Tag Usage
Do not confuse normalization with standardization or data-transformation.
Overview
A set of discrete data $x_i$ ($i=1,2,\ldots,n$) can be normalized "Normalization" refers to $X_i$ through the followingseveral related processes:
$$X_i = \frac{x_i}{\sum_{i=1}^n x_i}$$
("Feature scaling") A set of numbers whose maximum is $M$ and minimum is $m$ can be converted to the range from $0$ to $1$ by means of an affine transformation (which amounts to changing their units of measurement) $x \to (x-m)/(M-m)$.
A set of positive numbers $\{p_i\}$ representing probabilities or weights can be uniformly rescaled to sum to unity: divide each $p_i$ by the sum of all the $p_i$.
Analogously, a distribution (or indeed any non-negative function with a finite nonzero integral) can be normalized to have a unit integral by dividing its values by the integral.
A vector in a normed linear space is normalized (to unit length) by dividing it by its norm. This is a general procedure encompassing the two preceding operations as special examples.
A functionThe range from $\mathcal{f}(x)$$0$ to $1$ can be normalizedmade from $0$ to any desired limit $\alpha$ by multiplying a distributionpreviously unit-normalized value by $\mathcal{P}(x)$ in the following way:
$$\mathcal{P}(x) = \frac{\mathcal{f}(x)}{\int_{-\infty}^\infty \mathcal{f}(x) dx }$$$\alpha$.
This assumes that the improper integralOther kinds of operations exist having a similar intent of re-expressing values in the denominator cana predetermined range. Many of these are nonlinear and tend to be appropriately dealt with through limitsused in specialized settings.
