5 ms·

What does the "mean" really mean?

47 points by max_ 5y ago

bethecloud 5y ago
The mean is just another magic number
shoto_io 5y ago
Is this really meaningful?
- jwilber 5y ago
  Seems far to philosophical to be useful in my opinion.
- minikites 5y ago
  I think it's helpful to general number literacy to know about the different types of mean and in which situations they are applicable: https://en.wikipedia.org/wiki/Geometric_mean#Applications https://en.wikipedia.org/wiki/Geometric_mean#Applications https://en.wikipedia.org/wiki/Harmonic_mean#Examples https://en.wikipedia.org/wiki/Harmonic_mean#Examples
michael1999 5y ago
In many systems, the mean really does summarize the whole. E.g. kinematics works just fine using the mean point mass for solids (i.e. the centre-of-gravity) in place of the whole.
- kergonath 5y ago
  It's an approximation, though. You can't really describe a rotating body looking only at its centre of mass. So it does summarise the whole in the sense that we choose to ignore what it cannot describe and live with it, but it's more a matter of convenience than a profound meaning.
  - morei 5y ago
    No, that's exactly the point. Knowing the center of mass and the moment of inertia is sufficient: You don't know need to know the exact shape, or the variations in density inside the object etc etc.
    - kergonath 5y ago
      Yes, the centre of mass and the moment of inertia, i.e., not only the centre of mass. The moment of inertia depends on the shape of the object.
      - ndr 5y ago
        And so does the centre of mass' position...
- ummonk 5y ago
  And notably it can have a validity to it that other averages don’t. E.g. in this case you can calculate the motion of the center of mass from the forces applied to the object. The same can’t be said for other averages.
- xenocyon 5y ago
  > In many systems, the mean really does summarize the whole That's quite true, and the same statement also points to where the arithmetic mean isn't a good choice: where the goal isn't so much to summarize a whole as to give an insight into populations. For instance, if you are trying to figure out the "average" income in a city, the median provides better intuition than the mean. Furthermore, there are many situations where both the mean and median (and even the mode) fail to provide insight. For instance, the average person has one testicle, whether defined by mean or median (and even the mode is not much help here, with its equally useless value of 0). We should really think of summary stats as something to only be used when a distribution is well understood. In other cases, a simple frequency distribution diagram provides much better insight, and isn't very hard to do.
  - verygoodname 5y ago
    > For instance, the average person has one testicle, whether defined by mean or median There are two options: 1) The ratio between women and men is not exactly 1. In this case, the median number of testicles is not 1 (it is either 0, if there are more women than men, or 2 if there are more men than women). 2) The ratio between women and men is exactly 1. This is not true in real life, but let's pretend it is. In this case, any number between 0 and 2 is a median (since any number between 0 and 2 minimizes the sum of absolute deviations), and not just 1. TL;DR: The median number of testicles is almost surely not 1 (or, at least, not uniquely so).
    - corin_ 5y ago
      The median is specifically the mid point of the data set, so it won't be "any number between 0 and 2", it will be one specific number depending on the data set. If you look at ten people with 2, ten people with 0, and one person with 1, then the median will be 1. If you make it twelve people with 2, still ten with 0 and one with 1, suddenly the median is 2. Across an entire population, I would hazard a guess the median would be 0 or 2 unless the difference in numbers of each sex is small enough that people with 0<X<2 testicles might fall right in the middle of the distribution.
nanis 5y ago
> average is merely an abstraction which has meaning only within its mathematical set-up. And probability is just a normalized denumerably additive measure over a sigma algebra. We've done very well constructing theories on the basis of these definitions, but it is useful to remember that neither has empirical content.
- leereeves 5y ago
  Probability began as the study of games of chance, with a lot of empirical content. It wasn't until centuries later that the current abstract mathematical foundation in measure theory (a normalized denumerably additive measure over a sigma algebra) was developed. And even today probability is taught with a lot of empirical examples. (At least it was in my classes, both undergraduate and graduate.)
  - nanis 5y ago
    > And even today probability is taught with a lot of empirical examples. (At least it was in my classes, both undergraduate and graduate.) Can you mention some of these examples? FYI, I've taught a bunch of Stats/Econometrics classes at grad and undergrad level.
    - leereeves 5y ago
      My university treated stats and probability as separate disciplines, and I didn't have as many empirical examples in stats classes. We did use data sets in exercises, of course, but the focus was on correctly applying statistical methods, not whether the problem was empirically interesting or the answer was empirically correct. We had a lot more empirical work in probability. In introductory probability classes we had, among other things, famous examples like the Chevalier de Méré, birthday, Monty Hall, Bertrand's ballot, and coupon collector's problems; and a lot of calculating probabilities involving coins, dice, cards, and drawing balls from urns in various and increasingly complicated ways. In graduate probability classes we studied mathematical models of things like Brownian motion, birth-death processes, and queuing systems. Our work on these was theoretical, naturally, but grounded in empirical problems from the physical sciences.
derbOac 5y ago
Nice little paper, although maybe a little stilted. It's a good idea to think about these basic quantities a bit more deeply and why you might want to use one measure of central tendency more than another. It's good to have a solid rationale for it, and I thought this was a nice brief overview of some perspectives I wasn't aware of. For some reason I thought the mean could be thought of as a single number that is most representative of a sample or population in an information loss (algorithmic/kolgomorov complexity, maybe relative entropy?) sense, or maximum likelihood sense (maybe under some distributional constraints?). I might be misremembering that though, and it's difficult to figure out the right search terms to track it down.
- ummonk 5y ago
  Information loss / maximum likelihood only applies in special case distributions. E.g. take a distribution of variable X and the arithmetic mean might be the appropriate average. But then if you look at sqrt(X) then the appropriate average would be RMS.
- verygoodname 5y ago
  You can use "maximum entropy" arguments to convince yourself or others that a Gaussian distributional assumption is the best (e.g. if the only thing you know is that your distribution has support over ]-inf,+inf[ and that its variance is bounded, the maximum entropy distribution is a Gaussian) and, in such a case, the sample arithmetic mean will give you the maximum likelihood estimate of the population "location" parameter (and, yes, along with variance, it will contain all required information to perfectly summarize your sample, since these are sufficient statistics, under a Gaussian assumption). TL;DR: If you can safely assume additiveness/normality, then... yes, mostly. Otherwise, not necessarily.
  - a9h74j 5y ago
    I assume not only physicists have the rule: never trust a scalar or statistic unless you can also see the distribution (histogram) behind it.
humanistbot 5y ago
What I got out of this was: Everyone knows that the mean and median are different ways of summarizing a distribution, which have different purposes. But did you know there are many different kinds of means too? The one you think of when you use "mean" is the arithmetic mean. But there are weighted means, geometric means, and harmonic means. The straightforward arithmetic mean isn't always the best way of summarizing the center of a distribution, even though it is conceptually the simplest and the way we've always done it.
- thaumasiotes 5y ago
  > Everyone knows that the mean and median are different ways of summarizing a distribution, which have different purposes. But did you know there are many different kinds of means too? Sure. More than that, the median is one of those different kinds of means. A mean is a statement about the total aggregated value of several data points. It answers the question "if all of these values were the same value, what would that value be?". And obviously, the answer depends on the details of how you aggregate the individual values. If your function takes a set of values and returns a value that is at least the minimum of the set and at most the maximum, it's a mean.
dxbydt 5y ago
Notice on page 7 the author briefly alludes to weighted means. That part is very useful to programmers & data scientists. If you build up good intuition about how to choose those weights, you can use them for all sorts of programming/networking/graphics problems. For example - suppose you host some data on 3 cloud providers - GCP, AWS & Azure. You believe, on average, the latency is the same for each provider. However, the variance is different - say 2,3 and 4. So, how much traffic should you route to each provider ? "What does the mean really mean" is the same as the above question. If you believe average latency is the same, then how should it matter what the variance is ? Just route 1/3 of your traffic to each provider & be done. But that's exactly the wrong answer! Because the arithmetic mean is not what you want here. You want to make your own mean. You really want a weighted mean, but you can't just pick the weights willy-nilly. The sample mean has its own variance! So if you say you don't give a rat's ass about statistics, you will just route 1/3 traffic to each provider - what you are really doing is picking weights of 1/3, in turn giving your sample mean xbar a sample variance of exactly 1 ( easy to show ). But if you pick the right weights, the sample variance drops to 12/13, which is smaller than 1. So by picking your own weighted mean, you get to fiddle with the sample variance - you can dial it up or down! The smallest you can get it down to is 12/13, in which case the weights will have to be 6/13 to GCP, 4/13 to AWS & 3/13 to Azure. Fun stuff.
- lupire 5y ago
  Why doesn't more "GCP" lower variance even more?
kazinator 5y ago
What's with that "associative law" on page 6? It seems to be asserting that, for all possible k, if the first k input values x_1, ... x_k are replaced by the mean x, then the mean doesn't change. Sorry what? (1 2 3 4 5 6) -> arithmetic mean is x = 3.5. Ok, let k = 3; replace first 3 elements by 3.5: (3.5 3.5 3.5 4 5 6) -> arithmetic mean is 4.25 Unless I'm totally daft, there has to be a typo there somewhere? -- Edit, ah, ok. Returning to this, I see where my eyes fooled me. The term on the left goes to x_k. Thus x is a "sub-mean" representative value which is of the first k values. If the first k values are replaced by their "sub-mean", then the "master mean" is undisturbed. That is very intuitive. In the case of an arithmetic mean, it's basically a distribution situation: a certain total quantity is distributed into n slots. If we transfer an amount from any slot to any other, the mean does not change. We can choose a subset of the slots, and redistribute their subtotal among them that they have an equal value. That value will be their respective mean. Since we do not add anything to the total, the mean doesn't change. What isn't so obvious is that this generalizes to the other kinds of means, like geometric.
- deleted 5y ago
  [deleted]
- CrazyStat 5y ago
  You replace x_1, ..., x_k with the mean of those k values, not the mean of all the values. So you would be replacing the first 3 values with 2 in your example, not 3.5
- verygoodname 5y ago
  > What isn't so obvious is that this generalizes to the other kinds of means, like geometric. It's obvious if you realize that the geometric mean is just exp(mean(log(values))) and the harmonic mean is just 1/mean(1/values), where "mean" is the arithmetic mean (i.e. they are just the arithmetic mean performed in a transformed space).
MontyCarloHall 5y ago
This is extremely similar to Miguel de Carvalho’s “Mean, what do you mean?” published four years earlier: https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2016.pdf https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2016.pd... In addition to discussing Kolmogorov’s construction of the mean (first moment), he extends it to higher moments (e.g. variance), and provides some nice examples. Pity the authors didn’t cite the aforementioned paper. Note that I’m not implying any plagiarism happened; the subject is elementary enough that different people could easily arrive at similar manuscripts independently.
tingletech 5y ago
2020
deleted 5y ago
[deleted]
motohagiography 5y ago
Obligatory xkcd: https://xkcd.com/2435/ https://xkcd.com/2435/ (Geothmetic Meandian)
Gunax 5y ago
Looks like there is an error in the table at page 7. The power generalisation should be raised to the power of 1/α, right?
wrp 5y ago
The most thorough treatment of means and their applications in statistics is the monograph by Corrado Gini.[1] It is also available in a 1970 Russian translation. [1] Gini, Corrado. (1958) Le Medie. Torino: Unione Tipografico-Editrice Trinese.

47 points by max_ 5y ago

bethecloud 5y ago

The mean is just another magic number

shoto_io 5y ago

Is this really meaningful?

jwilber 5y ago
Seems far to philosophical to be useful in my opinion.
minikites 5y ago
I think it's helpful to general number literacy to know about the different types of mean and in which situations they are applicable: https://en.wikipedia.org/wiki/Geometric_mean#Applications https://en.wikipedia.org/wiki/Geometric_mean#Applications https://en.wikipedia.org/wiki/Harmonic_mean#Examples https://en.wikipedia.org/wiki/Harmonic_mean#Examples

michael1999 5y ago

In many systems, the mean really does summarize the whole. E.g. kinematics works just fine using the mean point mass for solids (i.e. the centre-of-gravity) in place of the whole.

kergonath 5y ago
It's an approximation, though. You can't really describe a rotating body looking only at its centre of mass. So it does summarise the whole in the sense that we choose to ignore what it cannot describe and live with it, but it's more a matter of convenience than a profound meaning.
- morei 5y ago
  No, that's exactly the point. Knowing the center of mass and the moment of inertia is sufficient: You don't know need to know the exact shape, or the variations in density inside the object etc etc.
  - kergonath 5y ago
    Yes, the centre of mass and the moment of inertia, i.e., not only the centre of mass. The moment of inertia depends on the shape of the object.
    - ndr 5y ago
      And so does the centre of mass' position...
ummonk 5y ago
And notably it can have a validity to it that other averages don’t. E.g. in this case you can calculate the motion of the center of mass from the forces applied to the object. The same can’t be said for other averages.
xenocyon 5y ago
> In many systems, the mean really does summarize the whole That's quite true, and the same statement also points to where the arithmetic mean isn't a good choice: where the goal isn't so much to summarize a whole as to give an insight into populations. For instance, if you are trying to figure out the "average" income in a city, the median provides better intuition than the mean. Furthermore, there are many situations where both the mean and median (and even the mode) fail to provide insight. For instance, the average person has one testicle, whether defined by mean or median (and even the mode is not much help here, with its equally useless value of 0). We should really think of summary stats as something to only be used when a distribution is well understood. In other cases, a simple frequency distribution diagram provides much better insight, and isn't very hard to do.
- verygoodname 5y ago
  > For instance, the average person has one testicle, whether defined by mean or median There are two options: 1) The ratio between women and men is not exactly 1. In this case, the median number of testicles is not 1 (it is either 0, if there are more women than men, or 2 if there are more men than women). 2) The ratio between women and men is exactly 1. This is not true in real life, but let's pretend it is. In this case, any number between 0 and 2 is a median (since any number between 0 and 2 minimizes the sum of absolute deviations), and not just 1. TL;DR: The median number of testicles is almost surely not 1 (or, at least, not uniquely so).
  - corin_ 5y ago
    The median is specifically the mid point of the data set, so it won't be "any number between 0 and 2", it will be one specific number depending on the data set. If you look at ten people with 2, ten people with 0, and one person with 1, then the median will be 1. If you make it twelve people with 2, still ten with 0 and one with 1, suddenly the median is 2. Across an entire population, I would hazard a guess the median would be 0 or 2 unless the difference in numbers of each sex is small enough that people with 0<X<2 testicles might fall right in the middle of the distribution.

nanis 5y ago

> average is merely an abstraction which has meaning only within its mathematical set-up. And probability is just a normalized denumerably additive measure over a sigma algebra. We've done very well constructing theories on the basis of these definitions, but it is useful to remember that neither has empirical content.

leereeves 5y ago
Probability began as the study of games of chance, with a lot of empirical content. It wasn't until centuries later that the current abstract mathematical foundation in measure theory (a normalized denumerably additive measure over a sigma algebra) was developed. And even today probability is taught with a lot of empirical examples. (At least it was in my classes, both undergraduate and graduate.)
- nanis 5y ago
  > And even today probability is taught with a lot of empirical examples. (At least it was in my classes, both undergraduate and graduate.) Can you mention some of these examples? FYI, I've taught a bunch of Stats/Econometrics classes at grad and undergrad level.
  - leereeves 5y ago
    My university treated stats and probability as separate disciplines, and I didn't have as many empirical examples in stats classes. We did use data sets in exercises, of course, but the focus was on correctly applying statistical methods, not whether the problem was empirically interesting or the answer was empirically correct. We had a lot more empirical work in probability. In introductory probability classes we had, among other things, famous examples like the Chevalier de Méré, birthday, Monty Hall, Bertrand's ballot, and coupon collector's problems; and a lot of calculating probabilities involving coins, dice, cards, and drawing balls from urns in various and increasingly complicated ways. In graduate probability classes we studied mathematical models of things like Brownian motion, birth-death processes, and queuing systems. Our work on these was theoretical, naturally, but grounded in empirical problems from the physical sciences.

derbOac 5y ago

Nice little paper, although maybe a little stilted. It's a good idea to think about these basic quantities a bit more deeply and why you might want to use one measure of central tendency more than another. It's good to have a solid rationale for it, and I thought this was a nice brief overview of some perspectives I wasn't aware of. For some reason I thought the mean could be thought of as a single number that is most representative of a sample or population in an information loss (algorithmic/kolgomorov complexity, maybe relative entropy?) sense, or maximum likelihood sense (maybe under some distributional constraints?). I might be misremembering that though, and it's difficult to figure out the right search terms to track it down.

ummonk 5y ago
Information loss / maximum likelihood only applies in special case distributions. E.g. take a distribution of variable X and the arithmetic mean might be the appropriate average. But then if you look at sqrt(X) then the appropriate average would be RMS.
verygoodname 5y ago
You can use "maximum entropy" arguments to convince yourself or others that a Gaussian distributional assumption is the best (e.g. if the only thing you know is that your distribution has support over ]-inf,+inf[ and that its variance is bounded, the maximum entropy distribution is a Gaussian) and, in such a case, the sample arithmetic mean will give you the maximum likelihood estimate of the population "location" parameter (and, yes, along with variance, it will contain all required information to perfectly summarize your sample, since these are sufficient statistics, under a Gaussian assumption). TL;DR: If you can safely assume additiveness/normality, then... yes, mostly. Otherwise, not necessarily.
- a9h74j 5y ago
  I assume not only physicists have the rule: never trust a scalar or statistic unless you can also see the distribution (histogram) behind it.

humanistbot 5y ago

What I got out of this was: Everyone knows that the mean and median are different ways of summarizing a distribution, which have different purposes. But did you know there are many different kinds of means too? The one you think of when you use "mean" is the arithmetic mean. But there are weighted means, geometric means, and harmonic means. The straightforward arithmetic mean isn't always the best way of summarizing the center of a distribution, even though it is conceptually the simplest and the way we've always done it.

thaumasiotes 5y ago
> Everyone knows that the mean and median are different ways of summarizing a distribution, which have different purposes. But did you know there are many different kinds of means too? Sure. More than that, the median is one of those different kinds of means. A mean is a statement about the total aggregated value of several data points. It answers the question "if all of these values were the same value, what would that value be?". And obviously, the answer depends on the details of how you aggregate the individual values. If your function takes a set of values and returns a value that is at least the minimum of the set and at most the maximum, it's a mean.

dxbydt 5y ago

Notice on page 7 the author briefly alludes to weighted means. That part is very useful to programmers & data scientists. If you build up good intuition about how to choose those weights, you can use them for all sorts of programming/networking/graphics problems. For example - suppose you host some data on 3 cloud providers - GCP, AWS & Azure. You believe, on average, the latency is the same for each provider. However, the variance is different - say 2,3 and 4. So, how much traffic should you route to each provider ? "What does the mean really mean" is the same as the above question. If you believe average latency is the same, then how should it matter what the variance is ? Just route 1/3 of your traffic to each provider & be done. But that's exactly the wrong answer! Because the arithmetic mean is not what you want here. You want to make your own mean. You really want a weighted mean, but you can't just pick the weights willy-nilly. The sample mean has its own variance! So if you say you don't give a rat's ass about statistics, you will just route 1/3 traffic to each provider - what you are really doing is picking weights of 1/3, in turn giving your sample mean xbar a sample variance of exactly 1 ( easy to show ). But if you pick the right weights, the sample variance drops to 12/13, which is smaller than 1. So by picking your own weighted mean, you get to fiddle with the sample variance - you can dial it up or down! The smallest you can get it down to is 12/13, in which case the weights will have to be 6/13 to GCP, 4/13 to AWS & 3/13 to Azure. Fun stuff.

lupire 5y ago
Why doesn't more "GCP" lower variance even more?

kazinator 5y ago

What's with that "associative law" on page 6? It seems to be asserting that, for all possible k, if the first k input values x_1, ... x_k are replaced by the mean x, then the mean doesn't change. Sorry what? (1 2 3 4 5 6) -> arithmetic mean is x = 3.5. Ok, let k = 3; replace first 3 elements by 3.5: (3.5 3.5 3.5 4 5 6) -> arithmetic mean is 4.25 Unless I'm totally daft, there has to be a typo there somewhere? -- Edit, ah, ok. Returning to this, I see where my eyes fooled me. The term on the left goes to x_k. Thus x is a "sub-mean" representative value which is of the first k values. If the first k values are replaced by their "sub-mean", then the "master mean" is undisturbed. That is very intuitive. In the case of an arithmetic mean, it's basically a distribution situation: a certain total quantity is distributed into n slots. If we transfer an amount from any slot to any other, the mean does not change. We can choose a subset of the slots, and redistribute their subtotal among them that they have an equal value. That value will be their respective mean. Since we do not add anything to the total, the mean doesn't change. What isn't so obvious is that this generalizes to the other kinds of means, like geometric.

deleted 5y ago
[deleted]
CrazyStat 5y ago
You replace x_1, ..., x_k with the mean of those k values, not the mean of all the values. So you would be replacing the first 3 values with 2 in your example, not 3.5
verygoodname 5y ago
> What isn't so obvious is that this generalizes to the other kinds of means, like geometric. It's obvious if you realize that the geometric mean is just exp(mean(log(values))) and the harmonic mean is just 1/mean(1/values), where "mean" is the arithmetic mean (i.e. they are just the arithmetic mean performed in a transformed space).

MontyCarloHall 5y ago

This is extremely similar to Miguel de Carvalho’s “Mean, what do you mean?” published four years earlier: https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2016.pdf https://www.maths.ed.ac.uk/~mdecarv/papers/decarvalho2016.pd... In addition to discussing Kolmogorov’s construction of the mean (first moment), he extends it to higher moments (e.g. variance), and provides some nice examples. Pity the authors didn’t cite the aforementioned paper. Note that I’m not implying any plagiarism happened; the subject is elementary enough that different people could easily arrive at similar manuscripts independently.

tingletech 5y ago

2020

deleted 5y ago

[deleted]

motohagiography 5y ago

Obligatory xkcd: https://xkcd.com/2435/ https://xkcd.com/2435/ (Geothmetic Meandian)

Gunax 5y ago

Looks like there is an error in the table at page 7. The power generalisation should be raised to the power of 1/α, right?

wrp 5y ago

The most thorough treatment of means and their applications in statistics is the monograph by Corrado Gini.[1] It is also available in a 1970 Russian translation. [1] Gini, Corrado. (1958) Le Medie. Torino: Unione Tipografico-Editrice Trinese.