In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is and give this definition with slightly different notation.
The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is the variance. The standard deviation of the distribution is the positive value (sigma). A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.Normal Distribution, Gale Encyclopedia of Psychology Their importance is partly due to the central limit theorem. It states that the average of many statistically independent samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.Lyon, A. (2014). Why are Normal Distributions Normal?, The British Journal for the Philosophy of Science.
Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.
However, normal distributions are frequently misused in contexts where the assumption that the data are normally distributed is not met and the normal distribution is a poor model.
A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's , and logistic distributions). (For other names, see Naming.)
The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.
Definitions Standard normal distribution The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when and , and it is described by this probability density function (or density): explicitly defines the standard normal distribution. In contrast, explicitly defines the standard normal curve (p. 33) and introduces the term standard normal distribution (p. 69).
The variable has a mean of 0 and a variance and standard deviation of 1. The density has its peak value at and inflection points at and .
Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss, for example, once defined the standard normal as which has a variance of , and Stephen Stigler once defined the standard normal as which has a simple functional form and a variance of
General normal distribution If is a standard normal deviate, then will have a normal distribution with expected value and standard deviation . This is equivalent to saying that the standard normal distribution can be scaled/stretched by a factor of and shifted by to yield a different normal distribution, called .
Conversely, if is a normal deviate with parameters and , then this distribution can be re-scaled and shifted via the formula to convert it to the standard normal distribution. This variate is also called the standardized form of .
In particular, the probability density function for can be written in terms of the standard normal distribution (with zero mean and unit variance):
The probability density must be scaled by so that the integral is still 1.
Notation The probability density function of the standard normal distribution is commonly denoted by the Greek letter phi, . The variant form is also used.
The cumulative distribution function of the standard normal distribution is commonly denoted by the capital Greek letter phi, .
The normal distribution is often referred to as or . When a random variable is normally distributed with mean and standard deviation , one may write
Alternative parameterizations Some authors advocate using the precision as the parameter defining the width of the distribution, instead of the standard deviation or the variance . The precision is normally defined as the reciprocal of the variance, . The formula for the distribution then becomes
This choice is claimed to have advantages in numerical computations when is very close to zero, and simplifies formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.
Alternatively, the reciprocal of the standard deviation might be defined as the precision, in which case 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 and . The dual expectation parameters for normal distribution are and .
Cumulative distribution function The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter , 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 for more.
The two functions are closely related, namely
For a generic normal distribution with density , mean and variance , the cumulative distribution function is
The probability that lies between and with is therefore
The complement of the standard normal cumulative distribution function, , is often called the Q-function, especially in engineering texts. It gives the probability that the value of a standard normal random variable will exceed : . Other definitions of the -function, all of which are simple transformations of , are also used occasionally.
The graph of the standard normal cumulative distribution function has 2-fold rotational symmetry around the point (0,1/2); that is, . Its antiderivative (indefinite integral) can be expressed as follows:
An asymptotic expansion of the cumulative distribution function for large can be derived using integration by parts:
where denotes the double factorial. For more, see .
Taylor series representation The Taylor series for the normal distribution can be derived by substituting into the Taylor series for the exponential function:
This series can be integrated term by term to obtain the Taylor series for the cumulative distribution function:
However, this series is ineffective for calculation due to slow convergence, except when is small.
Both of these series describe entire functions, which converge for all real and complex values of .
Recursive computation with Taylor series The recurrence relation for Hermite polynomials may be used to efficiently construct the Taylor series expansion about any point :
where:
Standard deviation and coverage
About 68% of values drawn from a normal distribution are within one standard deviation from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
More precisely, the probability that a normal deviate lies in the range between and is given by
To 12 significant digits, the values for are:
OEIS1 2 3 4 5 6
For large , one can use the approximation
Quantile function
The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
For a normal random variable with mean 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. The following table shows , not as defined above.
0.80 0.999 0.90 0.9999 0.95 0.99999 0.98 0.999999 0.99 0.9999999 0.995 0.99999999 0.998 0.999999999
For small , the quantile function has the useful asymptotic expansion
Using root finding to compute the quantile function Any of the described approaches for computing the cumulative distribution function can be used with Newton's method (or another root-finding algorithm such as Halley's method) to find the value of for which for some desired quantile . For example, starting with an initial, approximately correct guess , increasingly better approximations , , ... can be calculated iteratively using Newton's method with
Properties The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance. Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.Geary RC (1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184
The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share of stock. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
The value of the normal density is practically zero when the value lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and appropriate robust statistical inference methods applied.