Editing Normal distribution (section)

=== 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]].