Sebaran normal: Béda antarrépisi
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[[Gambar:Gaussian-pdf.png|thumb|300px|[[Probability density function]] of Gaussian distribution (bell curve).]]
'''Normal distribution''' (distribusi normal)
Biasa
Dina kaayaan sabenerna kumpulan distribusi mibanda bentuk anu sarupa, bedana ngan dina parameter ''location'' jeung ''scale'': [[nilai ekspektasi|mean]] jeung [[simpangan baku]]. '''Standard normal distribution'''
== Sajarah ==
Distribusi normal mimiti dikenalkeun ku [[Abraham de Moivre|de Moivre]] dina artikel taun [[1733]] (dicitak ulang edisi kaduana dina ''[[The Doctrine of Chances]]'', [[1738]]) dina kontek "pendekatan" [[sebaran binomial]] keur ''n'' anu loba. Hasil de Moivre diteruskeun ku [[Pierre Simon de Laplace|Laplace]] dina bukuna ''[[Analytical Theory of Probabilities]]'' ([[1812]]), mangsa kiwari disebut [[Theorem of de Moivre-Laplace]].
Laplace ngagunakeun distribusi normal keur [[analysis of errors]] dina percobaanna. [[Method of least squares]] nu kacida pentingna dikenalkeun ku [[Adrien Marie Legendre|Legendre]] dina taun [[1805]]. [[Carl Friedrich Gauss|Gauss]],
Istilah "bell curve" ngacu ka [[Jouffret]] nu ngagunakeun watesan "bell surface" dina taun [[1872]] keur [[multivariate normal distribution|bivariate normal]] dina komponen bebas (independent). Istilah "sebaran normal" "ditemukan" sacara sewang-sewangan ku [[Charles S. Peirce]], [[Francis Galton]] jeung [[Wilhelm Lexis]] kira-kira taun [[1875]] [Stigler]. This terminology is unfortunate, since it reflects and encourages the fallacy that "everything is Gaussian". (See the discussion of "occurrence" below).
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== Spesifikasi sebaran normal ==
Aya sababaraha jalan keur nangtukeun random variable. Anu paling ngagambarkeun
All of the [[cumulant]]s of the normal distribution are zero, except the first two.
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=== Fungsi probabiliti densiti ===
[[Fungsi dénsitas probabilitas]] dina '''sebaran normal''' numana
:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu )^2 / 2\sigma^2}}</math>
(Tempo
:<math>f(x) = {1 \over \sqrt{2\pi} }\,e^{-{x^2 / 2}}</math>
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For all normal distributions,
the density function is symmetric about its
=== Fungsi Sebaran Kumulatif ===
[[Fungsi sebaran kumulatif]] (saterusna disebut ''cdf'') hartina probabilitas
:<math>\Pr(X \le x) = \int_{-\infty}^x \frac{1}{\sigma\sqrt{2\pi}} e^{-(u-\mu)^2/(2\sigma^2)}\,du</math>
Standar normal cdf, sacara konvensional dilambangkeun ku <math>\Phi</math>,
:<math>\Phi(z) = \int_{-\infty}^z {1 \over \sqrt{2\pi} }\,e^{-{x^2 / 2}}\,dx</math>
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:<math>\Phi(z) = \frac{1}{2} \left(1 + \operatorname{erf}\,\frac{z}{\sqrt{2}}\right)</math>
The following graph shows the cumulative distribution function for values of ''z'' from -4 to +4:
[[Gambar:Cumulative_normal_distribution.png]]
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[[characteristic function|Fungsi karakteristik]] dihartikeun salaku [[nilai ekspektasi]]
<math>e^{itX}</math>.
Keur sebaran normal, ieu bisa ditembongkeun dina fungsi karakteristik
:<math>\phi_X(t)=E\left[e^{itX}\right]=\int_{-\infty}^{\infty} \frac{1} {\sigma\sqrt{2\pi}}\,e^{-{(x-\mu )^2 / 2\sigma^2}}\,e^{itx}\,dx = e^{i\mu t-\sigma^2 t^2/2}</math>
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== Pasipatan ==
# Lamun ''X'' ~ N(μ, σ<sup>2</sup>) sarta ''a'' sarta ''b''
# If ''X''<sub>1</sub> ~ N(μ<sub>1</sub>, σ<sub>1</sub><sup>2</sup>) and ''X''<sub>2</sub> ~ N(μ<sub>2</sub>, σ<sub>2</sub><sup>2</sup>), and ''X''<sub>1</sub> and ''X''<sub>2</sub> are ''independent'', then ''X''<sub>1</sub> + ''X''<sub>2</sub> ~ N(μ<sub>1</sub> + μ<sub>2</sub>, σ<sub>1</sub><sup>2</sup> + σ<sub>2</sub><sup>2</sup>).
# If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[Statistical independence|independent]] standard normal variables, then ''X''<sub>1</sub><sup>2</sup> + ... + ''X''<sub>''n''</sub><sup>2</sup> has a [[sebaran chi-kuadrat]] with ''n'' degrees of freedom.
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As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.
If ''X'' is a normal random variable with
:<math> Z = \frac{X - \mu}{\sigma} </math>
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:<math>\Pr(X<x) = \Phi\left(\frac{x-\mu}{\sigma}\right) = \frac{1}{2} \left(1+\mbox{erf}\,\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right)</math>
Conversely, if ''Z'' is a standard normal random variable, then
:<math>X=\sigma Z+\mu \,</math>
is a normal random variable with
The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one.
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This requires generating values from a uniform distribution, for which many methods are known. See also [[random number generator]]s.
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
=== The central limit theorem ===
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This is the so-called [[central limit theorem]].
The practical importance of the central limit
* [[Sebaran binomial]] mibanda parameter ''n'' sarta ''p'' ngadeukeutan kana normal keur ''n'' nu badag sarta ''p'' teu deukeut ka 1 atawa 0. ''Pendekatan'' sebaran normal mibanda
* A [[Poisson distribution]] with parameter λ is approximately normal for large λ. The approximating normal distribution has
▲* A [[Poisson distribution]] with parameter λ is approximately normal for large λ. The approximating normal distribution has mean μ = λ and standard deviation σ = √λ.
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution.
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''Approximately'' normal distributions occur in many situations, as a result of the [[central limit theorem]].
When there is
There are statistical methods to empirically test that assumption.
Effects can also act as '''multiplicative''' (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the [[logarithm]] of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called [[log-normal distribution|log-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
To summarize, here's a list of situations where approximate normality
is sometimes assumed. For a fuller discussion, see below.
* In counting problems (so the central limit
** Binomial random variables, associated to yes/no questions;
** Poisson random variables, associates to [[rare events]];
* In physiological
** The ''logarithm'' of
** 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;
** Other physiological
*
* Financial variables
** The ''logarithm'' of interest rates, exchange rates, and inflation; these variables behave like compound interest, not like simple interest, and so are multiplicative;
** Stock-market indices are supposed to be multiplicative too, but some
** Other financial variables may be normally distributed, but there is no
* Light intensity
** The intensity of laser light is normally distributed;
** Thermal light has a [[Bose-Einstein statistics|Bose-Einstein]] distribution on very short time scales, and a normal distribution on longer timescales due to the central limit
Of relevance to biology and economics is the fact that complex systems tend to display [[power law]]s rather than normality.
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=== Photon counts ===
Light intensity from a single source varies with time, and is usually assumed to be normally distributed. However, quantum mechanics interprets
=== Measurement errors ===
=== Physical characteristics of biological specimens ===
The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore
:Huxley, Julian: Problems of Relative Growth (1932)
Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further
The assumption that
* blood pressure of adult humans is supposed to be normally distributed, but only after separating males and females into different populations (
* The length of inert appendages such as hair, nails, teet, claws and shells is expected to be normally distributed if
=== Financial variables ===
Because of the exponential nature of [[interest]] and [[inflation]], financial indicators such as [[interest rate]]s, [[share|stock]] values, or [[commodity]] [[price]]s
[[Benoît Mandelbrot]], the popularizer of [[fractals]], has claimed that even the assumption of lognormality is flawed.
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=== Lifetime ===
Other examples of variables that are ''not'' normally distributed include the lifetimes of humans or mechanical devices. Examples of distributions used in this connection are the [[sebaran eksponensial]] (memoryless) and the [[Weibull distribution]]. In general, there is no
=== Test scores ===
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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.
* [[IQ|IQ scores]] and other ability scores are approximately normally distributed. For most IQ tests, the
''Criticisms: test scores are discrete variable associated with the number of correct/incorrect answers, and as such they are related to the binomial. Moreover (see [http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&selm=b26c3b%241s3c%40odds.stat.purdue.edu this 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 and '''additive''' variations of IQ, is subject to debate.''
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* [http://ce597n.www.ecn.purdue.edu/CE597N/1997F/students/michael.a.kropinski.1/project/tutorialMichael A. Kropinski's normal distribution tutorial]
* S. M.Stigler: ''Statistics on the Table'', Harvard University Press 1999, chapter 22. History of the term "normal distribution".
* [http://web.archive.org/19990117033417/members.aol.com/jeff570/mathword.html Earliest Known uses of some of the Words of Mathematics]. See: [http://web.archive.org/19991003084940/members.aol.com/jeff570/n.html] for "normal", [http://web.archive.org/19990508225359/members.aol.com/jeff570/g.html] for "Gaussian", and [http://web.archive.org/19990508224238/members.aol.com/jeff570/e.html] for "error".
* [http://web.archive.org/20000610213020/members.aol.com/jeff570/stat.html Earliest Uses of Symbols in Probability and Statistics]. See Symbols associated with the Normal Distribution.
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