which rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:[41]

\varphi\mathit{\Delta} = \frac{h}{\surd\pi}\, e^{-\mathrm{hh}\Delta\Delta},

where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares (NWLS) method.[42]

Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.[nb 5] It was Laplace who first posed the problem of aggregating several observations in 1774,[43] although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e−t ²dt = √π in 1782, providing the normalization constant for the normal distribution.[44] 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.[45]

It is of interest to note that in 1809 an American mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss.[46] His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.[47]

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:[48] "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

Naming

Since its introduction, the normal distribution has been known by many different names: the law of error, the law of facility of errors, Laplace's second law, Gaussian law, etc. By the end of the 19th century some authors[nb 6] had started using the name normal distribution, where the word "normal" was used as an adjective — the term was derived from the fact that this distribution was seen as typical, common, 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."[49] Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.[50]

“ Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. — Pearson (1920) ”

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

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

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".[51]

When the name is used, the "Gaussian distribution" was named after Carl Friedrich Gauss, who introduced the distribution in 1809 as a way of rationalizing the method of least squares as outlined above. The related work of Laplace, also outlined above has led to the normal distribution being sometimes called Laplacian, especially in French-speaking countries. Among English speakers, both "normal distribution" and "Gaussian distribution" are in common use, with different terms preferred by different communities.

See also

Notes

  1. ^ The designation "bell curve" is ambiguous: there are many other distributions which are "bell"-shaped: the Cauchy distribution, Student's t-distribution, generalized normal, logistic, etc.
  2. ^ For example, this algorithm is given in the article Bc programming language.
  3. ^ De Moivre first published his findings in 1733, in a pamphlet "Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem Expansi" that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example Walker (1985).
  4. ^ "It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — Gauss (1809, section 177)
  5. ^ "My custom of terming the curve the Gauss–Laplacian or normal curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from Pearson (1905, p. 189)
  6. ^ Besides those specifically referenced here, such use is encountered in the works of Peirce, Galton and Lexis approximately around 1875.

Citations

  1. ^ Casella & Berger (2001, p. 102)
  2. ^ Gale Encyclopedia of Psychology — Normal Distribution
  3. ^ Cover, T. M.; Thomas, Joy A (2006). Elements of information theory. John Wiley and Sons. p. 254. 
  4. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf. Retrieved 2011-06-02. 
  5. ^ Halperin & et al. (1965, item 7)
  6. ^ McPherson (1990, p. 110)
  7. ^ Bernardo & Smith (2000)
  8. ^ a b c Patel & Read (1996, [2.1.4])
  9. ^ Fan (1991, p. 1258)
  10. ^ Patel & Read (1996, [2.1.8])
  11. ^ Scott, Clayton; Robert Nowak (August 7, 2003). "The Q-function". Connexions. http://cnx.org/content/m11537/1.2/. 
  12. ^ Barak, Ohad (April 6, 2006). "Q function and error function". Tel Aviv University. http://www.eng.tau.ac.il/~jo/academic/Q.pdf. 
  13. ^ Weisstein, Eric W., "Normal Distribution Function" from MathWorld.
  14. ^ Bryc (1995, p. 23)
  15. ^ Bryc (1995, p. 24)
  16. ^ WolframAlpha.com
  17. ^ part 1, part 2
  18. ^ Galambos & Simonelli (2004, Theorem 3.5)
  19. ^ Bryc (2995, p. 35)
  20. ^ Bryc (1995, p. 35)
  21. ^ Bryc (1995, p. 27)
  22. ^ Lukacs & King (1954)
  23. ^ Patel & Read (1996, [2.3.6])
  24. ^ http://www.allisons.org/ll/MML/KL/Normal/
  25. ^ "Stat260: Bayesian Modeling and Inference Lecture Date: February 8th, 2010, The Conjugate Prior for the Normal Distribution, Lecturer: Michael I. Jordan|". http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf. 
  26. ^ Cover & Thomas (2006, p. 254)
  27. ^ Amari & Nagaoka (2000)
  28. ^ Mathworld entry for Normal Product Distribution
  29. ^ a b Krishnamoorthy (2006, p. 127)
  30. ^ Krishnamoorthy (2006, p. 130)
  31. ^ Krishnamoorthy (2006, p. 133)
  32. ^ Huxley (1932)
  33. ^ Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press. pp. 592–593. 
  34. ^ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90 70754 3 39. http://www.waterlog.info/pdf/freqtxt.pdf. 
  35. ^ Johnson et al. (1995, Equation (26.48))
  36. ^ Kinderman & Monahan (1976)
  37. ^ Johnson et al. (1994, page 85)
  38. ^ Le Cam (2000, p. 74)
  39. ^ De Moivre (1733), Corollary I — see Walker (1985, p. 77)
  40. ^ Stigler (1986, p. 76)
  41. ^ Gauss (1809, section 177)
  42. ^ Gauss (1809, section 179)
  43. ^ Laplace (1774, Problem III)
  44. ^ Pearson (1905, p. 189)
  45. ^ Stigler (1986, p. 144)
  46. ^ Stigler (1978, p. 243)
  47. ^ Stigler (1978, p. 244)
  48. ^ Maxwell (1860), p. 23
  49. ^ Peirce, C. S. (c. 1909 MS), Collected Papers v. 6, paragraph 327.
  50. ^ Kruskal & Stigler (1997)
  51. ^ "Earliest uses… (entry STANDARD NORMAL CURVE)". http://jeff560.tripod.com/s.html. 

References

External links

Continuous
Discrete