Normal distribution

normally distributedGaussian distributionnormalGaussianstandard normal distributionbell curvestandard normalnormalitynormal curveGaussian random variable
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.wikipedia
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Central limit theorem

Lyapunov's central limit theoremlimit theoremscentral limit
The normal distribution is useful because of the central limit theorem.
In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.

Student's t-distribution

Student's ''t''-distributiont-distributiont''-distribution
However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

Standard normal deviate

standard normal variablestandard-normal
If Z is a standard normal deviate, then will have a normal distribution with expected value \mu and standard deviation \sigma.
A standard normal deviate is a normally distributed deviate.

Logistic distribution

logisticbell-shaped curvelogistical
However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
It resembles the normal distribution in shape but has heavier tails (higher kurtosis).

Mode (statistics)

The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

Cauchy distribution

LorentzianCauchyLorentzian distribution
However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).
It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

Exponential family

exponential familiesnatural parameternatural parameters
Normal distributions form an exponential family with natural parameters and, and natural statistics x and x 2.
The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.

68–95–99.7 rule

3-sigma68-95-99.7 rulethree sigma rule
This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.
a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively; more accurately, 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, respectively.

Error function

complementary error functionerfcomplementary Gaussian error function
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 [-x, x]; that is The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
In statistics, for nonnegative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and variance 1/2, erf(x) describes the probability of Y falling in the range [−x, x].

Propagation of uncertainty

error propagationtheory of errorspropagation of error
Moreover, 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.
For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation

Log-normal distribution

lognormallog-normallognormal distribution
Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.


probit functioninverse distribution function of a standard normal distribution
The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution, which is commonly denoted as N(0,1).

Standard deviation

standard deviationssample standard deviationSD
The value of the normal distribution is practically zero when the value x 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).
This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches (7.62 cm) of the mean (67–73 inches (170.18–185.42 cm)) – one standard deviation – and almost all men (about 95%) have a height within 6 inches (15.24 cm) of the mean (64–76 inches (162.56–193.04 cm)) – two standard deviations.


In particular, the quantile z_{0.975} is 1.96; therefore a normal random variable will lie outside the interval in only 5% of cases.
In probability and statistics, 1.96 is the approximate value of the 97.5 percentile point of the normal distribution.


Q factor (digital communications)
The complement of the standard normal CDF, is often called the Q-function, especially in engineering texts.
In statistics, the Q-function is the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|tail distribution function]] of the standard normal distribution.

Statistical hypothesis testing

hypothesis testingstatistical teststatistical tests
These values are used in hypothesis testing, construction of confidence intervals and Q-Q plots.


averagesample medianmedian-unbiased estimator
For a sample of size N = 2n + 1 from the normal distribution, the efficiency for large N is


ΦPhi (letter)Φ φ
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 (phi). The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter \Phi (phi), is the integral


outliersstatistical outliersconservative estimate
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 the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter \Phi (phi), is the integral
This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution.

Stable distribution

stable distributionsGeneralized Central Limit TheoremLévy alpha-stable distribution
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.
Stable distributions have 0 < α ≤ 2, with the upper bound corresponding to the normal distribution, and α = 1 to the Cauchy distribution.


unimodalunimodal distributionunimodal function
Figure 1 illustrates normal distributions, which are unimodal.

Fourier transform

continuous Fourier transformFourierFourier transforms
The Fourier transform of a normal density f with mean \mu and standard deviation \sigma is
The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion).

Probability density function

probability densitydensity functiondensity
The probability density of the normal distribution is
The standard normal distribution has probability density


cumulant generating functioncumulant-generating functioncumulants
The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero.
As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.