Standard deviation is a form of variance that has the same units as the mean. Whereas the variance has square units.

This means the quantity of mean/variance depends on your choice of units, whereas the quantity of mean/standard deviation. Is constant no matter the units.

Why do we care about variance? Because it is a nice and linear property. This means it is easy to calculate and manipulate.

Moreover, the variance / standard deviation very nicely describe a Gaussian distribution (bell curve). This is a very important distribution because of the law of large numbers. Because we see bell curves so often, it is nice to have tools (std.dev) that work well with these curves.

It should be noted that, in optimization problems, there can be reasons to try and minimize something other than variance. We tend to pick the variance / std.dev because it is familiar, easy to work with, and very efficient. Notably, the derivative of variance tends to be linear, which makes it pretty efficient to use in gradient descent.

This is the right answer but maybe not to the right question. For most applications the linearity of variance does not really matter (e.g. when taking descriptives of a sample), at least in the way they are used/taught. Not that many appliers of statistics do computations where the mathematical properties of variance have any significance, and even when it does (e.g. ANOVA), most appliers don't really understand where it comes from.

I think the OP's frustration in how computation of standard deviation as a quantity is just taught by rote in a cookbook manner is quite understandable. If one's not thought more in depth about distributions and estimation of their parameters, it really makes little sense.

Whether the statistics should be taught more in depth is another question. The current state of the mathematics of statistics and probability is such that it needs quite heavy tools to do rigorously. And countless of hours of getting to know the quirky inconsistent notation.

But until it is taught more rigorously, more intuitive measures of dispersion like MAD make a lot more sense for most applications. And I don't think it would be an unreasonable ask for the statisticians to lay out the "porcelain" for working with the more intuitive, if somewhat more analytically inconvenient measures. A bit like we don't require programmers to understand how transistors work.

Try this?

https://web.stanford.edu/class/ee486/doc/chap5.pdf

Edit: standard deviation is just a fancy name for root mean squared over the sample offsets. The square is there to deal with the fact that there are positive and negative offsets. The mean is the whole point of the exercise, and the root is to negate the fact that we squared earlier.

But why not use the abs function or operator rather than squaring and rooting?

This is how I always find maths and particularly statistics - I always just want to answer ‘but why’ and then people lose interest in explaining after a while.

> But why not use the abs function or operator rather than squaring and rooting?

You can. It's called the mean absolute deviation. Which one you use depends on your application. If you just want to give a summary of how spread out the data is then either would work. Lots of people say that we should use the mean absolute deviation as the default rather than the standard deviation.

And suddenly another small subarea of maths starts to make sense for me. Thank you, and thanks to everyone commenting in this subthread.

I too have a problem with math tools passed down without any surrounding context - without telling why are we using this formula, instead of any other variant from the family of formulas that would satisfy the same goals. I'd have much easier time dealing with statistics in school if someone told me that a) deviation with ABS instead of root-square is also a thing, and b) we use the root-square one because it amplifies offsets from the mean.

By the way, statistics as its own discipline is taught rather differently from the "here is a toolkit of magic tables and formulas and how to use it" method used when teaching it to non-statisticians. The magical bits actually make sense.

>I too have a problem with math tools passed down without any surrounding context

Also one of my peeves. However over the years I've come to realize that in addition to teachers who omit the context, there is also a class of student that actively doesn't want to hear the context. I'm not sure exactly why this is, but I see it in my immediate family quite a bit. A sort of <cover ears, lalalala...too much detail> kind of thing perhaps due to difficulty taking on too much information?

The mean absolute deviation should be always introduced to people before introducing standard deviation. When trying to figure out what the heck is the standard deviation I also stumbled upon the mean absolute deviation. Now, to me, the standard deviation is the mean absolute deviation. It's just a version that is more sensitive to outliers. Otherwise, I still don't feel like I intuitively (visually) understand the standard deviation.

I feel like the standard deviation is a black box to basically everyone without statistics degree. It also bothers me that it is so widespread, because almost nobody know how to interpret this value. It seems to me like most scientists treat the standard deviation as a magical number that allows to compare spread between datasets and otherwise doesn't mean anything on it's own. I feel like a lot more insight could be gained from the mean absolute deviation.

I'm also curious how are scientific insights affected by defaulting to something that is extra-sensitive to outliers. The effect can't be big per single research, but in collective?

Okay, we have deviations from the mean. They are (0.1, 0.1, -0.1, -0.1). Or we have deviations from the mean (0.2, 0, 0, -0.2). I think is is pretty intuitive that in the second case the deviation is larger. In the first case there are only two differing values that are 0.2 apart. In the second case there are three differing values and some are 0.2 apart and others are 0.4 apart. Also, in the first case there are two pairs of equal value and in the second case there is only one pair that is of equal value. The squaring and rooting does this for you. Of course, other functions could do a similar thing for you. The squaring and rooting has nice mathematical properties especially if you consider the variance, i.e., omitting the square root. That is much more differentiable than an absolute value.

Also, consider that the correlation coefficient consists of similar formulas. They all look like the theorem of pythagoras. This ties in nicely with the fact that observables can be seen as having angles in between them where the correlation coefficient is the cosine of these angles.

I still don’t understand why we pick a metric because it’s easy to work with. Isn’t it either the right metric or not?

The standard deviation is not arbitrary, it tells us what the final normal distribution would look like if you averaged a lot of independent distributions.

You can read up on the central limit theorem if you want to understand more.

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

Normal distribution can be parameterized with almost any measures of location and dispersion. The reason why standard deviation (which is just variance in hiding) is used is because it makes the equations (that non-statisticians practically never need to look at) a bit nicer given the other (mostly arbitrary) conventions used in the maths.

Put simply, standard deviation is arbitrary for most people.

I don't want to keep going on because it sounds like I'm trolling, but when you tell me 'it tells us what the final normal distribution would look like if you averaged a lot of independent distributions' I just want to say 'why do I care about that? where did that requirement come from?'

Economics of stocks is a good example, when you put a lot of stocks in one portfolio you can calculate its risk (standard deviation) and returns (average value), and see how much each piece of stock contributed to that. This is because the results of many variables looks like a normal distribution with the combined average and standard deviation. Now it gets a bit more complicated in the real world since stocks are not independent from each other, but it shouldn't be hard to understand why knowing the standard deviation of something is useful.

> Now it gets a bit more complicated in the real world since stocks are not independent from each other

This is also where I get frustrated.

I'm taught about the CLT but every place I think I might use it, the samples aren't really IDD when you look hard enough, so it doesn't apply.

Computer benchmarking (which is where I'm usually trying to apply statistics) is the classic example. One iteration changes the state of the computer for the next iteration. They aren't remotely IDD, but people still try to claim the CLT. Seems like all their stats beyond that point is broken.

Right, that is how statistics usually is done. Note however that you don't need IDD, independence is enough. So you assume independence, from that CLT follows which means that standard deviation is all that matters. Then the real world deviates from this based on how much your distributions fails to be independent.

I'd assume that computer benchmarks have very low correlation between runs, unless you leak ram or your components don't keep a stable temperature. So in that case the CLT seems to apply just fine.

I can't understand this. Surely either samples are independent, or they are not?

And they are not in a computer program. The state of, to just give one example, the processor cache depends on what happened in the previous run.

We have all this rigour in how statistical techniques are applied, and we make black or white decisions based on the numbers that come out of them, but at their very foundation we're saying 'whatever I'll just pretend this rule applies it's probably fine'. How can we build on that flawed foundation? I don't get it!

Statistical techniques are a model of reality: Reality is not rigorous, but when our models are rigorous, imprecision enters the computation at one place only. If our models aren't rigorous, imprecision occurs at more places, blurring the answer even more.

"All models are wrong, but some models are useful." Just remember that there's some imprecision in the answer a model gives you, and keep that fact in mind when applying that answer to make real-world decisions.

I think it begins with binomial expansion and coin tossing.

e.g. https://socratic.org/questions/what-is-the-difference-betwee...

> Isn’t it either the right metric or not?

There are many "right" metrics - this is one of them. The big picture is we want a measure/idea of how spread out the data is. One can come up with many ways to do this, and absolute value and root mean square are two of the most common. They correspond to L0 and L2 norms. There is also the L-infinity norm (just use the farthest point).

The L2 norm is the most complicated of the 3, but as people pointed out, it is one of the easiest to use analytically because of its nice mathematical properties.

Have a look at https://en.wikipedia.org/wiki/Statistical_dispersion for other measures.

>Isn’t it either the right metric or not?

To illustrate, perhaps an example closer to home:

If someone asked you to tell them "how many lines of code in this project?" you'd start wondering things like "do I count comments?", "do I count dependent libraries?", "perhaps some metric that is equivalent of a statement count but not strictly counting lines?" and on and on.

Statistical measures are like that. Someone with a decent knowledge of the problem space came up with the best stab at how to characterize the data.

I think you're right that there is an arbitrariness to this algorithm, and there are other ways of normalizing your data. And you could probably build entire new branches of statistics off of them (like non Euclidean geometry), and they'd probably even be better in some way. But, you'd be speaking a different language from everybody else.

Speaking of language, have you ever thought about how all names and grammatical structures are arbitrary as well? There are parallels.

No, standard deviation is special. If you have a large enough amount of independent distributions which gets averaged or added in some way, then the only thing which matters are their standard deviations and average values for what the final distribution looks like.

I'm well out of my wheelhouse, so anyone else chiming in would be appreciated. I'm not sure what you mean by "looks like", but I would imagine other "nonstandard" measures of deviation would also result in a consistent model.

My larger point is that, something can be arbitrary and still be meaningful. If people agree on a common meaning, it formalizes a communication protocol that allows for much better information sharing.

"Looks like" as in the difference approaches zero as the number of times approaches infinity.

This means that any distribution with standard deviation of X will behave as a normal distribution with standard deviation X when you have enough of them. Or in other words, any distribution with standard deviation of X will behave like any other distribution with standard deviation of X when you have enough of them. There is no other measure like this as there is no other equivalent to the central limit theorem. Therefore it makes sense to have this as a universal measure of random processes.

The mathematical representations and operations people see at school were picked because they are easy to work on most common problems. There is not this "correct math" thing, you pick what works best.

Start with the Normal or Multivariate normal. We start with this because it is the distribution that emerges naturally when we have lots of little errors adding up around a basically consistent effect (see also [0]). So basically any situation where something ought to happen but 101 things could also happen to cause small changes will turn out to be normally distributed. An exceedingly common occurrence + one that almost inevitably turns up when dealing with a practical error prone process.

Now that we have a normal distribution, the mean is an obvious metric to pick because it captures the notion of 'middle' in a useful sense. Then we have proofs that we can characterise the normal with the mean and 1 other parameter (normal normal) or a matrix (multivariate normal). We call that the standard deviation^.

The exact formula wasn't a coincidence. The normal can obviously be characterised by the mean and a statistic from another formula. There were a bunch of experiments tried (eg, using |x| instead of sqrt[x^2]) but it turned out that sqrt[x^2] had some other nice property that minimised some sort of error so they went with it as a standard. I forget what one, might be error of estimating the true parameters from a sample or similar.

We could characterise the the normal as an infinite sum or something quirky, but when people say 'easy to work with' they mean instead of a function or something quirky we can simply pick a number.

Standard Deviation isn't as important when working with non-normal distributions, although I think it still turns out to be useful. But its importance is that it characterises a normal apart from the information captured in the mean. I'm not a mathematician, YMMV, could be wrong, standard disclaimers.

[0] https://en.wikipedia.org/wiki/Central_limit_theorem

^ I'm not going to edit this but it occurs to me that we call it the Variance. Same thing as std. dev in my opinion.

' There is a fairly subtle observation to make - Normal is characterised by mean and std. dev, but the most efficient ( https://en.wikipedia.org/wiki/Efficiency_(statistics) ) unbiased estimator of the std. dev of the population is the adjusted std. dev of the sample. Therefore, accounting for mean, you can't get a more efficient characterisation of the normal than mean & std. dev. Ie, if you picked a formula other than std. dev then the most efficient estimators to characterise it would still be mean and std. dev. Don't recall if there are equally efficient choices but I think that proves there are none better.

That might have been the logic for why std. dev was chosen. Just a guess.

> how I always find maths and particularly statistics - I always just want to answer ‘but why’ and then people lose interest in explaining after a while.

Unless you try to solve more problems yourself, you won't understand the advantages or disadvantages of some methods in solving the problems. So the real answer is: construct the examples, compute and compare, don't approach it "philosophically."

All the methods used today survived because they were the solutions to some problems. It doesn't mean that everybody applies them properly: for that you have to get some experience yourself.

Learning about the historical development of the methods is also for me satisfying experience: e.g. logarithms aren't a concept devised for philosophical purposes, Napier developed them to save astronomers time and limit "slippery errors" of (their) calculations:

https://www.thocp.net/reference/sciences/mathematics/logarit...

I think you just described the common core method. The main problem is teachers don't understand all the methods they are teaching and kids pick up on this. Then the parents don't understand all the methods and the few intellectually curios help their kids figure it out; but he vast majority of people just tell their kids it's stupid.

I didn't address the topic of what and how children learn, I've given some ideas to the one who wrote:

"I have a PhD in computer science but I cannot understand long division"

For that, personal experience in devising the examples, evaluating them and comparing the results is the only way to get the understanding. One can't complain that one doesn't understand Greek texts if one personally never tried to learn the Greek letters. One could have learned "about" Greek but at the end one hasn't done the inevitably necessary preconditions to actually read Greek texts.

The error in thinking is believing that because one already learned about something else somewhere else he should somehow "understand" something without doing the necessary work on that something. The solution is "make your own homework" to fill the gaps.

The topic of what, how and why children learn in schools is much more complex.

Edit: Also, now reading the Wikipedia article: https://en.wikipedia.org/wiki/Long_division I do understand why people don't understand it: the taught notation for the process, used in many countries, is utterly confusing to me, so my reason for confusion when looking at various notations is not that I don't understand the nature of the method but that I haven't spent time analyzing what was built on top of the basically simple idea -- I can also "understand" all the notations but I'd also need more time for that. Still, having enough experience, I can claim that I understand the idea and the process even if I never learn all the notations and conventions in different countries (which part of the process you write where, what you write and what you don't when etc). The idea is simple, the conventions enforced don't have to be. Those who only learned the conventions maybe never invested any time to figure out what's behind the conventions: what is actually being done.

That exists too: https://en.wikipedia.org/wiki/Average_absolute_deviation

Standard deviation is mathematically easier to work with though and has nice properties e.g. you can compute the standard deviation of combining a bunch of independent things by combining their individual standard deviations.

The abs has a problem in that it has no derivative at 0. I imagine that's why people go with squaring and rooting.

Because the derivative at 0 of abs is undefined.

Why do I care about that though?

Taylor series, calculating maxima/minima, etc. also standard deviations compose in a very nice way. All throughout math, we use squared terms instead of absolute value all the time because it’s convenient.

1. Long division uses a similar idea to Euclid's algorithm for GCD. Are you familiar with that? Regardless, both algorithms are divide and conquer algorithms.

6240 / 5

= divide(6240, 5)

= divide(1240, 5) + 1000

= divide(240, 5) + 1000 + 200

= divide(40, 5) + 1000 + 200 + 40

= 1000 + 200 + 40 + 8

= 1248

----------

2. Regarding standard deviation, one of these bullets might help:

- The normal distribution has exactly one shape, centered at x=0. But it's useful to apply two transformations to it: translation and horizontal stretch. To translate a distribution left/right, change its mean. To horizontally stretch a normal distribution, change its standard deviation.

- Mean has units of length. Standard deviation has units of length. They tell you where the normal distribution is offset, and how wide it is. Mean and standard deviation are just measuring sticks/rulers for normal distributions.

- When people talk about the standard deviation with any arbitrary data, they're usually assuming the data is normally distributed. If the data is not normally distributed, standard deviation no longer refers to the width of the normal distribution, so we lose that visualization.

- With non-normally distributed data, the standard deviation is still useful as an analytical tool because taking the (sqrt of the) summed squares still gives us a number that grows as the data spreads further apart or if the distribution grows wider. There's a center point for the data (the mean), so to ensure you're measuring the overall spread of the data, you subtract the center point off each data point before squaring them. In other words, you're squaring deviations from the mean. And unlike the sum of absolute differences, the sum of squared differences (variance) is differentiable. Differentiability is a great property, so this is the standard way to compute a sum of deviations from the center.

GP must be talking about the shortcut procedure to do it that is taught in schools.

also maybe they are specific about they way they do long division in the US? it was a surprise to me in college too and i have a french background. it's not how the french system does long division...

Now you’re talking about something different. The proof of the division algorithm is built up from a bunch of other machinery. It relies on the fact that the ring of integers is a Euclidean domain.

Take a course in ring theory if you want to see how all of this machinery gets built up.