Editing Normal distribution (section)

==Occurrence and applications==
The occurrence of normal distribution in practical problems can be loosely classified into four categories:
# Exactly normal distributions;
# Approximately normal laws, for example when such approximation is justified by the [[central limit theorem]]; and
# Distributions modeled as normal – the normal distribution being the distribution with [[Principle of maximum entropy|maximum entropy]] for a given mean and variance.
# Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

=== Exact normality ===
[[File:QHarmonicOscillator.png|thumb|The ground state of a [[quantum harmonic oscillator]] has the Gaussian distribution.]]
A normal distribution occurs in some [[physical theory|physical theories]]:
* The [[Maxwell–Boltzmann distribution#Distribution for the velocity vector|velocity distribution]] of independently moving and perfectly elastic spheres, which is a consequence of [[Maxwell's theorem|Maxwell's Dynamical Theory of Gases, Part I (1860)]].{{sfnp|Maxwell|1860|p=23}}
* The [[ground state]] [[wave function]] in [[Position and momentum spaces#Quantum mechanics|position space]] of the [[quantum harmonic oscillator]].<ref>{{cite book |last1=Larkoski |first1=Andrew J. |title=Quantum Mechanics: A Mathematical Introduction |date=2023 |publisher=Cambridge University Press |location=United Kingdom |pages=120-121 |url=https://www.google.com/books/edition/Quantum_Mechanics/iKmnEAAAQBAJ?hl=en&gbpv=1&dq=normal%20distribution&pg=PA120&printsec=frontcover |access-date=30 May 2025}}</ref>
* The position of a particle that experiences [[diffusion]]. If initially the particle is located at a specific point (that is its probability distribution is the [[Dirac delta function]]), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the [[diffusion equation]]&nbsp;<math display=inline>\frac{\partial}{\partial t} f(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2} f(x,t)</math>. If the initial location is given by a certain density function <math display=inline>g(x)</math>, then the density at time ''t'' is the [[convolution]] of ''g'' and the normal probability density function.

=== Approximate normality ===
''Approximately'' normal distributions occur in many situations, as explained by the [[central limit theorem]]. When the outcome is produced by many small effects acting ''additively and independently'', its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
* In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where [[infinitely divisible]] and [[Indecomposable distribution|decomposable]] distributions are involved, such as
** [[binomial distribution|Binomial random variables]], associated with binary response variables;
** [[Poisson random variables]], associated with rare events;
* [[Thermal radiation]] 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 theorem.

=== Assumed normality ===
[[File:Fisher iris versicolor sepalwidth.svg|thumb|right|Histogram of sepal widths for ''Iris versicolor'' from Fisher's [[Iris flower data set]], with superimposed best-fitting normal distribution]]
{{Blockquote|I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.|{{harvtxt |Pearson |1901 }}}}
There are statistical methods to empirically test that assumption; see the above [[#Normality tests|Normality tests]] section.
* In [[biology]], the ''logarithm'' of various variables tend to have a normal distribution, that is, they tend to have a [[log-normal distribution]] (after separation on male/female subpopulations), with examples including:
** Measures of size of living tissue (length, height, skin area, weight);<ref>{{harvtxt |Huxley |1932 }}</ref>
** 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.
* 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 [[Benoit Mandelbrot]] have argued that [[Levy skew alpha-stable distribution|log-Levy distributions]], which possesses [[heavy tails]] would be a more appropriate model, in particular for the analysis for [[stock market crash]]es. The use of the assumption of normal distribution occurring in financial models has also been criticized by [[Nassim Nicholas Taleb]] in his works.
* [[Propagation of uncertainty|Measurement errors]] in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.<ref>{{cite book|last=Jaynes|first=Edwin T.|year=2003|title=Probability Theory: The Logic of Science|publisher=Cambridge University Press|pages=592–593|url=https://books.google.com/books?id=tTN4HuUNXjgC&pg=PA592|isbn=9780521592710}}</ref>
* In [[Standardized testing (statistics)|standardized testing]], results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the [[Intelligence quotient|IQ test]]) or 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.
[[File:FitNormDistr.tif|thumb|220px|Fitted cumulative normal distribution to October rainfalls, see [[distribution fitting]] ]]
* Many scores are derived from the normal distribution, including [[percentile rank]]s (percentiles or quantiles), [[normal curve equivalent]]s, [[stanine]]s, [[z-scores]], and T-scores. Additionally, some [[Psychological statistics|behavioral statistical]] procedures assume that scores are normally distributed; for example, [[t-tests]] and [[Analysis of variance|ANOVAs]]. [[Bell curve grading]] assigns relative grades based on a normal distribution of scores.
* In [[hydrology]] the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the [[central limit theorem]].<ref>{{cite book|last=Oosterbaan|first=Roland J. | editor-last=Ritzema |editor-first=Henk P.|chapter=Chapter 6: Frequency and Regression Analysis of Hydrologic Data|year=1994 | edition=second revised|title=Drainage Principles and Applications, Publication 16|publisher=International Institute for Land Reclamation and Improvement (ILRI)|location=Wageningen, The Netherlands|pages=175–224|chapter-url=http://www.waterlog.info/pdf/freqtxt.pdf|isbn=978-90-70754-33-4}}</ref> The blue picture, made with [[CumFreq]], illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].

=== Methodological problems and peer review ===
[[John Ioannidis]] [[Why Most Published Research Findings Are False|argued]] that using normally distributed standard deviations as standards for validating research findings leave [[falsifiability|falsifiable predictions]] about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.<ref>Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005</ref>