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Probability density function The red line is the standard normal distribution
Cumulative distribution function Colors match the image above
notation:
parameters:	μ ∈ R — mean (location) σ2 ≥ 0 — variance (squared scale)
support:	x ∈ R   if σ2 > 0 x = μ   if σ2 = 0
pdf:
cdf:
mean:	μ
median:	μ
mode:	μ
variance:	σ2
skewness:	0
kurtosis:	0
entropy:
mgf:
cf:
Fisher information:

History

.

The bean machine is a device invented by Sir Francis Galton to demonstrate how the normal distribution appears in nature.^ We will now learn how to find the probability that a score will fall between any two values on a normal probability distribution.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

^ On the right hand side, upper panel, a normal distribution will appear with the mu and sigma you have set.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

^ Sigma (standard deviation) is a measure, or determiner, of how spread out the normal distribution is.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

Definition

.The constant 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 of the distance between the inflection points of the curve) also equal to one.^ Again, we are creating a correspondence between the idea of probability and the area under a curve.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

^ Total Area under the Normal curve .

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

^ The normal curve has two inflection points.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

.It is traditional^[7] in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters ƒ or p.^ One is symbolized by the Greek letter mu; and the other is symbolized by the Greek letter sigma.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

^ The second parameter of the normal distribution is what's called its standard deviation, and it is symbolized by the small Greek letter sigma.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

The alternative glyph φ is also used quite often, however within this article we reserve “φ” to denote characteristic functions.

.This yields the classic “bell curve” shape (provided that a < 0 so that the quadratic function is concave).^ The most generally used bell shaped curve is called the normal probability distribution.

• Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES]

Notice that f(x) > 0 everywhere. One can adjust a to control the “width” of the bell, then adjust b to move the central peak of the bell along the x-axis, and finally adjust c to control the “height” of the bell. For f(x) to be a true probability density function over R, one must choose c such that (which is only possible when a < 0).

Characterization

Probability density function

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

Cumulative distribution function

See also: Error function, Q-function, and Standard normal table

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

Characteristic function

Moment generating function

Properties

1. The family of normal distributions is closed under linear transformations. That is, if X is normally distributed with mean μ and variance σ², then a linear transform aX + b (for some real numbers a ≠ 0 and b) is also normally distributed:

   

   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]

   

2. The converse of (1) is also true: if X₁ and X₂ are independent and their sum X₁ + X₂ is distributed normally, then both X₁ and X₂ must also be normal. This is known as the Cramér’s theorem.
3. Normal distribution is infinitely divisible: for a normally distributed X with mean μ and variance σ² we can find n independent random variables {X₁, …, X_n} each distributed normally with means μ/n and variances σ²/n such that

   

4. Normal distribution is stable (with exponent α = 2): if X₁, X₂ are two independent N(μ, σ²) random variables and a, b are arbitrary real numbers, then

   

   where X₃ is also N(μ, σ²). This relationship directly follows from property (1).
5. The Kullback–Leibler divergence between two normal distributions X₁ ∼ N(μ₁, σ²₁ )and X₂ ∼ N(μ₂, σ²₂ )is given by:^[13]

   

   The Hellinger distance between the same distributions is equal to

   

6. The Fisher information matrix for normal distribution is diagonal and takes form

   

7. Normal distributions belongs to 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.
8. Of all probability distributions over the reals with mean μ and variance σ², the normal distribution N(μ, σ²) is the one with the maximum entropy.
9. The family of normal distributions forms a manifold with constant curvature −1. The same family is flat with respect to the (±1)-connections ∇^(e) and ∇^(m).^[14]

Standardizing normal random variables

Moments

• Central moments are the moments of X around its mean μ. Thus, central moment of order p is the expected value of (X − μ)^p. Using standardization of normal distribution, this expectation will be equal to σ^p·E[Z^p], where Z is standard normal.

  

  Here n!! denotes the double factorial, that is the product of every other number from n to 1; and 1_{…} is the indicator function.
• Central absolute moments are the moments of |X − μ|. They coincide with regular moments for all even orders, but are nonzero for all odd p’s.

  

• Raw moments and raw absolute moments are the moments of X and |X| respectively. The formulas for these moments are much more complicated, and are given in terms of confluent hypergeometric functions ₁F₁ and U.

  

  These expressions remain valid even if p is not integer.
• First two cumulants are equal to μ and σ² respectively, whereas all higher-order cumulants are equal to zero.

Order	Raw moment	Central moment	Cumulant
1		0
2
3		0	0
4			0
5		0	0
6			0
7		0	0
8			0

Central limit theorem

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

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

Standard deviation and confidence intervals

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

1	0.682689492137
2	0.954499736104
3	0.997300203937
4	0.999936657516
5	0.999999426697
6	0.999999998027

0.80	1.281551565545
0.90	1.644853626951
0.95	1.959963984540
0.98	2.326347874041
0.99	2.575829303549
0.995	2.807033768344
0.998	3.090232306168
0.999	3.290526731492
0.9999	3.890591886413
0.99999	4.417173413469

Related and derived distributions

• If X is distributed normally with mean μ and variance σ², then
  • The exponent of X is distributed log-normally: e^X ~ lnN(μ, σ²).
  • The absolute value of X has folded normal distribution: IXI ~ N_f (μ, σ²).
  • The square of X, scaled down by the variance σ², has the non-central chi-square distribution with 1 degree of freedom: X²/σ² ~ χ²₁(μ). If μ = 0, the distribution is called simply chi-square.
  • Variable X restricted to an interval [a, b] is called the truncated normal distribution.
  • (X − μ)⁻² has a Lévy distribution with location 0 and scale σ⁻².
• If X₁ and X₂ are two independent standard normal random variables, then
  • Their sum and difference is distributed normally with mean zero and variance two: X₁ ± X₂ ∼ N(0, 2).
  • Their product Z = X₁ · X₂ follows an (unnamed?) distribution with density function ^[15]

    

    where K₀ is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at z = 0, and has the characteristic function φ_Z(t) = (1 + t²)^−1/2.
  • Their ratio follows the standard Cauchy distribution: X₁ ÷ X₂ ∼ Cauchy(0, 1).
  • Their Euclidean norm has the Rayleigh distribution (also known as chi distribution with 2 degrees of freedom):

    

• If X₁, X₂, …, X_n are independent standard normal random variables, then the sum of their squares has the chi-square distribution with n degrees of freedom: .
• If X₁, X₂, …, X_n are independent normally distributed random variables with means μ and variances σ², then their sample mean is independent from the sample standard deviation, which can be demonstrated using the Basu’s theorem or Cochran’s theorem. The ratio of these two quantities will have the Student’s t-distribution with n − 1 degrees of freedom:

  

• If X₁, …, X_n, Y₁, …, Y_m are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom:

  

Descriptive and inferential statistics

Scores

Normality tests

• “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(p_k), x_(k)), where plotting points p_k are equal to p_k = (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)), p_k), where . For normally distributed data this plot should lie on a 45° line between (0,0) and (1,1).
  • Wilk–Shapiro test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
  • Normal probability plot (rankit plot)

• Moment tests:
  • D’Agostino’s K-squared test
  • Jarque–Bera test

• Empirical distribution function tests:
  • Kolmogorov–Smirnov test
  • Lilliefors test
  • Anderson–Darling test

Estimation of parameters

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

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

• is the uniformly minimum variance unbiased (UMVU) estimator for μ, by the Lehmann–Scheffé theorem.
• is a consistent estimator of μ, that is converges in probability to μ as n → ∞.
• has normal final sample distribution:

  

  which implies that the standard error of is equal to , that is if one wishes to decrease the standard error by a factor of 10, one must increase the number of samples by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and number of trials in Monte Carlo simulation.
• is a biased estimator of σ², whereas s² is unbiased. On the other hand, is a superior estimator in terms of the mean squared error (MSE) criterion.
• is a consistent and asymptotically normal estimator:

  

• has a distribution proportional to chi-squared in finite sample:

  

• is independent from , by Cochran’s theorem. The normal distribution is the only distribution whose sample mean and sample variance are independent.
• The ratio

  

  has Student’s t-distribution. This t-statistic is ancillary, and is used for testing the hypothesis H₀:μ = μ₀ and in construction of confidence intervals.
• The 1−α confidence intervals for μ and σ² are:

  

  where q^… denotes the quantile function. For large n it is possible to replace the quantiles of t- and χ²-distributions with the normal quantiles. For example, the approximate 95% confidence intervals will be given by

  

  where 1.96 is the 97.5%-th quantile of the standard normal distribution.

Occurrence

1. Exactly normal distributions;
2. Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
3. Distributions modeled as normal — the normal distribution being one of the simplest and most convenient to use, frequently researchers are tempted to assume that certain quantity is distributed normally, without justifying such assumption rigorously. In fact, the maturity of a scientific field can be judged by the prevalence of the normality assumption in its methods.^{[citation needed]}

Exact normality

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

Approximate normality

• In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as
  • Binomial random variables, associated with |binary response variables;
  • Poisson random variables, associated with rare events;
• Thermal light has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normality

• In biology:
  • The logarithm of measures of size of living tissue (length, height, skin area, weight)^[16];
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
  • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).
• In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoît Mandelbrot argue that log-Levy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes.
• Measurement errors in physical experiments are often assumed to be normally distributed. This assumption allows for particularly simple practical rules for how to combine errors in measurements of different quantities. However, whether this assumption is valid or not in practice is debatable. A famous remark of Lippmann says: “Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics.” ^[17]
• In standardized testing, results can be made to have a normal distribution. This is done by either selecting the number and difficulty of questions (as in the IQ test), or by transforming the raw test scores into “output” scores by fitting them to the normal distribution. For example, the SAT’s traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

Generating values from normal distribution

• The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ⁻¹(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in Hart (1968) and in the erf article.
• A simple approximate approach that is easy to program is as follows: simply sum 12 uniform (0,1) deviates and subtract 6 — the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).^[18]
• The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1]. Then two random variables X and Y

  

  will both have the standard normal distribution, and be independent. This formulation arises because for a bivariate normal random vector (X Y) the squared norm X² + Y² will have the chi-square distribution with two degrees of freedom, which is an easily-generated exponential random variable corresponding to the quantity −2ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.
• Marsaglia polar method is a modification of the Box–Muller method algorithm, which does not require computation of functions sin() and cos(). In this method U and V are drawn from the uniform (−1,1) distribution, and then S = U² + V² is computed. If S is greater or equal to one then the method starts over, otherwise two quantities

  

  are returned. Again, X and Y here will be independent and standard normally distributed.
• The ziggurat algorithm (Marsaglia & Tsang 2000) is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases where the combination of those two falls outside the “core of the ziggurat” a kind of rejection sampling using logarithms, exponentials and more uniform random numbers has to be employed.
• There is also some investigation into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally-distributed data.

Numerical approximations of the normal cdf

• Abramowitz & Stegun (1964) give the approximation for Φ(x) with the absolute error |ε(x)| < 7.5·10⁻⁸ (algorithm 26.2.17):

  

  where ϕ(x) is the standard normal pdf, and b₀ = 0.2316419, b₁ = 0.319381530, b₂ = −0.356563782, b₃ = 1.781477937, b₄ = −1.821255978, b₅ = 1.330274429.
• Hart (1968) lists almost a hundred of rational function approximations for the erfc() function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by West (2009) combines Hart’s algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
• Marsaglia (2004) suggested a simple algorithm^[19] based on the Taylor series expansion

  

  for calculating Φ(x) with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when x = 10).
• The GNU Scientific Library calculates values of the standard normal cdf using Hart’s algorithms and approximations with Chebyshev polynomials.

Software for calculation of normal distribution

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

See also

Statistics portal

• Multivariate normal distribution — normally distributed vectors
• Complex normal distribution — complex normal random variables and vectors
• Matrix normal distribution — normally distributed matrices
• Gaussian process — normal stochastic processes
  • Wiener process
  • Brownian bridge
  • Ornstein-Uhlenbeck process
• Gaussian q-distribution
• Pearson distribution — generalized family of probability distributions that extend the Gaussian distribution to include different skewness and kurtosis values
• Wrapped normal distribution

• Behrens–Fisher problem
• Bell curve grading
• Central limit theorem - average of a sufficiently large number of identically distributed independent random variables each with finite mean and variance will be approximately normally distributed
• Data transformation (statistics) - simple techniques to transform data into normal distribution
• Erdős-Kac theorem, on the occurrence of the normal distribution in number theory
• Gaussian blur, convolution using the normal distribution as a kernel
• Gaussian function
• Normally distributed and uncorrelated does not imply independent (an example of two normally distributed uncorrelated random variables that are not independent; this cannot happen in the presence of joint normality)
• Probit function
• Standard normal table
• Sum of normally distributed random variables

Notes

References

External links

• Mathworld: Normal Distribution
• Intuitive derivation.
• Is normal distribution due to Karl Gauss? Euler, his family of gamma functions, and place in history of statistics
• Maxwell demons: Simulating probability distributions with functions of propositional calculus
• Visualization of normal distribution
• XLL Excel Addin function for Normal Dist. Random Number Generator

• Normal distribution table
• Public Domain Normal Distribution Table
• Distribution Calculator – Calculates probabilities and critical values for normal, t, chi-square and F-distribution.
• Java Applet on Normal Distributions
• Free Area Under the Normal Curve Calculator from Daniel Soper's Free Statistics Calculators website.
• Interactive Graph of the Standard Normal Curve Quickly Visualize the one and two-tailed area of the Standard Normal Curve
• Javascript calculator which calculates the probability that a value randomly chosen from a Normal Distribution is greater than, less than or between chosen values
• Standard Normal Distribution Table for the iPhone

From Wikiversity

Histograms

Some normal distributions with various parameters.

Normal probability plots

. A normality probability plot - data falling around this straight indicates the degree of departure from normality.^ We will now learn how to find the probability that a score will fall between any two values on a normal probability distribution. Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES] ^ You are exploring a statistical model in which you assume that data comes from normal probability distributions and that collecting data amounts to taking a sample from a normal population. Normal Distribution 28 January 2010 0:35 UTC www.psych.utah.edu [Source type: FILTERED WITH BAYES] For more information, see * Q-Q plot

Symbols

1. Population mean = μ (Mu)
2. Population variance = σ² = (Sigma squared)

Descriptive indicators of normality

• Skewness
• Kurtosis

Inferential tests of normality

• Kolmogorov-Smirnov test
• Shapiro–Wilk test

Dealing with non-normality

Transformations

• Standardise data in SPSS

Non-parametric statistics

• Non-parametric statistics

Here are sentences from other pages on Normal distribution, which are similar to those in the above article.