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Comment by smitty1e | Hacker News Reader

"Normalized".

That's what we call data that have been scaled to fit on a tidy axis.

One problem with the word normalised is that it implies that there is some true original value from before normalisation that isn't necessarily between 0 and 1. Of course if that's true then the word is ideal, but if not then it's confusing.

Genuine question, i'm no math expert: if you're using the value by multiplying it with some other value, does that not imply it qualifies as normalised?

i.e. say you have a width of 500 and you want to move half way across so you have this value of 0.5 to get 250. By dividing 250 by 500, aren't we in fact normalising it?

I think generally normalized quantities maintain their original units whereas in your ratio case the unit is dropped. Units in the abstract sense I guess, such as this is a measurement along this vector or this is a ratio of any vector.

I think it's the opposite way round: normalisation results in units being dropped.

The typical, most common, definition of normalisation is value ÷ max possible value, giving a result in [0,1]. (More general definitions of normalisation exist e.g. if you rescale so the standard deviation is a fixed value, or even use non linear rescaling, that could count, but never mind all that.) The parent comment's example of "position along width ÷ total width" certainly fits that bill.

Whenever you divide something by the max of that something, the max is going to have the same units as the original value and you're bound to end up cancelling them. Or put another way, if you rescale 10cm into 0.5, it's certainly not 0.5cm so the units are either dropped or, at least, changed e.g. you could argue you've got 0.5x where x is the unit equal to 20cm.

Hmm when I think of 'normalizing' I don't think of dividing by a max at all - in my experience it is more taking a quantity (perhaps in English measurement) and transferring to a more 'standard' unit (say metric).

In general I don't think normalization always includes a sense of being in a bounded interval. From a mathematical perspective you could perhaps say normalization is achieved by multiplying your quantity by a 1D operator. You can't change the dimensionality this way, but are certainly changing 'units' a la mm in the x direction -> m in the x direction for example. I guess what I'm saying is that 'normalized' and the like are not the best fit for the SO question.

If I could take my own shot at the SO challenge from a mathematical perspective it would perhaps be sigmoid. Where the result of a normalization function takes a 1D value and maps it to a similar 1D value, the sigmoid takes a 1D value and maps it to a similar 1D value between (0, 1). So if I want to drop the previous information and only keep the resulting map, I can say 'this is my sigmoided value' - IE it is impossible for it to be outside of that range. Unfortunately sigmoid also connotates a differentiable curve which is extraneous information...

https://en.wikipedia.org/wiki/Normalization_(statistics)

https://en.wikipedia.org/wiki/Sigmoid_function

Of course the literal meaning of "normalisation" is to make more "normal", and that can mean almost anything at all. Even the Wikipedia article you linked to starts with "normalization can have a range of meanings". If you have heard that word most used with one meaning and I have heard it most used with another meaning then that doesn't invalidate either of those definitions.

The Wikipedia article you linked to gives two very broad definitions in the lead. The definition covered in the first paragraph is "adjusting values measured on different scales to a notionally common scale", which seems to be what you're talking about.

The definition covered in the second paragraph is "the creation of shifted and scaled versions of statistics", in particular "some types of normalization involve only a rescaling, to arrive at values relative to some size variable". I'm not comfortable with the use of "of statistics" in that second definition: the very first example is in the article standard score [1], which is about a rescaled element of the population, not a rescaled statistic. In any case, outside of statistics, a rescaling is a common meaning for this word, and a rescaled statistic is clearly just a special case of this. I think it was clear from the context that we were talking about this meaning originally. By far the most common case of this is a linear rescaling (including translation) to [0,1] but I was already up front that this is just a special case.

As for your sigmoid comment, I may have misunderstood but it sounds like you're saying that if you have a variable in the range in [0,1] then it can be described as the result of a sigmoid function. My objection to this is the same as my original objection to calling such as variable "normalised": it is a confusing variable name to use unless you actually did get it by applying a sigmoid function to something, not just because it holds a value that could hypothetically be obtained from a sigmoid function (but you didn't).

[1] https://en.wikipedia.org/wiki/Standard_score

> if you're using the value by multiplying it with some other value, does that not imply it qualifies as normalised

Certainly, "normalisation" can mean something more general than "rescale to the interval [0,1]".

For example, "rescaling to the interval [0, 20]" might make sense in some contexts and would usually count as a type of normalisation, and would still involve muliplying by a number. But that would be multiplying by (20 / max possible value) rather than the typical case of multiplying by (1 / max possible value).

So normalisation can mean something more general than the most usual common case, but it's not just any old "multiplying it with some other value". It has to specifically for the purpose of rescaling to some fixed, more useful/sensible range.

> say you have a width of 500 and you want to move half way across so you have this value of 0.5 to get 250

This is a great example of multiplication that isn't normalisation! You've multiplied by 0.5 to get the midpoint, but that process isn't normalisation because you've not ended up with some more sensible range. You started with [0,infinity) (the set of all possible widths) and ended up with that same infinite range.

> By dividing 250 by 500

Hang on, I'm confused about your example: are you asking about 500*0.5 (=250) or 250/500 (=0.5)? If it's the latter than that's normalisation, and not even something fancy or general but classic linear rescaling to the interval [0,1]. Yes, that's certainly normalisation.

----

Going back to my original comment: I was talking about values that were in the range 0 to 1 that hadn't reached that range by being multiplied by anything at all; they were just naturally in that range to begin with. For example, a probability would fit this bill.

> If it's the latter than that's normalisation.

I'm very clumsily trying to say that if:

  width / maxWidth = normalWidth

then:

  normalWidth * maxWidth = width.

And so, even though we aren't necessary calculating the normal in the second equation, we can still name our variable a 'normal' as rearranging proves that is indeed what the value is.

Edit: formatting

That's what I use, too: "normalized floats." I would prefer a better word, though. I work a lot with audio, and "signed normalized floats" doesn't exactly roll off the tongue.

This would be like calling numbers constrained between 0 and 100 "multiplied". Yes, in some common situations you normalize data and it ends up between 0 and 1. And in some common situations you multiply and end up with numbers between 0 and 100 (like percentages).

But normalization doesn't always result in numbers between 0 and 1 and multiplication doesn't always result in numbers between 0 and 100.

Similarly, not all numbers constrained between 0 and 1 have been normalized and not all numbers constrained between 0 and 100 have been multiplied.

Add in the fact that the normalization statisticians most commonly use is z-score standardization (subtracting by the mean and dividing by standard deviation, resulting in data centered around 0), and you're going to end up with a lot of confusion. In fact, this example highlights that while normalization does mean "to scale" it doesn't mean that the result will always be bounded to a particular interval.

This has a nice confusion factor because IEEE calls such floats denormalized. Because they may have extra precision and different format.

Perhaps I'm misunderstanding you, but with floating point, denormalized/denormal numbers aren't just numbers with a magnitude less than 1, they're numbers so small that it's no longer possible to adjust the exponent in the usual way, so leading zeroes are used in the mantissa. This reduces precision.

https://en.wikipedia.org/wiki/IEEE_754-1985#Examples

https://wiki.sei.cmu.edu/confluence/display/c/FLP05-C.+Do+no...

That's a good suggestion. 'Normalizing' is also used to refer to taking a vector and scaling it to make its length equal to 1. Google tells me it's also used in an analogous way elsewhere in maths.

https://mathworld.wolfram.com/NormalizedVector.html

I don't think "normalised" is specific enough. Normalising can refer to division by values other than the maximum. I'm also not convinced it implies non-negativity.

(My vote's with "proportion" BTW.)

https://en.wikipedia.org/wiki/Normalization_(statistics)

That isn't the right term. The standard meaning of "normalisation" in most scientific and engineering domains is shifting and scaling to get the sample mean to 0 and sample standard deviation to 1. I.e. it will almost always include both negative numbers and numbers larger than 1.

FWIW, I'd call that the z-score.

I use this term for https://github.com/VCVRack/Rack. Sometimes I say "`x` is a normalized float between 0 to 1" to somewhat reveal the definition of the term.

"Normalized from 0 to 1" is what I say as a non-native english speaker

I am trying to think of what the opposite word is of normalization.

In the spaces I work in, people just say denormalized / denormalization.

That was easier than I imagined, thanks :)

Weirdization or maybe WeirdAlization if you like Weird Al.

Normalized could also imply demeaned.

You could also call that centred to clear up confusion, or mean-subtracted to be more specific. I think if you're specifically dealing with statistical quantities it should be obvious, e.g. standardised if you also scale to unit variance.

Personally if I saw normalised I would assume scaled by a constant, not offset (as in subtraction).

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