Talk:Mean

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated B-class, High-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
High Importance
 Field: Probability and statistics
One of the 500 most frequently viewed mathematics articles.
WikiProject Statistics (Rated B-class, Top-importance)
WikiProject icon

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

B-Class article B  This article has been rated as B-Class on the quality scale.
 Top  This article has been rated as Top-importance on the importance scale.
 

Just curious... Why not include the simple arithmetic process of finding the mean of a set of numbers? Some people don't understand Summation.

We already have a worked example for the arithmetic mean. I reverted your additions, for several reasons: First of all, you talk about two "methods" when in fact there is no principled difference between those "methods", it's just a difference in notation. Second, the bit about rounding the mean value to the nearest integer is just plain wrong: the mean age of a group is a well-defined concept. If you don't want a fractional answer, the concept you're looking for might be the median. --MarkSweep 05:45, 10 October 2005 (UTC)

There is a difference, Mark. I used the phrase "simple" arithmetic process. Someone may want to find the mean of a set of numbers, but if they do not understand Summation, then they do not understand the process used. Perhaps you could write out the process in summation so it would be easier. I know this site isn't a math lesson, but it would simply be helpful.

And if you have a number of people, you can't have eight tenths of a person, unless they are in a casket with no head. My reasoning for rounding is that having only part of a person is impossible. In other situations, such as classroom grades, having a mean of 75.8 would be totally reasonable. In the given situation, it was not.

I must respectfully disagree with your edit, but I will leave it.

can i know the reason[edit]

Hi,

Can I know the reason for revertig the link Dynamic arithmetic mean, median, mode calculator

Was it irrevlevant?

The calculator link did not strike me as particularly helpful, and the page has a lot of ads on it. --MarkSweep (call me collect) 06:22, 4 November 2005 (UTC)


Not helpfull??? Ok, I will not bother wiki with more links....

Better reason: the calculator is incorrect. For example, the median of the list (1,2,2,3,4,4,5) is 3. It isn't 4 as your link suggests. So thanks for not bothering us with more disreputable links. Silly rabbit 23:10, 23 November 2005 (UTC)

Relationship between different types of mean[edit]

I hase described these here:

http://en.wikipedia.org/wiki/Talk:Average#Relationship_between_different_types_of_mean

I would be very happy if someone could check them and add them to either article.

thanks.

The Rational Mean[edit]

The Rational Mean is the unifying concept which embraces all known Means and the most simple principle for generating high-order root-solving methods, as shown at: The Rational Mean

So, it is disturbing to realize so many people seem unaware of this basic arithmetical principles and methods. —Preceding unsigned comment added by [[User:{{{1}}}|{{{1}}}]] ([[User talk:{{{1}}}|talk]] • [[Special:Contributions/{{{1}}}|contribs]])


Population mean[edit]

Population mean redirects here, but this page seems to say that population mean is another term for the expected value. Looking at the Arithmetic mean, and *its* definition of a population mean, I don't think this page is correct. Fresheneesz 01:06, 15 February 2007 (UTC)

I added a section on Sample means, which also redirects here with no explanation of how the sample mean is a random variable, etc. I added it under Examples of Means, but perhaps a new section should be started called "population and sample means"? -- Vince 15:32, 17 April 2007 (UTC)

ok, created that section. I placed it at the bottom, but perhaps it can be fit in the article better. This takes care of the problem of sample mean and population mean redirecting to this article, which may have left some statistics students confused. --Vince |Talk| 21:22, 17 April 2007 (UTC)

More motivation and elementary discussion seems to be required[edit]

This article really needs a little motivation for people who may not be the most mathematically inclined readers. It shouldn't be too hard to introduce arithmetic means in a way that an elementary school student can understand (in principle anyway). I don't want to try to tackle this now, because I'm already involved in other projects, but it is something to consider. Silly rabbit 10:38, 23 April 2007 (UTC)

I don't think this page could get any more or less mathematical. See also arithmetic mean, and Wikibooks:Statistics, which both show really simple examples. +mt 05:43, 18 February 2008 (UTC)

maybe there could be another article with name such as "mean: a very elementary treatment" with lots of simple examples, since mean is a elementary school subject? kjetil1001Kjetil Halvorsen 16:43, 8 January 2010 (UTC) —Preceding unsigned comment added by Kjetil1001 (talkcontribs)

Doubt[edit]

There are things like Upper Quartile, Lower Quartile, Inter Queartile etc. But does anyone know about Decile? I have heard of this before. Does anyone know what it is? I have no idea. Indianescence (talk) 09:52, 21 January 2008 (UTC)

Quartiles and deciles are related, but different statistical measures. The best way to understand them is to follow examples, and calculate them yourself. An elementary stats book is recommended. +mt 05:43, 18 February 2008 (UTC)

Moving Average[edit]

Should this page reference Moving average? Yellowking (talk) 15:48, 6 July 2008 (UTC)

I think so, and have added it to the "Other means" section. +mt 18:31, 6 July 2008 (UTC)

Most general method for finding an average?[edit]

In the properties section there was the statement: "The most general method for defining a mean or average, y, takes any function of a list g(x_1, x_2, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the mean replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y)."

I removed this statement, because it is wrong. Median and maximum cannot be represented this way. It also seems stupid to me, to take a function of n variables and use only the values for arguments of the form (y, y, ..., y). HenningThielemann (talk) 18:38, 17 August 2008 (UTC)

Well, Im not even sure if I understand the statement anyway. Removal due to ambiguity if nothing else. As for seeming "stupid", bare in mind that ALL of statistics is merely defined. Arithmetic means have their use - but only in the uses we use them for. Likewise with anything else. I know it sounds circular, but only because it is... the methods, means, functions, are all pretty much man made, and are used because we know of no better way of describing data. So no, I dont think any notion is any more or less "stupid" than any other. --98.247.99.158 (talk) 23:25, 14 February 2010 (UTC)

Math/Stat properties of the mean[edit]

It would have been nice to include the reason the arithmetic mean is so popular for many statisticians (e.g., maximum likelihood estimator, BLUE, minimum mean square error, etc.) 198.102.213.253 (talk) 22:36, 28 January 2009 (UTC) Allen Fleishman, PhD


The first section after the Introduction[edit]

It seems to talk a lot about mode, median and range, and very little about the mean itself. Needs some improvement. —Preceding unsigned comment added by 92.16.2.42 (talk) 12:12, 9 April 2009 (UTC)

There are several types and interpretations of "mean". I'm guessing you were looking for arithmetic mean? +mt 14:40, 9 April 2009 (UTC)

graphics[edit]

i think it would be very useful to visualize the concept with such a graphic: [[1]]

midmean[edit]

i've also seen the term midmean which is the arithmetic mean of the central 50% of a data set. (the midmean is influenced less by extreme outlier values.) should this be mentioned? – ishwar  (speak) 14:07, 10 November 2009 (UTC)

nonlinear means[edit]

I will start in the near future to add a section about nonlinear means, such as the Frechet mean and others. Ideas welcome. Kjetil Halvorsen 16:27, 17 November 2009 (UTC) —Preceding unsigned comment added by Kjetil1001 (talkcontribs)

I think the new addition is confusing. "Linear mean" would typically refer to a mean that is linear as a function, not a mean that is defined on a vector space. At any rate, it is not true that the means thus far presented in the article are all linear (in either sense of the word). The geometric mean does not rely on the linear space structure, for instance. I would favor just presenting the Frechet mean alongside the others rather than dubbing it "non-linear" and the others "linear". Sławomir Biały (talk) 17:39, 17 November 2009 (UTC)
Also, the term "non-linear mean" should probably be found in a reliable source before it can be used in the article. Google doesn't seem to turn up much that conforms with the usage here. Sławomir Biały (talk) 17:49, 17 November 2009 (UTC)
There is prior art: the book "Shape and shape theory" by Kendall, Barden, Carne, Le, chapter 9, section 9.1:"concept of mean

in non-linear spaces" discusses the general problem of defining a mean on non-linear spaces. But I can change the title to: Means on non-linear spaces" and word the discussion better. Kjetil Halvorsen 19:41, 17 November 2009 (UTC) —Preceding unsigned comment added by Kjetil1001 (talkcontribs)

Well, exercise some care at any rate. The earlier version of the section falsely claimed that the means already defined were all "linear", but this was not true (in either sense of the word). It may be better to start a separate article (means in nonlinear spaces?) first, and then include a summary here. Sławomir Biały (talk) 20:32, 17 November 2009 (UTC)
I just briefly checked the source. I believe it is entirely about Frechet means. We have an article on that, so the best thing to do is to expand that article first. Sławomir Biały (talk) 20:39, 17 November 2009 (UTC)
 : OK, I go for something like that. But I should mention that the means possiblbe for non-linear spaces are not only
   Frechet means, they are extrinsic means, Karcher means, and in some settings Procrustes means. A general discussion of the problem 
    and different possible solutions seem better that just formulating one solution.   Kjetil Halvorsen 03:36, 18 November 2009 (UTC)  —Preceding unsigned comment added by Kjetil1001 (talkcontribs)  

"Assumed mean"?[edit]

begin quote from article[edit]

Assumed mean[edit]

Main article: Assumed mean formula

The assumed mean is used when calculating the mean where using the arithmetic mean is tiring.

end quote from article[edit]

What does that mean? I have two guesses as to the meaning of the phrase "assumed mean":

  • The mean of a set of numbers is one of those numbers. Thus that set of numbers "assumes" its mean.
  • One assumes that the mean is a particular specified number.

But those are just guesses. Can someone explain? Michael Hardy (talk) 04:53, 22 November 2009 (UTC)

I cannot explain, but I can shift the questions somewhat. Apparently, there is a formula known as the Assumed mean formula. I don't know what this is, who calls it that, or what it's good for. Google books wasn't much help either (the stub needs a lot of help, too). It doesn't seem sufficiently noteworthy to be included in the list here, so I have removed it. Sławomir Biały (talk) 09:15, 22 November 2009 (UTC)
As far as I'm aware it's a technique people in statistics used to use before computers were so common, and it might actually still be useful for accuracy with large quantities that differ little. You simply chose a nice number which you could subtract from every number to give a smallish remainder and just work with those remainders, at the end you add back the amount you first thought of. It was especially useful for getting a reasonably accurate value of the standard deviation with one pass over the data. Dmcq (talk) 12:41, 22 November 2009 (UTC)
I just had a look at the assumed mean formula article and it looks like an undigested formula and is less than useless. The main thing that the assumed mean method had going for it was the bit in Standard deviation#Simplification of formula where you can get the standard deviation without subtracting the actual mean. Dmcq (talk) 12:58, 22 November 2009 (UTC)

I think I've figured out what it's about. Maybe I'll see if I can clean up that article. It's quite unclearly written. Michael Hardy (talk) 04:52, 4 December 2009 (UTC)

Calculation of means[edit]

There should be a section about algorithms for calculation of means?, such as discussing the formulas for calculating the means iteratively by updating, usefull for very large amount of data, and numerical stability issues, such as ordering before summing, summing the smallest entries first, usefull when calculating means of huge quantities of data of wildly differing size? Opinions? --Kjetil Halvorsen 16:59, 8 January 2010 (UTC)


Disapproval[edit]

There is far too much information to consolidate mean and average. They are not the same thing. The average is just a central tendency... and includes a wide variety of discussion topics, all leading to their own in-depth pages. I see no reason to change this. You might as well consolidate average, mean, harmonic mean, geometric mean, etc, etc, etc into one article - but who would read it given its length and the variable subject matter? People want the information that pertains to the topic they look up - no matter how inconvenient it may be to the database manager. Convenience of information. Wikis have links to other articles for a reason. --98.247.99.158 (talk) 23:17, 14 February 2010 (UTC)

Reasoning; vote; time frame; and request for resolution[edit]

Reasoning

The above vote for disapproval seems to have hit at least one nail on the head, "People want the information that pertains to the topic they look up". My self evaluation concerning math, according to what I have read, is very elementary and thus a good reason to keep the articles separate.

Vote

It seems to me the "average" person, seeking information on the subject of Average, does not need to be redirected to such an in-depth article as Mean. Although certain information probably should be moved (and certainly any duplicate), from "Average" to "Mean" I vote do not merge or combine as a whole.

Time frame

  • I would like to inquire as to the normal length of time concerning the process to make a decision, to either move or not, so that template(s) can be removed? The last discussion in the "Average" article was in January, 2010 and the last discussion here was in February, 2010. While I think enough time should be alloted to allow consensus I have an aversion to permanent, semi-permanent, or other long term templates on an article.

Request for resolution

  • If there is no actual reason would someone please stir the pot or let it settle so readers do not have to be subjected to lingering templates? Thank you, Otr500 (talk) 10:00, 19 June 2010 (UTC)

I've removed the merge template as the consensus at Talk:Average was against a merge. Dmcq (talk) 11:20, 19 June 2010 (UTC)

quadratic mean[edit]

Why the quadratic mean is not explained? --Arnaugir (talk) 14:55, 7 March 2012 (UTC)

If words or phrases are in blue you can click on them to get to an article about them. Quadratic mean will get you to root mean square which is the same thing. Dmcq (talk) 15:35, 7 March 2012 (UTC)
I know about that, I mean why it isn't explained along with other types of means.--Arnaugir (talk) 21:02, 7 March 2012 (UTC)
Why should this article say anything more about it? There's an article about it, this article deals with means in general. The link takes you to an article that says more about it. Dmcq (talk) 23:01, 7 March 2012 (UTC)
Root mean square is widely used in electronics, electrical engineering, statistics, rotational dynamics, and lots of other areas, but it is seldom called the "quadratic mean", and seldom regarded as a plain "mean" so it is better placed in its own article, with just a link from the article on plain "mean" because it is unlikely that anyone wanting to know about plain "means" would be looking for RMS. I suppose we could also include "cubic mean" and "quartic mean" but these are also rare terms. Dbfirs 08:07, 8 March 2012 (UTC)
There's also standard deviation and what happens when one finds a linear regression line which looks like the same sort of thing even though its purpose is different. Dmcq (talk) 11:29, 8 March 2012 (UTC)
Yes, that's what I was thinking of in statistics. In general, the quadratic mean involves a "second moment of ....". There are also applications of "third moment" (cubic mean for skewness) and fourth moment (quartic mean for kurtosis) in both statistics and in mechanics, but I've forgotten the details other than moment of inertia and second moment of area. I expect we have mechanics articles somewhere? Dbfirs 21:49, 8 March 2012 (UTC)

Fractions[edit]

Under what circumstances might it not be appropriate to use normal calculations for a mean of "fractional parts of real numbers". The "Mean of circular quantities is a rather quirky method that seems non-standard to me, but might have applications in some fields. Can anyone explain to me when it might be needed for fractions? How are they "circular"? Would it be more appropriate to claim that the method could be used in modular arithmetic? Dbfirs 19:23, 15 April 2013 (UTC)

I think it meant modular arithmetic. Maybe the author was trying to dumb it down and ended up making it opaque. Duoduoduo (talk) 20:32, 15 April 2013 (UTC)
Thanks, I'll try to make it more transparent. Dbfirs 19:43, 16 April 2013 (UTC)

Reason for removing "move" tag[edit]

There has been a tag on subsection "Small sample sizes" of section "Population and sample means" to move portions of it to Sample mean since February 2013, but as far as I can see no reason is given. I'm removing it for two reasons: (1) No reason given and no subsequent discussion; and (2) the move would be inappropriate since this subsection is not about the sample mean and as far as I can see never even mentions it (except one place where it meant to say population mean) -- it's about inferring confidence interval bounds for the population mean. Duoduoduo (talk) 13:47, 26 April 2013 (UTC)

Section "Population and sample means" was tagged with template "Main article: Sample mean". Subsection "Small sample sizes" is a detail that belongs to that main article, not in "Mean" itself. Fgnievinski (talk) 19:52, 26 April 2013 (UTC)
But like I said, the subsection "Small sample sizes" has nothing to do with the sample mean. Why would we put some material into an article that the material has nothing to do with? Duoduoduo (talk) 20:10, 26 April 2013 (UTC)
The section (not subsection) title and the main article tag seem misleading; I've modified both and added a move portions tag too. See and let me know if you agree. Thanks. Fgnievinski (talk) 20:00, 27 April 2013 (UTC)

distinctions in the lead[edit]

I'm not quite sure, what the difference between the following two is supposed to be nor does it seem the article actually explains that somewhere clearly. Moreover at other places in the text they seem to be treated as the same.

It cites a rather reputable source (Feller) for that distinction, but i have the feeling either article fails to explain Feller's distinction clearly or Feller has been misread. Unfortunately i don't have access to Feller to check it myself.--Kmhkmh (talk) 04:43, 8 May 2013 (UTC)

The exact quote is "The terms mean, average, and mathematical expectation are synonymous". That really doesn't support what is written in the article. I'll change it accordingly. Dingo1729 (talk) 17:52, 9 May 2013 (UTC)
Thanks for checking the source. Unfortunately, the source's statement that "average" is a synonym for these conflicts with our article Average, which says In colloquial language average usually means the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However it can alternatively mean the median, the mode, or some other central or typical value. Moreover, I think that "average" is never used for continuous distributions, whereas "mean" and "expectation" are. So I think we need to get a quote from a different source that words this more carefully. Duoduoduo (talk) 19:50, 9 May 2013 (UTC)
I see your points. Feller actually says in his introduction (a couple of hundred pages before) that he is only considering discrete sample spaces, so he's ignoring continuous distributions. Of course we're not quoting directly so we have some freedom in how we say things. Would "mean, mathematical expectation (and sometimes average)" work? Or we could simply leave out "average" altogether. Dingo1729 (talk) 20:18, 9 May 2013 (UTC)
I like that wording. I'll put it in. Duoduoduo (talk) 20:23, 9 May 2013 (UTC)
I added "expected value" since that's so much more common than "mathematical expectation". It's still sourced to the same reference because Feller defines (on the same page) the "mean or expected value...". The links from both terms are to the same place, which might confuse, but unlinking either one seems wrong. Dingo1729 (talk) 16:31, 11 May 2013 (UTC)

Thanks for picking that up, but I'm still a bit wary about the result. Originally the article tried to distinguish between various kinds of means in the lead and this distinction gets a bit lost. Moreover I'd assume Feller says his sentence presumably in particular context (probability theory) and this is a context our article does not have! While in the context of probability/probability distribution Feller's line matches my personal experience, it does however not outside probability theory. For instance I've never seen the term expectation value/mathematical expectation being applied to descriptive statistics or general math (without a probability context), there I've only seen the usage of mean or average. Similarly I've never heard of geometric expectation value. So if I'm not completely mistaken about the usage of these terms, the quote by Feller is rather misleading as a first line in the lead as it doesn't provide context and neglects that article (and reader) operate in somewhat different context and hence are likely to misread the quote somewhat.--Kmhkmh (talk) 08:14, 12 May 2013 (UTC)

Statistics or Mathematics?[edit]

I changed the top of the article to say "This article is about the mathematical concept"; Duoduoduo reverted to "..statistical.." with the comment only used in statistics/probability context. Surely that can't be right. The Pythagorean means predate statistics by a couple of millenia. Then there's the mean value theorem. And the article contains "Mean of a function" and "Fréchet mean", which I wouldn't naturally place in Statistics. I was thinking of Statistics as a branch of Mathematics, so Mathematics would cover all these meanings. Anyone else have an opinion? Dingo1729 (talk) 18:16, 9 May 2013 (UTC)

Okay, I'm convinced. I'll change it back to "This article is about the mathematical concept". As for the first sentence, as you point out it refers to various areas of math, so "probability theory" is overly restrictive. See if you like my expanded version. Duoduoduo (talk) 19:57, 9 May 2013 (UTC)
I agree, see also my comment above regarding the lead.--Kmhkmh (talk) 09:05, 12 May 2013 (UTC)

Using a very small sample[edit]

I'm wondering what to do about this section of the article. It's more technical than seems appropriate for this article and I'm not happy with the claims it makes. To begin with, for a Bayesian there is no problem whatsoever with a single sample (or even zero samples) implying a confidence interval for the mean. It all comes from the prior. So I don't like the tone of the section which states precise confidence intervals as though this would be agreed by every statistician. A Bayesian might well come up with a confidence interval, but it would have little or no connection with the calculations here. The whole section reeks of murky hidden assumptions in the analysis. Though I can't put my finger on just what those assumptions are. So I'd really like to delete the section, or move it to another article (where?). And I'd like to state that this is a theoretical calculation which has no bearing on reality, and may show that there are problems with non-Bayesian statistics. But the section is well referenced and I can't find a balancing criticism of it. Any suggestions? — Preceding unsigned comment added by Dingo1729 (talkcontribs) 04:28, 18 September 2013 (UTC)

Since I'm not getting any feedback I think I'll move the section to Standard error of the mean#Correction for finite population as was suggest by (I think) Fgnievinski . Dingo1729 (talk) 04:02, 20 September 2013 (UTC)

Axiomatization/Original Research[edit]

I've deleted a large section from the article and moved it here. It's unreferenced and I can't find any references; I'm not really sure what to search for. I've copied the section here in case someone can salvage it. I'm usually pretty supportive of borderline original research in math articles, but this seems too much.Dingo1729 (talk) 21:11, 28 September 2013 (UTC)


Properties[edit]

All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

Weighted and unweighted means[edit]

Weighted mean[edit]

A weighted mean M is a function which maps tuples of positive numbers to a positive number

M : (0,\infty)^n \to (0,\infty)

such that the following properties hold:

  • "Fixed point": M(1,1,...,1) = 1
  • Homogeneity: Mx1, ..., λ xn) = λ M(x1, ..., xn) for all λ and xi. In vector notation: Mx) = λ Mx for all n-vectors x.
  • Monotonicity: If xi ≤ yi for each i, then MxMy

It follows

  • Boundedness: min xMx ≤ max x
  • Continuity:  \lim_{x\to y} M x = M y
  • There are means which are not differentiable. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a median), but is not differentiable.
  • All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
    • If f is bijective, then the generalized f-mean satisfies the fixed point property.
    • If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
    • In general a generalized f-mean will miss homogeneity.

The above properties imply techniques to construct more complex means:

If C, M1, ..., Mm are weighted means and p is a positive real number, then A and B defined by

 A x = C(M_1 x, \dots, M_m x) ,
 B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)} ,

are also weighted means.

Unweighted mean[edit]

Intuitively spoken, an unweighted mean is a weighted mean with equal weights. Since our definition of weighted mean above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.

Thus we define M to be an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same.

Symmetry: Mx = Mx) for all n-tuples x and permutations π on n-tuples.

Analogously to the weighted means, if C is a weighted mean and M1, ..., Mm are unweighted means and p is a positive real number, then A and B defined by

 A x = C(M_1 x, \dots, M_m x) ,
 B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)} ,

are also unweighted means.

Converting unweighted mean to weighted mean[edit]

An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean M and weight the numbers by natural numbers a_1,\dots,a_n. (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by

A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).

Means of tuples of different sizes[edit]

If a mean M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely

  • Given an arbitrary tuple x, which is partitioned into y1, ..., yk, then
M x \in \mathrm{convexhull}(M y_1, \dots, M y_k).
(See Convex hull.)

Dingo1729 (talk) 23:13, 28 September 2013 (UTC)