# What is the difference between normalization and re-scaling?

This site does not describe the nature of the tag. Does it differ from re-scaling? Many authors use the two terms interchangeably.

Re-scaling means that you multiply your variable by a factor, i.e. $$x \to x/a$$. A normalization is a specific kind of re-scaling, where the factor $$a$$ is such that the values of $$x$$ become of order one. Its form depends on the context and what you are trying to do. Examples are $$a= \langle x \rangle$$ or $$a= \textrm{max}(x)$$. Perhaps the most common one is when you do $$x \to (x-\langle x \rangle)/\sigma$$, because if $$x$$ follows a Gaussian distribution, the re-scaled variable follows a normal distribution (average 0 and std 1). Note, this last transformation actually involves a re-scaling, but also a translation.
• "Your variable" means a random variable your aim to re-scale. $x$ is the random variable. $a$ is the factor. Feb 22, 2021 at 1:30
• In which sense $x \to (x-\langle x \rangle)/\sigma$ is not a re-scaling (after a translation) and, more specifically, a normalization? Feb 23, 2021 at 16:45
• When you say that "Re-scaling the data entails observed data minus minimum score and division by maximum Range of data" that's just an example of a possible re-scaling. There are many ways in which something can be re-scaled. And note that my answer applies only to mathematical statistics after the sentence "if $x$ follows a Gaussian distribution". Before that is general. Feb 24, 2021 at 10:10