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| {{other uses|Standard score|Normalizing constant}}
| | Jayson Berryhill is how I'm called and my wife doesn't like it at all. I've usually cherished living in Alaska. Since I was eighteen I've been working as a bookkeeper but quickly my wife and I will start our personal business. To perform lacross is the thing I love most of all.<br><br>Here is my blog: are psychics real ([http://www.khuplaza.com/dent/14869889 please click the next post]) |
| {{refimprove|date=July 2012}}
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| In [[statistics]] and applications of statistics, '''normalization''' can have a range of meanings.<ref name=Dodge/> 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 distribution]]s 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 [[quantile]]s of the different measures are brought into alignment.
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| 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).
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| In theoretical statistics, parametric normalization can often lead to [[pivotal quantity|pivotal quantities]] – functions whose [[sampling distribution]] does not depend on the parameters – and to [[ancillary statistic]]s – pivotal quantities that can be computed from observations, without knowing parameters.
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| == Examples ==
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| There are various 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]].
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| ! Name !! Formula !! Use
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| | [[Standard score]] || <math>\frac{X - \mu}{\sigma}</math> || Normalizing errors when population parameters are known. Works well for populations that are [[normally distributed|normal distribution]]
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| | [[Student's t-statistic]] || <math>\frac{X - \overline{X}}{s}</math> || Normalizing [[Residual (statistics)|residuals]] when population parameters are unknown (estimated).
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| | [[Studentized residual]] || <math>\frac{\hat \epsilon_i}{\hat \sigma_i} = \frac{X_i - \hat \mu_i}{\hat \sigma_i}</math> || Normalizing residuals when parameters are estimated, particularly across different data points in [[regression analysis]].
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| | [[Standardized moment]] || <math>\frac{\mu_k}{\sigma^k}</math> || Normalizing moments, using the standard deviation <math>\sigma</math> as a measure of scale.
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| | [[Coefficient of variation|Coefficient of<br> variation]] || <math>\frac{\sigma}{\mu}</math> || Normalizing dispersion, using the mean <math>\mu</math> as a measure of scale, particularly for positive distribution such as the [[exponential distribution]] and [[Poisson distribution]].
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| Note that some other ratios, such as the [[variance-to-mean ratio]] <math>\left(\frac{\sigma^2}{\mu}\right)</math>, are also done for normalization, but are not nondimensional: the units do not cancel, and thus the ratio has units, and are not scale invariant.
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| ==Other Types==
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| Other non-dimensional normalizations that can be used with no assumptions on the distribution include:
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| * Assignment of [[percentiles]]. This is common on standardized tests.
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| * Normalization by adding and/or multiplying by constants so values fall between 0 and 1. This used for [[probability density function|probability density functions]], with applications in fields such as physical chemistry in assigning probabilities to {{math|{{abs|''ψ''}}<sup>2</sup>}}.
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| == See also ==
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| * [[Standard score]]
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| ==References==
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| {{reflist|refs=
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| <ref name=Dodge> [[Yadolah Dodge|Dodge, Y]] (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9 (entry for normalization of scores)</ref>
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| }}
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| == External links ==
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| *[http://www.qsarworld.com/qsar-statistics-normalization.php Statistical Normalization]
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| *[http://www.qsarworld.com/qsar-statistics-mean-shifting.php Mean shifting]
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| [[Category:Statistical ratios]]
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| [[Category:Statistical terminology]]
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| {{statistics-stub}}
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Jayson Berryhill is how I'm called and my wife doesn't like it at all. I've usually cherished living in Alaska. Since I was eighteen I've been working as a bookkeeper but quickly my wife and I will start our personal business. To perform lacross is the thing I love most of all.
Here is my blog: are psychics real (please click the next post)