Normalization

A normalization is applied in general on a set of data with numerical values. The set of values have key values describing the aggregated set of numerical values. The key values can be means, deviation value, threshold value for the list. In the context of statistics, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. For probability distributions, normalization may refer to a linear transformation of the data set to bring the entire probability distributions e.g. to an expectation value of 0 and a variance of 1. This is possible if the variance exists (i.e, is finite). 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 for thresholds can be used in a very simple case for regulatory thresholds e.g. in ecotoxicology. Regulatory thresholds (e.g. provide by risk management agencies or public health agencies) might be different for toxicants in the environment and they might be reduced according to new scientific evidence and the risk assessment.

Examples
The following examples show normalization with a
 * the arithmetic mean,
 * the standard deviation or
 * regulatory threshold (e.g. threshold for a toxic substance in a product)

Learning Tasks
Normalize the scores of an assessment of students according to the mean value.
 * data $$ D = (10,40,98,30,50,65,34,59,33,44,55,29) $$
 * calculate the arithmetic means $$\mu$$ of the data $$D$$.
 * normalized data $$ N=(10-\mu,40-\mu,98-\mu,30-\mu,50-\mu,65-\mu,34-\mu,59-\mu,33-\mu,44-\mu,55-\mu,29-\mu) $$ to means $$\mu$$. Calculate the arithmetic means of $$N$$. What is the result.
 * What does the normalized vector $$ N $$ tell you as teacher about the assessment?
 * Add new values to $$D$$ or modify the values in $$D$$ and calculate the arithmetic mean of $$N$$ again. What do see and why do you get these results?

Threshold Normalization
In this case the threshold normalized value 3.0 means that the measured value exceeds the allowed threshold value by a factor of 3.
 * measure value is 300mg with an allow regulatory threshold of 100mg.
 * measure value is 7.5mg with an allow regulatory threshold of 2.5mg.