Normalization vs. scaling

What is the difference between data 'Normalization' and data 'Scaling'? Till now I thought both terms refers to same process but now I realize there is something more that I don't know/understand. Also if there is a difference between Normalization and Scaling, when should we use Normalization but not Scaling and vice versa?

• Normalising typically means to transform your observations ${\bf x}$ into $f({\bf x})$ (where $f$ is a measurable, typically continuous, function) such that they look normally distributed. Some examples of transformations for normalising data are power transformations. Scaling simply means $f({\bf x})=c{\bf x}$, $c\in {\mathbb R}$, this is, multiplying your observations by a constant $c$ which changes the scale (for example from nanometers to kilometers). – user10525 Sep 3 '12 at 9:04
• Related / also of interest: whats-the-difference-between-normalization-and-standardization. – gung Sep 3 '12 at 13:54
• normalisation is also a scaling method, same as standardisation – user26070 May 24 '13 at 10:44

I am not aware of an "official" definition and even if there it is, you shouldn't trust it as you will see it being used inconsistently in practice.

This being said, scaling in statistics usually means a linear transformation of the form $f(x) = ax+b$.

Normalizing can either mean applying a transformation so that you transformed data is roughly normally distributed, but it can also simply mean putting different variables on a common scale. Standardizing, which means subtracting the mean and dividing by the standard deviation, is an example of the later usage. As you may see it's also an example of scaling. An example for the first would be taking the log for lognormal distributed data.

But what you should take away is that when you read it you should look for a more precise description of what the author did. Sometimes you can get it from the context.

Scaling is a personal choice about making the numbers feel right, e.g. between zero and one, or one and a hundred. For example converting data given in millimeters to meters because it's more convenient, or imperial to metric.

While normalisation is about scaling to an external 'standard' - the local norm - such as removing the mean value and dividing by the sample standard deviation, e.g. so that your sorted data can be compared with a cummulative normal, or a cummulative Poisson, or whatever.

So if a lecturer or manager wants data 'normalised' it means "re-scale it my way" ;-)

I don't know if you mean exactly this, but I see a lot of people referring to Normalization meaning data Standardization. Standardization is transforming your data so it has mean 0 and standard deviation 1:

x <- (x - mean(x)) / sd(x)


I also see people using the term Normalization for Data Scaling, as in transforming your data to a 0-1 range:

x <- (x - min(x)) / (max(x) - min(x))


It can be confusing!

Both techniques have their pros and cons. When scaling a dataset with too many outliers, your non-outlier data might end up in a very small interval. So if your dataset has too many outliers, you might want to consider Standardizing it. Nonetheless, when you do that you will end up with negative data (sometimes you don't want that) and unbounded data (you might not want that either).

Centering means substacting the mean of the random variable from the variables. I.e x -xi

Scalelling means dividing variable by its standard deviation. I.e xi /s

Combination of the two is called normalization or standization. I.e x-xi/s

• The question is a duplicate. – Michael Chernick Aug 2 '17 at 3:22

protected by kjetil b halvorsenAug 2 '17 at 9:44

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