What does "normalization" mean and how to verify that a sample or a distribution is normalized?

I understand by computing $$ \frac>> $$ we get normalized data, but here it's asking to verify whether a distribution is normalized or not.

• data-transformation
• terminology
• normalization
• standardization

$\begingroup$ What it means for a distribution to be normalized is not so simple (and it's usually not the distribution itself being normalized, but the random variable). For example, in the case of the uniform, some people may mean "linearly rescaled so as to get a standard uniform" (i.e. to get $a=0$ and $b=1$) . while another person might mean "linearly rescaled so as to get mean 0 and sd 1". For the uniform, I'd normally assume the first, but as you see from an answer below, other people may take it to mean something else. The best option is to ask the person using the term to be less ambiguous. $\endgroup$

$\begingroup$ The more conventional terms are standardized (to achieve a mean of zero and SD of one) and normalized (to bring the range to the interval $[0,1]$ or to rescale a vector norm to $1$). Thus the re-expression $X\to (X-\text)/SD$ is a standardization whereas multiplying a density $f$ by a constant $C$ to make $\int_<-\infty>^\infty Cf(x)dx=1$ is a normalization, because $\int f(x)dx$ is the $L^1$ norm of $f$. $\endgroup$

$\begingroup$ Please don't cross-post, @Ada. That is against SE policy. If you post a Q on 1 site & then think you should have posted it on another, flag your Q & ask the moderators to migrate it for you. $\endgroup$

4 Answers 4

Unfortunately, terms are used differently in different fields, by different people within the same field, etc., so I'm not sure how well this can be answered for you here. You should make sure you know the definition that your instructor / the textbook is using for "normalized". However, here are some common definitions:

Centered: $$ X- $$ Standardized: $$ \frac>> $$ Normalized: $$ \frac <\max(X)-\min(X)>$$ Normalizing in this sense rescales your data to the unit interval. Standardizing turns your data into $z$-scores, as @Jeff notes. And centering just makes the mean of your data equal to $0$.

It is worth recognizing here that all three of these are linear transformations; as such, they do not change the shape of your distribution. That is, sometimes people call the $z$-score transformation "normalizing" and believe, because of $z$-scores' association with the normal distribution, that this has made their data normally distributed. This is not so (as @Jeff also notes, and as you could tell by plotting your data before and after). Should you be interested, you could change the shape of your data using the Box-Cox family of transformations, for example.

With respect to how you could verify these transformations, it depends on what exactly is meant by that. If they mean simply to check that the code ran properly, you could check means, SDs, minimums, and maximums.