Sebaran normal

Normal distribution(distribusi normal) mangrupa hal anu penting dinaprobability distributiondi loba widang. 
Biasa ogé disebutGaussian distribution, hususna dina widangfisikajeungrékayasa. 
Dina kaayaan sabenerna kumpulan distribusi mibanda bentuk anu sarupa, bédana ngan dina paraméterlocationjeungscale:meanjeungsimpangan baku.Standard normal distributionnyaéta distribusi normal anu mibanda nilaimeansarua jeung nol sarta nilai standar deviasi sarua jeung hiji. Sabab bentuk grafikdénsitas probabilitasmangrupabell, sering disebutbell curve.
Distribusi normal mimiti dikenalkeun kude Moivredina artikel taun1733(dicitak ulang edisi kaduana dinaThe Doctrine of Chances,1738) dina kontek "pendekatan"sebaran binomialkeurnanu loba.  Hasil de Moivre diteruskeun kuLaplacedina bukunaAnalytical Theory of Probabilities(1812), mangsa kiwari disebutTheorem of de Moivre-Laplace.
Laplace ngagunakeun distribusi normal keuranalysis of errorsdina percobaanna.Method of least squaresnu kacida pentingna dikenalkeun kuLegendredina taun1805.Gauss, ogé ngakukeun yén manéhna geus maké métodeu  anu sarua ti mimiti taun1794, justified it rigorously in1809by assuming a normal distribution of the errors.
Istilah "bell curve" ngacu kaJouffretnu ngagunakeun watesan "bell surface" dina taun1872keurbivariate normaldina komponen bébas (independent). Istilah "sebaran normal" "ditemukan" sacara sewang-sewangan kuCharles S. Peirce,Francis GaltonjeungWilhelm Lexiskira-kira taun1875[Stigler]. This terminology is unfortunate, since it reflects and encourages the fallacy that "everything is Gaussian". (See the discussion of "occurrence" below).
Yen sebaran disebut sebarannormalatawaGaussian, ngagantikeun sebarande Moivrean,
is just an instance ofStigler's law of eponymy:
"No scientific discovery is named after its original discoverer".
Aya sababaraha jalan keur nangtukeun random variable. Anu paling ngagambarkeun nyaéta probability density function (plot at the top), which represents how likely éach value of the random variable is. The cumulative density function is a conceptually cléaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the moments, thecumulants, thecharacteristic function, themoment-generating function, and the cumulant-generating function. Some of these are very useful for théoretical work, but not intuitive. Seeprobability distributionfor a discussion.
All of thecumulantsof the normal distribution are zero, except the first two.
Fungsi dénsitas probabilitasdinasebaran normalnu mana méan μ jeung simpangan baku σ (sarua jeung,varianσ) mangrupa contofungsi Gauss,
(Tempo ogéfungsi eksponensialjeungpi.) Lamunvariabel acakXmibanda distribusi ieu, bisa dituliskeunX~ N(μ, σ). Lamun μ = 0 jeung σ = 1, distribusi disebut distribusi standar normal, rumusna
Gambar di luhur nunjukkeun grafik probability density function tina sebaran normal nu mana μ = 0 jeung sababaraha nila σ.
For all normal distributions,
the density function is symmetric about its méan value. About 68% of the aréa under the curve is within one standard deviation of the méan, 95.5% within two standard deviations, and 99.7%  within three standard deviations. Theinflection pointsof the curve occur at one standard deviation away from the méan.
Fungsi sebaran kumulatif(saterusna disebutcdf) hartina probabilitas di mana nilai variabelXleuwih leutik tinimbangx, jeung digambarkeun dina watesan fungsi densiti nyaéta
Standar normal cdf, sacara konvensional dilambangkeun kuΦ{\displaystyle \Phi }, mangrupa nilai cdf umum di-evaluasi kuμ=0{\displaystyle \mu =0}jeungσ=1{\displaystyle \sigma =1},
The standard normal cdf can be expressed in terms of aspecial functioncalled theerror function, as
The following graph shows the cumulative distribution function for values ofzfrom -4 to +4:
On this graph, we see the probability that a standard normal variable has a value less than 0.25 is approximately equal to 0.60.
Fungsi karakteristikdihartikeun salakunilai ekspektasieitX{\displaystyle e^{itX}}.
Keur sebaran normal, ieu bisa ditembongkeun dina fungsi karakteristik nyaéta
saperti nu katempo ku kuadrat eksponen nu lengkep.
As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.
IfXis a normal random variable with méan μ and variance σ, then
is a standard normal random variable:Z~N(0,1). 
An important consequence is that the cdf of a general normal distribution is therefore
Conversely, ifZis a standard normal random variable, then
is a normal random variable with méan μ and variance σ.
The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one. 
Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.
For computer simulations, it is often useful to generate values that have a normal distribution.
There are several methods; the most basic is to invert the standard normal cdf. More efficient methods are also known.
One such method is theBox-Muller transform. 
The Box-Muller transform takes twouniformly distributedvalues as input and maps them to two normally distributed values.
This requires generating values from a uniform distribution, for which many methods are known. See alsorandom number generators.
The Box-Muller transform is a consequence of Property 3 and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which is éasy to generate).
The normal distribution has the very important property that
under certain conditions, the distribution of a sum of a large number ofindependent variablesis approximately normal.
This is the so-calledcentral limit theorem.
The practical importance of the central limit théorem is that the normal distribution can be used as an approximation to some other distributions.
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.
Approximatelynormal distributions occur in many situations, as a result of thecentral limit theorem.
When there is réason to suspect the presence of a large number of small effectsacting additively, it is réasonable to assume that observations will be normal. 
There are statistical methods to empirically test that assumption.
Effects can also act asmultiplicative(rather than additive) modifications. In that case, the assumption of normality is not justified, and it is thelogarithmof the variable of interest that is normally distributed. The distribution of the directly observed variable is then calledlog-normal.
Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the théory of errors (see below).
To summarize, here's a list of situations where approximate normality 
is sometimes assumed. For a fuller discussion, see below.
Of relevance to biology and economics is the fact that complex systems tend to displaypower lawsrather than normality.
Light intensity from a single source varies with time, and is usually assumed to be normally distributed. However, quantum mechanics interprets méasurements of light intensity asphotoncounting. Ordinary light sources which produce light by thermal emission, should follow aPoisson distributionorBose-Einstein distributionon very short time scales. On longer time scales (longer than thecoherence time), the addition of independent variables yields an approximately normal distribution. The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution.
Repéated méasurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it isassumedthat the remaining error must be the result of a large number of very smalladditiveeffects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Note that this is thecentralassumptionof the mathematicaltheory of errors.
The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore méasures of body size should at most follow a lognormal rather than normal distribution. Despite common claims of normality, the sizes of plants and animals is approximately lognormal. The evidence and an explanation based on modéls of growth was first published in the classic book
Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further maké the joint distribution of sizes deviate from lognormality.
The assumption that linéar size of biological specimens is normal léads to a non-normal distribution of weight (since weight/volume is roughly the 3rd power of length, and gaussian distributions are only preserved by linéar transformations), and conversely assuming that weight is normal léads to non-normal lengths. This is a problem, because there is noa prioriréason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.
Because of the exponential nature ofinterestandinflation, financial indicators such asinterest rates,stockvalues, orcommoditypricesmaké good examples ofmultiplicativebehaviour. As such, they should not be expected to be normal, but lognormal.
Benoît Mandelbrot, the popularizer offractals, has claimed that even the assumption of lognormality is flawed.
Other examples of variables that arenotnormally distributed include the lifetimes of humans or mechanical devices. Examples of distributions used in this connection are thesebaran eksponensial(memoryless) and theWeibull distribution. In general, there is no réason thatwaiting timesshould be normal, since they are not directly related to any kind of additive influence.
The IQ score of an individual for example can be seen as the result of many small additive influences: many genes and many environmental factors all play a role.
Criticisms: test scores are discrete variable associated with the number of correct/incorrect answers, and as such they are related to the binomial. Moreover (seethis USENET post), raw IQ test scores are customarily 'massaged' to force the distribution of IQ scores to be normal. Finally, there is no widely accepted model of intelligence, and the link to IQ scores let alone a relationship between influences on intelligence andadditivevariations of IQ, is subject to debate.