statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

and applications of statistics, normalization can have a range of meanings. 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

. A different approach to normalization of probability distributions is

quantile normalization In statistics, quantile normalization is a technique for making two distributions identical in statistical properties. To quantile-normalize a test distribution to a reference distribution of the same length, sort the test distribution and sort t ...

, where the

quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...

s 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 In the natural sciences, especially in atmospheric and Earth sciences involving applied statistics, an ''anomaly'' is a persisting deviation in a physical quantity from its expected value, e.g., the systematic difference between a measurement and ...

. Some types of normalization involve only a rescaling, to arrive at values relative to some size variable. In terms of

levels of measurement Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scal ...

, 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.

Examples

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

, 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 :Statistical ratios. Note that some other ratios, such as the

variance-to-mean ratio In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a pro ...

\left(\frac\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

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

. * 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 physical chemistry in assigning probabilities to .

References

{{reflist, refs= Dodge, Y (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN, 0-19-920613-9 (entry for normalization of scores) Statistical ratios Statistical data transformation Equivalence (mathematics)