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Normal distribution

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

Normal distribution
Probability density function
Probability density function for the normal distribution
The red curve is the standard normal distribution
Cumulative distribution function
Cumulative distribution function for the normal distribution
Notation
Parameters = mean (location)
= variance (squared scale)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF
Fisher information

Kullback-Leibler divergence

The parameter is the mean or expectation of the distribution (and also its median and mode); and is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.[3]

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. Many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

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

Contents

DefinitionsEdit

Standard normal distributionEdit

The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when   and  , and it is described by this probability density function:

 

The factor   in this expression ensures that the total area under the curve   is equal to one.[note 1] The factor   in the exponent ensures that the distribution has unit variance (i.e. the variance is equal to one), and therefore also unit standard deviation. This function is symmetric around  , where it attains its maximum value   and has inflection points at   and  .

Authors differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance  , that is

 

Stigler[4] goes even further, defining the standard normal with variance  :

 

General normal distributionEdit

Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor   (the standard deviation) and then translated by   (the mean value):

 

The probability density must be scaled by   so that the integral is still 1.

If   is a standard normal deviate, then   will have a normal distribution with expected value   and standard deviation  . Conversely, if   is a normal deviate with parameters   and  , then   will have a standard normal distribution. This variate is called the standardized form of  

Every normal distribution is the exponential of a quadratic function:

 

where   and  . In this form, the mean value is  , and the variance is  . For the standard normal distribution,  ,  , and  .

NotationEdit

The probability density of the standard Gaussian distribution (standard normal distribution) (with zero mean and unit variance) is often denoted with the Greek letter   (phi).[5] The alternative form of the Greek letter phi,  , is also used quite often.

The normal distribution is often referred to as   or  .[6] Thus when a random variable   is distributed normally with mean   and variance  , one may write

 

Alternative parameterizationsEdit

Some authors advocate using the precision   as the parameter defining the width of the distribution, instead of the deviation   or the variance  . The precision is normally defined as the reciprocal of the variance,  .[7] The formula for the distribution then becomes

 

This choice is claimed to have advantages in numerical computations when   is very close to zero and simplify formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.

Also the reciprocal of the standard deviation   might be defined as the precision and the expression of the normal distribution becomes

 

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution.

Normal distributions form an exponential family with natural parameters   and  , and natural statistics x and x2. The dual, expectation parameters for normal distribution are η1 = μ and η2 = μ2 + σ2.

Cumulative distribution functionEdit

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

 

The related error function   gives the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range  ; that is

 

These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below.

The two functions are closely related, namely

 

For a generic normal distribution with density  , mean   and deviation  , the cumulative distribution function is

 

The complement of the standard normal CDF,  , is often called the Q-function, especially in engineering texts.[8][9] It gives the probability that the value of a standard normal random variable   will exceed  :  . Other definitions of the  -function, all of which are simple transformations of  , are also used occasionally.[10]

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

 

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

 

where   denotes the double factorial.

An asymptotic expansion of the CDF for large x can also be derived using integration by parts; see Error function#Asymptotic expansion.[11]

Standard deviation and coverageEdit

 
For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.

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

More precisely, the probability that a normal deviate lies in the range between   and   is given by

 

To 12 significant figures, the values for   are:[12]

        OEIS
1 0.682689492137 0.317310507863
3 .15148718753
OEISA178647
2 0.954499736104 0.045500263896
21 .9778945080
OEISA110894
3 0.997300203937 0.002699796063
370 .398347345
OEISA270712
4 0.999936657516 0.000063342484
15787 .1927673
5 0.999999426697 0.000000573303
1744277 .89362
6 0.999999998027 0.000000001973
506797345 .897

Quantile functionEdit

The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:

 

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

 

The quantile   of the standard normal distribution is commonly denoted as  . These values are used in hypothesis testing, construction of confidence intervals and Q-Q plots. A normal random variable   will exceed   with probability  , and will lie outside the interval   with probability  . In particular, the quantile   is 1.96; therefore a normal random variable will lie outside the interval   in only 5% of cases.

The following table gives the quantile   such that   will lie in the range   with a specified probability  . These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:.[13][14] NOTE: the following table shows  , not   as defined above.

         
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

For small  , the quantile function has the useful asymptotic expansion  

PropertiesEdit

The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance.[15][16] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.[17][18]

The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.

The value of the normal distribution is practically zero when the value   lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied.

The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the Cauchy distribution and the Lévy distribution.

Symmetries and derivativesEdit

The normal distribution with density   (mean   and standard deviation  ) has the following properties:

  • It is symmetric around the point   which is at the same time the mode, the median and the mean of the distribution.[19]
  • It is unimodal: its first derivative is positive for   negative for   and zero only at  
  • The area under the curve and over the  -axis is unity (i.e. equal to one).
  • Its density has two inflection points (where the second derivative of   is zero and changes sign), located one standard deviation away from the mean, namely at   and  [19]
  • Its density is log-concave.[19]
  • Its density is infinitely differentiable, indeed supersmooth of order 2.[20]

Furthermore, the density   of the standard normal distribution (i.e.   and  ) also has the following properties:

  • Its first derivative is  
  • Its second derivative is  
  • More generally, its nth derivative is   where   is the nth (probabilist) Hermite polynomial.[21]
  • The probability that a normally distributed variable   with known   and   is in a particular set, can be calculated by using the fact that the fraction   has a standard normal distribution.

MomentsEdit

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

If   has a normal distribution, these moments exist and are finite for any   whose real part is greater than −1. For any non-negative integer  , the plain central moments are:[22]

 

Here   denotes the double factorial, that is, the product of all numbers from   to 1 that have the same parity as  

The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer  

 

The last formula is valid also for any non-integer   When the mean   the plain and absolute moments can be expressed in terms of confluent hypergeometric functions   and  [citation needed]

 
 

These expressions remain valid even if   is not integer. See also generalized Hermite polynomials.

Order Non-central moment Central moment
1    
2    
3    
4    
5    
6    
7    
8    

The expectation of   conditioned on the event that   lies in an interval   is given by

 

where   and   respectively are the density and the cumulative distribution function of  . For   this is known as the inverse Mills ratio. Note that above, density   of   is used instead of standard normal density as in inverse Mills ratio, so here we have   instead of  .

Fourier transform and characteristic functionEdit

The Fourier transform of a normal density   with mean   and standard deviation   is[23]

 

where   is the imaginary unit. If the mean  , the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and standard deviation  . In particular, the standard normal distribution   is an eigenfunction of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable   is closely connected to the characteristic function   of that variable, which is defined as the expected value of  , as a function of the real variable   (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable  .[24] The relation between both is:

 

Moment and cumulant generating functionsEdit

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

 

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

 

Since this is a quadratic polynomial in  , only the first two cumulants are nonzero, namely the mean   and the variance  .


Zero-variance limitEdit

In the limit when   tends to zero, the probability density   eventually tends to zero at any  , but grows without limit if  , while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when  .

However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function"   translated by the mean  , that is   Its CDF is then the Heaviside step function translated by the mean  , namely

 


Maximum entropyEdit

Of all probability distributions over the reals with a specified mean   and variance  , the normal distribution   is the one with maximum entropy.[25] If   is a continuous random variable with probability density  , then the entropy of   is defined as[26][27][28]

 

where   is understood to be zero whenever  . This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified variance, by using variational calculus. A function with two Lagrange multipliers is defined:

 

where   is, for now, regarded as some density function with mean   and standard deviation  .

At maximum entropy, a small variation   about   will produce a variation   about   which is equal to 0:

 

Since this must hold for any small  , the term in brackets must be zero, and solving for   yields:

 

Using the constraint equations to solve for   and   yields the density of the normal distribution:

 

The entropy of normal distribution equals to

 

Operations on normal deviatesEdit

The family of normal distributions is closed under linear transformations: if   is normally distributed with mean   and standard deviation  , then the variable  , for any real numbers   and  , is also normally distributed, with mean   and standard deviation  .

Also if   and   are two independent normal random variables, with means  ,   and standard deviations  ,  , then their sum   will also be normally distributed,[proof] with mean  and variance  .

In particular, if   and   are independent normal deviates with zero mean and variance  , then   and   are also independent and normally distributed, with zero mean and variance  . This is a special case of the polarization identity.[29]

Also, if  ,   are two independent normal deviates with mean   and deviation  , and  ,   are arbitrary real numbers, then the variable

 

is also normally distributed with mean   and deviation  . It follows that the normal distribution is stable (with exponent  ).

More generally, any linear combination of independent normal deviates is a normal deviate.

Infinite divisibility and Cramér's theoremEdit

For any positive integer  , any normal distribution with mean   and variance   is the distribution of the sum of   independent normal deviates, each with mean   and variance  . This property is called infinite divisibility.[30]

Conversely, if   and   are independent random variables and their sum   has a normal distribution, then both   and   must be normal deviates.[31]

This result is known as Cramér’s decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.[32]

Bernstein's theoremEdit

Bernstein's theorem states that if   and   are independent and   and   are also independent, then both X and Y must necessarily have normal distributions.[33][34]

More generally, if  , ...,   are independent random variables, then two distinct linear combinations   and  will be independent if and only if all   are normal and  , where   denotes the variance of  .[33]

Other propertiesEdit

  1. If the characteristic function   of some random variable   is of the form  , where   is a polynomial, then the Marcinkiewicz theorem (named after Józef Marcinkiewicz) asserts that   can be at most a quadratic polynomial, and therefore   is a normal random variable.[32] The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero cumulants.
  2. If   and   are jointly normal and uncorrelated, then they are independent. The requirement that   and   should be jointly normal is essential; without it the property does not hold.[35][36][proof] For non-normal random variables uncorrelatedness does not imply independence.
  3. The Kullback–Leibler divergence of one normal distribution   from another   is given by:[37]
     

    The Hellinger distance between the same distributions is equal to

     
  4. The Fisher information matrix for a normal distribution is diagonal and takes the form
     
  5. The conjugate prior of the mean of a normal distribution is another normal distribution.[38] Specifically, if   are iid   and the prior is  , then the posterior distribution for the estimator of   will be
     
  6. The family of normal distributions not only forms an exponential family (EF), but in fact forms a natural exponential family (NEF) with quadratic variance function (NEF-QVF). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.
  7. In information geometry, the family of normal distributions forms a statistical manifold with constant curvature  . The same family is flat with respect to the (±1)-connections ∇  and ∇ .[39]

Related distributionsEdit

Central limit theoremEdit

 
As the number of discrete events increases, the function begins to resemble a normal distribution
 
Comparison of probability density functions,   for the sum of   fair 6-sided dice to show their convergence to a normal distribution with increasing  , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where   are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance   and   is their mean scaled by  

 

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

The theorem can be extended to variables   that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:

  • The binomial distribution   is approximately normal with mean   and variance   for large   and for   not too close to 0 or 1.
  • The Poisson distribution with parameter   is approximately normal with mean   and variance  , for large values of  .[40]
  • The chi-squared distribution   is approximately normal with mean   and variance  , for large  .
  • The Student's t-distribution   is approximately normal with mean 0 and variance 1 when   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, improvements of the approximation are given by the Edgeworth expansions.

Operations on a single random variableEdit

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

Combination of two independent random variablesEdit

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

  • Their sum and difference is distributed normally with mean zero and variance two:  .
  • Their product   follows the "product-normal" distribution[41] with density function   where   is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at  , and has the characteristic function  .
  • Their ratio follows the standard Cauchy distribution:  .
  • Their Euclidean norm   has the Rayleigh distribution.

Combination of two or more independent random variablesEdit

  • If   are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with   degrees of freedom
 
  • If   are independent normally distributed random variables with means   and variances  , then their sample mean is independent from the sample standard deviation,[42] which can be demonstrated using Basu's theorem or Cochran's theorem.[43] The ratio of these two quantities will have the Student's t-distribution with  degrees of freedom:
 
  • If  ,   are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with ( ,  ) degrees of freedom:[44]
 

Operations on the density functionEdit

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.

ExtensionsEdit

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

  • The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector XRk is multivariate-normally distributed if any linear combination of its components k
    j=1
    aj Xj
    has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
  • Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
  • Complex normal distribution deals with the complex normal vectors. A complex vector XCk is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
  • Matrix normal distribution describes the case of normally distributed matrices.
  • Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element hH is said to be normal if for any constant aH the scalar product (a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
  • Gaussian q-distribution is an abstract mathematical construction that represents a "q-analogue" of the normal distribution.
  • the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. Note that this distribution is different from the Gaussian q-distribution above.

A random variable X has a two-piece normal distribution if it has a distribution

 
 

where μ is the mean and σ1 and σ2 are the standard deviations of the distribution to the left and right of the mean respectively.

The mean, variance and third central moment of this distribution have been determined[45]

 
 
 

where E(X), V(X) and T(X) are the mean, variance, and third central moment respectively.

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

  • Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
  • The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.


Statistical InferenceEdit

Estimation of parametersEdit

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   from a normal   population we would like to learn the approximate values of parameters   and  . The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:

 

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

 

Sample meanEdit

Estimator   is called the sample mean, since it is the arithmetic mean of all observations. The statistic   is complete and sufficient for  , and therefore by the Lehmann–Scheffé theorem,   is the uniformly minimum variance unbiased (UMVU) estimator.[46] In finite samples it is distributed normally:

 

The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix  . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of   is proportional to  , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.

From the standpoint of the asymptotic theory,   is consistent, that is, it converges in probability to   as  . The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:

 

Sample varianceEdit

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

 

The difference between   and   becomes negligibly small for large n's. In finite samples however, the motivation behind the use of   is that it is an unbiased estimator of the underlying parameter  , whereas   is biased. Also, by the Lehmann–Scheffé theorem the estimator   is uniformly minimum variance unbiased (UMVU),[46] which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator   is "better" than the   in terms of the mean squared error (MSE) criterion. In finite samples both   and   have scaled chi-squared distribution with (n − 1) degrees of freedom:

 

The first of these expressions shows that the variance of   is equal to  , which is slightly greater than the σσ-element of the inverse Fisher information matrix  . Thus,   is not an efficient estimator for  , and moreover, since   is UMVU, we can conclude that the finite-sample efficient estimator for   does not exist.

Applying the asymptotic theory, both estimators   and   are consistent, that is they converge in probability to   as the sample size  . The two estimators are also both asymptotically normal:

 

In particular, both estimators are asymptotically efficient for  .

Confidence intervalsEdit

By Cochran's theorem, for normal distributions the sample mean   and the sample variance s2 are independent, which means there can be no gain in considering their joint distribution. There is also a converse 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   and s can be employed to construct the so-called t-statistic:

 

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;[47] similarly, inverting the χ2 distribution of the statistic s2 will give us the confidence interval for σ2:[48]

 
 

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 confidence level 1 − α, meaning that the true values μ and σ2 fall outside of these intervals with probability (or significance level) 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   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.


Normality testsEdit

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. Many 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  . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
    • Shapiro-Wilk 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)
  • Moment tests:
  • Empirical distribution function tests:

Bayesian analysis of the normal distributionEdit

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

  • Either the mean, or the variance, or neither, may be considered a fixed quantity.
  • When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
  • Both univariate and multivariate cases need to be considered.
  • Either conjugate or improper prior distributions may be placed on the unknown variables.
  • An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data, but more complex.

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

Sum of two quadraticsEdit

Scalar formEdit

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

 

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

  1. The factor   has the form of a weighted average of y and z.
  2.   This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that   is one-half the harmonic mean of a and b.
Vector formEdit

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

 

where

 

Note that the form xA x is called a quadratic form and is a scalar:

 

In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since  , only the sum   matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form  

Sum of differences from the meanEdit

Another useful formula is as follows:

 

where  

With known varianceEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows   with known variance σ2, the conjugate prior distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ2. Then if   and   we proceed as follows.

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

 

Then, we proceed as follows:

 

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean   and precision  , i.e.

 

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

 

That is, to combine n data points with total precision of (or equivalently, total variance of n/σ2) and mean of values  , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

 

With known meanEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows   with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ2 is as follows:

 

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

 

where

 

Then:

 

The above is also a scaled inverse chi-squared distribution where

 

or equivalently

 

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

 

With unknown mean and unknown varianceEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows   with unknown mean μ and unknown variance σ2, a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution. Logically, this originates as follows:

  1. From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
  2. From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
  3. Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
  4. To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
  5. This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Note that each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
  6. This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

 

The update equations can be derived, and look as follows:

 

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for   is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new "interaction term" needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

[Proof]

The prior distributions are

 

Therefore, the joint prior is

 

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

 

Writing it in terms of variance rather than precision, we get:

 

where  

Therefore, the posterior is (dropping the hyperparameters as conditioning factors):

 

In other words, the posterior distribution has the form of a product of a normal distribution over p(μ | σ2) times an inverse gamma distribution over p2), with parameters that are the same as the update equations above.

Occurrence and applicationsEdit

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

  1. Exactly normal distributions;
  2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
  3. Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
  4. Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normalityEdit

Certain quantities in physics are distributed normally, as was