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normalization statistics wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 history toggle history subsection 1 1 standard score z score 1 2 student s t statistic 1 3 feature scaling 1 4 batch normalization 2 examples 3 other types 4 see also 5 references toggle the table of contents normalization statistics 10 languages বাংলা español فارسی italiano polski português русский simple english 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia statistical procedure for other uses see normalizing constant in statistics and applications of statistics normalization can have a range of meanings 1 in the simplest cases normalization of ratings means adjusting values measured on different scales to a notionally common scale often prior to averaging in more complicated cases normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment in the case of normalization of scores in educational assessment there may be an intention to align distributions to a normal distribution a different approach to normalization of probability distributions is quantile normalization where the quantiles of the different measures are brought into alignment in another usage in statistics normalization refers to the creation of shifted and scaled versions of statistics where the intention is that these normalized values allow the comparison of corresponding normalized values for different datasets in a way that eliminates the effects of certain gross influences as in an anomaly time series some types of normalization involve only a rescaling to arrive at values relative to some size variable in terms of levels of measurement such ratios only make sense for ratio measurements where ratios of measurements are meaningful not interval measurements where only distances are meaningful but not ratios in theoretical statistics parametric normalization can often lead to pivotal quantities functions whose sampling distribution does not depend on the parameters and to ancillary statistics pivotal quantities that can be computed from observations without knowing parameters history edit standard score z score edit see also standard score the concept of normalization emerged alongside the study of the normal distribution by abraham de moivre pierre simon laplace and carl friedrich gauss from the 18th to the 19th century as the name standard refers to the particular normal distribution with expectation zero and standard deviation one that is the standard normal distribution normalization in this case standardization was then used to refer to the rescaling of any distribution or data set to have mean zero and standard deviation one 2 while the study of normal distribution structured the process of standardization the result of this process also known as the z score given by the difference between sample value and population mean divided by population standard deviation and measuring the number of standard deviations of a value from its population mean 3 was not formalized and popularized until ronald fisher and karl pearson elaborated the concept as part of the broader framework of statistical inference and hypothesis testing 4 5 in the early 20th century student s t statistic edit see also student s t statistic william sealy gosset initiated the adjustment of normal distribution and standard score on small sample size educated in chemistry and mathematics at winchester and oxford gosset was employed by guinness brewery the biggest brewer in ireland back then and was tasked with precise quality control it was through small sample experiments that gosset discovered that the distribution of the means using small scaled samples slightly deviated from the distribution of the means using large scaled samples the normal distribution and appeared taller and narrower in comparison 6 this finding was later published in a guinness internal report titled the application of the law of error to the work of the brewery and was sent to karl pearson for further discussion which later yielded a formal publishment titled the probable error of a mean in the year of 1908 7 under guinness brewery s privacy restrictions gosset published the paper under the pseudonym student gosset s work was later enhanced and transformed by ronald fisher to the form that is used today 8 and was alongside the names student s t distribution referring to the adjusted normal distribution gosset proposed and student s t statistic referring to the test statistic used in measuring the departure of the estimated value of a parameter from its hypothesized value divided by its standard error popularized through fisher s publishment titled applications of student s distribution 6 feature scaling edit see also feature scaling the rise of computers and multivariate statistics in mid 20th century necessitated normalization to process data with different units hatching feature scaling a method used to rescale data to a fixed range like min max scaling and robust scaling this modern normalization process especially targeting large scaled data became more formalized in fields including machine learning pattern recognition and neural networks in late 20th century 9 10 batch normalization edit see also batch normalization batch normalization was proposed by sergey ioffe and christian szegedy in 2015 to enhance the efficiency of training in neural networks 11 examples edit there are different types of normalizations in statistics nondimensional ratios of errors residuals means and standard deviations which are hence scale invariant some of which may be summarized as follows note that in terms of levels of measurement these ratios only make sense for ratio measurements where ratios of measurements are meaningful not interval measurements where only distances are meaningful but not ratios see also category statistical ratios name formula use standard score x μ σ displaystyle frac x mu sigma normalizing errors when population parameters are known works well for populations that are normally distributed 12 student s t statistic β β 0 s e β displaystyle frac widehat beta beta _ 0 operatorname s e widehat beta the departure of the estimated value of a parameter from its hypothesized value normalized by its standard error studentized residual ε i σ i x i μ i σ i displaystyle frac hat varepsilon _ i hat sigma _ i frac x_ i hat mu _ i hat sigma _ i normalizing residuals when parameters are estimated particularly across different data points in regression analysis standardized moment μ k σ k displaystyle frac mu _ k sigma k normalizing moments using the standard deviation σ displaystyle sigma as a measure of scale coefficient of variation σ μ displaystyle frac sigma mu normalizing dispersion using the mean μ displaystyle mu as a measure of scale particularly for positive distribution such as the exponential distribution and poisson distribution min max feature scaling x x x min x max x min displaystyle x frac x x_ min x_ max x_ min feature scaling is used to bring all values into the range 0 1 this is also called unity based normalization this can be generalized to restrict the range of values in the dataset between any arbitrary points a displaystyle a and b displaystyle b using for example x a x x min b a x max x min displaystyle x a frac left x x_ min right left b a right x_ max x_ min note that some other ratios such as the variance to mean ratio σ 2 μ textstyle left frac sigma 2 mu right are also done for normalization but are not nondimensional the units do not cancel and thus the ratio has units and is not scale invariant other types edit other non dimensional normalizations that can be used with no assumptions on the distribution include assignment of percentiles this is common on standardized tests see also quantile normalization normalization by adding and or multiplying by constants so values fall between 0 and 1 this is used for probability density functions with applications in fields such as quantum mechanics in assigning probabilities to ψ 2 see also edit normal score ratio distribution standard score feature scaling references edit dodge y 2003 the oxford dictionary of statistical terms oup isbn 0 19 920613 9 entry for normalization of scores stigler stephen m 2002 statistics on the table the history of statistical concepts and methods 3 printing ed cambridge mass harvard univ press isbn 978 0 674 00979 0 lang niklas august 23 2023 what is the z score data basecamp retrieved march 13 2025 fisher r a january 1 2017 statistical methods for research workers gyan books isbn 978 9351286585 pearson karl november 1 1901 liii on lines and planes of closest fit to systems of points in space the london edinburgh and dublin philosophical magazine and journal of science 2 11 559 572 doi 10 1080 14786440109462720 issn 1941 5982 1 2 brown angus 2008 the strange origins of the student s t test physiology news 13 16 doi 10 36866 pn 71 13 retrieved march 13 2025 student 1908 the probable error of a mean biometrika 6 1 1 25 doi 10 2307 2331554 issn 0006 3444 jstor 2331554 rohlf f james sokal robert r 2012 statistical tables 4th ed new york n y freeman isbn 978 1 4292 4031 4 duda richard o hart peter e stork david g 2001 pattern classification 2nd ed new york wiley isbn 978 0 471 05669 0 bishop christopher m 2006 pattern recognition and machine learning information science and statistics new york springer isbn 978 0 387 31073 2 ioffe sergey szegedy christian march 2 2015 batch normalization accelerating deep network training by reducing internal covariate shift arxiv 1502 03167 cs lg freedman david pisani robert purves roger february 20 2007 statistics fourth international student edition w w norton company isbn 9780393930436 retrieved from https en wikipedia org w index php title normalization_ statistics oldid 1328048565 categories statistical ratios statistical data transformation equivalence mathematics hidden categories articles with short description short description matches wikidata use american english from march 2021 all wikipedia articles written in american english use mdy dates from march 2021 this page was last edited on 17 december 2025 at 16 51 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents normalization statistics 10 languages add topic