4 ms·

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

by 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.

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.