Talk:Average

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[edit] No more redirect

Good, glad to see that this is no longer a redirect. It should never have been a redirect to arithmetic mean. At the least it should discuss the median, the mode and the subtle misuse of averages of "convenient" types in advertising and propaganda. -- Derek Ross

Yip, I'll be sure to cite How to lie with statistics in the further information section. ;-) --snoyes 02:53 Mar 1, 2003 (UTC)
Excellent! My favourite book on arithmetic! -- Derek Ross

A mean is only one particular type of average. A weighted average could refer to a weighted median as well, so I don't think that a redirect is the right solution to use for the weighted average article. -- Derek Ross

[edit] Properties of Median

"Also note that 1/2 of the scores, namely{1,2,2}, have values <= median and the other half, namely{2,3,9}, have values >= median"

This is not true, 1,2,2,2 (4 values) are <= median and 2,2,2,3,9 (5 values) have values >= median. This means that 2/3 of the population are <= median, and 5/6 are >= median. Not 50/50. -- PRB

There are six values (1,2,2,2,3,9) in the sorted list. The list can be split in half giving two sorted lists of three values (1,2,2) and (2,3,9). The median is the mean of the largest value in the smaller list and of the smallest number in the larger list. When there are an odd number of values in the original list, the median will be the centremost number in the sorted list. -- Derek Ross | Talk

I always thought the median was ((highest-lowest)/2)+lowest, i.e. halfway between the lowest and the highest. So the median of {1,2,2,2,3,9} would be 5. If that's not the median, what is it? - Montréalais 09:37, 14 March 2006 (UTC)

I'm not sure that there is a name for ((highest-lowest)/2)+lowest but perhaps that's just ignorance on my part. The median of {1,2,2,2,3,9} is definitely 2 though. And in fact the median of {1,2,2,2,3,9 000 000} comes to 2 too ! -- Derek Ross | Talk 15:41, 14 March 2006 (UTC)
The name is midrange. Bo Jacoby 14:13, 26 March 2007 (UTC).

I think the definition of median is very vague and should be more precise. "middle", "higher half" and "lower half" are very vague terms. For example, one might think in a sequence of 1,23,24,25...40 the median is 23, because it is the number that separates the "lower half" (values <= 20 by some definition) from the "higher half." For folks looking for concise definitions of terms, the language is confusing.

"Median - the middle value that separates the higher half from the lower half of the data set"

[edit] Merge with central tendency

It looks like average and central tendency mean the same thing. If thats the case, they should be merged. The article on central tendency is so small that it would be an easy merge. Anyone agree? Fresheneesz 23:31, 18 March 2006 (UTC)

Go for it. -- Derek Ross | Talk 23:45, 18 March 2006 (UTC)

[edit] Relationship between different types of mean

It would be worth noting the relationship between averages:

H^2=AG (or perhaps HA=G^2)

and

G=A-V/2 (approximately)

where:

H=harmonic mean

A=arithmetic mean

G=geometric mean

V=variance

I think there is another relationship:

V=(A^2-G^2) or perhaps SD=(A^2-G^2)

The last relationship was alluded to in a footnote to Corporate Finance by (from memory) Brierly and Miers. I spent a lot of time trying to figure out the relationship, as I wanted to be able to calculate the variance for published time series where the monthly daily A and G means are published, but not V.

I do not know how to do the fancy mathematical format stuff - please could someone else do it for me? SURE BUT WHO IZ THIS?

The above means you can calculate different kinds of average even when you do not have access to the original data.

[edit] Technical tag

This article is ridiculously too complex for a subject so basic. I believe it needs to be completely rewritten to be understood by the average reader. -- Mwalcoff 03:59, 6 September 2007 (UTC)

This article is very difficult to understand even though it is describing relatively simple mathematical concepts. I have a college level math education (Calculus II) and cannot decipher parts of this article ("symetric with permutations of the list"??). The section "Measures of central tendency" is especially problematic. What this section describes is actually rather simple, but it seems to be written in a sort of obfuscated math-speak. I'm reluctant to edit it myself for fear of introducing some generalization that the wiki math geeks will object to. Could someone with a good knowledge of the subject please rewrite this section in a way that is more understandable to the general public? Thanks! Kaldari 19:42, 29 October 2007 (UTC)


If I were to rewrite this section I would begin as follows:

An average or mean is a method that creates a representative member of a list. For example, what single value best represents the kind of values in the list of numbers 2 and 8? There are many different possible answers to this question.

The most common type of average is the arithmetic average, sometimes simply called the mean. The arithmetic average of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. It is then simple to find that A = (2 + 8)/2 = 5. It is also obvious that switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. If we increase the number terms in the list of terms for which we want an average we get, for example, that the arithmetic average of 2 and 8 and 11 is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. It is again simple to find that A = (2 + 8 + 11)/3 = 7. Again we see that changing the order of the three members of the list does not change the result, eg., A = (8 + 11 + 2)/3 = 7. This summation method is easily generalized for lists with any number of elements.

There are many other kinds of averages. However, they can all be understood in the same manner. For example, sometimes it is informative to consider the geometric average. Here, instead of adding numbers we multiply them. Thus, the geometric average of 2 and 8 is obtained by solving for G in the following equation: 2 * 8 = G * G. Thus, the geometric average of 2 and 8 is G = sqrt( 2* 8) = 4. And again it is seen that changing the order of the members of the list to be averaged does not change the result: G = sqrt(8*2) = 4.

In finance people are often interested in the annualized return which is a different kind of average. To begin with an example consider two years in which the return in the first year is minus 10% and the return in the second year is plus 60%. Then the annualized return, R, would be obtained by solving the equation: (1 - 10%) * (1 + 60%) = (1 + R) * (1 + R). The value of R that makes this equation true is R = 12%. It is again to be noted that changing the order to find the annualized return of 60% and -10% gives the same result as the annualized return of -10% and 60%. This method can be generalized (see list below) to examples where the periods are not all of one year duration.

It should now be obvious that it would be easy to come up with many other ways of combining the elements of a list in a manner that does not change when the order of the list is changed. For each of them one can define an average based on that method.

Another often mentioned method of obtaining an average is a mode, M. Here the method for finding a mode is to take the list and set all numbers in the list equal to the most common value in the list. Thus, if the list is 1, 2, 2, 3, 3, 3, 4 then the method would be to transform this list to 3, 3, 3, 3, 3, 3, 3. If we instead began with the list M, M, M, M, M, M, M and set all its members equal to its most common member (requiring no transformation at all) then upon equating the two results we would find M = 3.

A final average worth discussing is the median, m. Its method is to order the list according to its magnitude and then repeatedly remove the pair consisting of the highest and lowest value till either one or two values are left. If two values are left replace them with their arithmetic average. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this list replace them by their arithmetic average (3 + 7)/2 = 5. Now do the same for the equal sized list consisting of all the same value M: M, M, M, M. It is already ordered. We remove the two end values to get M, M. We take their arithmetic average to get M. Finally, set this result equal to our previous result to get M = 5.

All averages (including esoteric ones like the Heronian mean described below) can be thought of as examples of this general method for obtaining averages. A number of averages, including the ones discussed above, that have been found to be useful in some circumstance or other are listed below along with their formal solutions.

[Insert list of averages or means.]

Amirab 05:37, 30 October 2007 (UTC)

Sorry, Amirab, but that's still way too complicated. The kind of person who needs to know what an average is is either quite young or uneducated with no math skills beyond arithmatic (and probably a reading level around that of a fifth-grader). -- Mwalcoff 01:27, 31 October 2007 (UTC)

Perhaps there are more than one kind of person who needs help understanding what an average is. The previous entries on the topic show that even those who contributed to the page did not have a full understanding of its meaning, which is not surprising since even technical mathematical writings appear unaware of the unifying generalization presented here. I believe that the page should not focus on one kind of person to the exclusion of others. I would be glad to see how to make the first example I suggest any simpler and I am sure the above suggestion can be improved by augmentation and reordering. But I would not be glad to see the increasing level of sophistication of the content of the rest of the discussion omitted because it is beyond the interests of those to whom only the first example is helpful. Amirab 16:07, 31 October 2007 (UTC)

This is a HUGE improvement. I have added your wording with some editing (mostly to keep wording consistent with other articles). Thanks for taking the time to write that up. I'm sure many school-kids will be grateful. Kaldari 22:27, 31 October 2007 (UTC)
I agree it is a great improvement. I may fiddle with it a little. Thanks for your work on this page. -- Mwalcoff 02:33, 1 November 2007 (UTC)

[edit] Annualization using geometric average return in finance

The annualization example is misleading. The geometric average is used in financial reporting, but it's known to be an approximation that is not a true average, because each percentage is based on a different quantity. It's perfectly true that, starting with $100, a 50% loss can be averaged with a 50% gain to calculate a 0% return on a $100 investment. But financial reporting uses the average of sequential rates of return, which does not provide a true average. For instance, a 50% loss on $100 leaves $50 at the end of the first period. The return for the second period is based on the second period starting value of $50. So if there were a 50% gain in the second period, this would be a $25 gain, for a total of $75 at the end of two periods. The geometric average return for the two periods is -13.4%. The average dollar loss is $12.50 per year or 12.5%.

In the example given in the article, the 10.08% return is a compound interest return. The rate appears higher than the expected 10% return because after the first year, the interest is added to the capital. In other words, the size of the investment increases, but the rate is based on the size of the original investment.

There seems to be a lot of misunderstanding in the general public about how financial averages are calculated. I don't think this article should include how the geometric average is used in finance, because finance uses the geometric average as an approximation, not as a true average. --64.181.90.156 17:58, 9 October 2007 (UTC)

I do not see that the annualization example is misleading, nor do I agree that geometric average is not a true average, nor do I agree that a geometric average is an approximation of something else that is more financially correct.
Define a factor as one plus a return, then factors muliply the original investment in an order independent manner.
The annualized factor is the geometric average of the yearly factors. 18 October 2007 (Amirab) —Preceding unsigned comment added by Amirab (talk • contribs) 17:18, 18 October 2007 (UTC)
Not sure if this well help clarify things, but here's something from a book I have (Essentials of Investments by Bodie et al, p 133). This is from a section about measuring rates of return, particularly in mutual funds: "The geometric return is also called a time-weighted average return because it ignores the quarter-to-quarter variation in funds under management....In fact, an investor will obtain a larger cumulative return if high returns are earned in those periods when additional sums have been invested, while the lower returns are realized when less money is at risk....The appeal of the time-weighted return is that in some cases we wish to ignore variation in money under management." It goes on to say that mutual funds are required to report time-weighted returns because the funds themselves don't control the amount of money under management. But the rate of return that's reported seems to be a compound return, which will vary depending on the frequency of compounding. I'll leave a message on Wikipedia:WikiProject Mathematics and see if someone there can help us with this. --Foggy Morning 01:18, 28 October 2007 (UTC)
To lose 50% in one year and gain 50% in the next year gives the same final result as losing 13.4% in two consecutive years:
$100.00 − 50.0% = $100.00 − $50.00 = $50.00; $50.00 + 50.0% = $50.00 + $25.00 = $75.00.
$100.00 − 13.4% = $100.00 − $13.40 = $86.60; $86.60 − 13.4% = $86.60 − $11.60 = $75.00.
Therefore it is quite reasonable to take 13.4% to be the average yearly loss, expressed as a percentage. I don't understand the objection, and no "approximation" is involved (other than rounding the decimal numbers to a finite precision). If we define the constant percentage that gives the same final result to be the average, it is the "true" average. If you take another definition, then the other definition becomes the "true" average. One kind of average is not a priori more true than the other, but one may be more useful than the other. Taking the arithmetic average of the percentages is simple but misleading, and completely useless when large percentages are involved.
Of course, any averaging method for presenting a rate of change by giving the average change will have to indicate the period over which this change takes place. That could be one year, as in the example, or a month, or whatever. An average rate of change of −13.4% per year is equivalent to an average rate of change of −1.2% per month.  --Lambiam 07:44, 28 October 2007 (UTC)
Lambiam, many thanks for answering! That's a beautifully clear explanation. What I'm not sure about is whether it's a good idea to bring up the annualization of financial returns in this article about average. In that example, the loss over 2 years is $25 on a $100 initial investment, or $12.50 loss per year. That's -12.5% of the original investment per year, but 13.4% due to price volatility at the end of the first year.
Finance uses the geometric mean to calculate certain returns. This is from the book I have -- "The geometric average of the quarterly returns is equal to the single per-period return that would give the same cumulative performance as the sequence of actual returns...[e.g.]...
rG = ((1+0.10) x (1+0.25) x (1-0.20) x (1+0.25))1/4 - 1 = 0.0829, or 8.29% "
As best I understand it, this is a quarterly compounded return, not a simple rate of return. And it seems to be averaging percent losses with percent gains.
I'm not at all opposed to annualization of financial returns being covered in Wikipedia -- I'd really like it to be covered. But I'm worried about it being here on the average page. Finance uses compounding and averages in ways that make sense within the world of finance but don't always make sense taken out of context, I think.
One thing that I think would be very helpful on this page is something about the arithmetic average of percentages. You mentioned that they're useless when large percentages are involved. Why is that? Can that be explained here in this article? Lambiam? or Amirab? :) --Foggy Morning 02:12, 1 November 2007 (UTC)

[edit] heronian mean

It is stated that the heronian mean cannot be expressed in one way, but in another. Please express it as a g-function. Bo Jacoby 11:05, 5 November 2007 (UTC).

I wonder if the Heronian mean should be mentioned at all here; in any case, I suspect that the generalization to multisets of sizes other than 2 is OR. Is there any published reliable source for this?
Properties of "true" averages I'd like to see mentioned (but I have no sources I can cite) are:
  • Symmetry: The average is really of a multiset; if presented as a list, permuting the list leaves the average invariant.
  • Monotonicity: If x1 ≤ x1', average[x1, x2, ..., xn] ≤ average[x1', x2, ..., xn].
  • Stability: Extension of the multiset with the average leaves the latter invariant: if average[x1, ..., xn] = a, also average[x1, ..., xn, a] = a.
The generalization of the Heronian mean does not have that property: Her[24, 726] = 294, but Her[24, 726, 294] = 287.  --Lambiam 20:47, 5 November 2007 (UTC)


I added the g-function and fixed the summation for the Heronian mean. Stability for weighted means would require that the extension with the average would also be with zero weight or all the weights will have to be renormalized. Amirab 22:26, 5 November 2007 (UTC)

Well, weighted mean is not even symmetric; the input is not a list of values but a list of values-with-weights. What about the question whether this generalized Heronian mean is an instance of Wikipedia:original research? Does any source outside Wikipedia handle this?  --Lambiam 23:56, 5 November 2007 (UTC)

Thank you for improving. A few comments:

  1. Quote: "An easy way to get a representative value from a list is to randomly pick any number from the list. However, the word 'average' is usually reserved for more sophisticated methods that are generally found to be more useful." Wrong. Taking samples is very useful.
  2. An important property of averages is the independence of units of measurement: average(constant·list)=constant·average(list). It applies to all generalized means. The property of independence of zero point, average(constant+list)=constant+average(list), is true for arithmetic mean, mode, and median, but not more generally.
  3. If the heronian mean is the arithmetic mean of the geometric means of all possible pairs of numbers taken from the list, then the lower limit of summation, 'j=i', must be corrected to 'j=i+1'. I have not heard about heronian mean elsewhere and I request an explanation of its value.
  4. The geometric mean of (−1,+1) is not even a real number. The useful application of geometric mean is restricted to positive numbers.
  5. The weighted mean is equal to the arithmetic mean of a list constructed by repeating each element of the original list the number of times indicated by the corresponding weight. List=(0,1), weights=(2,3). New list=(0,0,1,1,1). Arithmetical mean(new list)=(0+0+1+1+1)/5. Weighted mean(list, weights)=(2·0+3·1)/(2+3). So a weighted list is merely a shorthand notation for a list.

Bo Jacoby 11:53, 6 November 2007 (UTC).

Re 5. In general, the weights in a weighted mean do not need to be whole numbers.  --Lambiam 16:41, 6 November 2007 (UTC)
If the weights are rational numbers they can be changed into whole numbers by multiplication by a common denominator. If they are irrational numbers they can be approximated by rational numbers. So the important case is that of whole numbers. Bo Jacoby 09:01, 8 November 2007 (UTC).
Re 3. If you use "j=i+1", then the formula for the generalized Heronian mean no longer generalizes the original meaning of the Heronian mean of two values. The generalization given in Wikipedia is not the only one possible; another approach is to take a weighted sum of the arithmetic and geometric means, which is pursued in at least one published paper.[1].  --Lambiam 16:41, 6 November 2007 (UTC)
What is the heronian mean of (1,4)? 2 or 2.33? What is the use of the heronian mean? Bo Jacoby 09:01, 8 November 2007 (UTC).
It is 7/3. The Heronian mean is used in a formula for the volume of a frustum; see Heronian mean and Frustum.  --Lambiam 10:05, 8 November 2007 (UTC)

I am now curious to know if there exists a continuous symmetric function, g, of a list, x1 .. xn, of real non-negative numbers and some number m such that g(x1 .. xn) = g(m .. m) but where it is not the case that min(x1 ... xn) <= m <= max(x1 .. xn) ? I am also curious if "monotonicity" &/or "stability" need to be assumed or can be proved from different versions of the definition of an average. Amirab 16:32, 6 November 2007 (UTC)

Assuming that by "not the case" you mean "not necessarily the case" in the sense that there exists a counterexample list x1 .. xn falsifying the implication: constant functions do not have the min/max bounding property, and neither does, e.g., the sum of the cosines of the values.  --Lambiam 16:52, 6 November 2007 (UTC)

Lambiam, can you give me an example where there does not exist an angle between the min and max angles in the list that is input into the sum of cosinces? Or are you just using the property that cos(x+2*N*pi) = cos(x) to claim that the angle you want to pick is 2*N*pi away from one that does fall in the desired range? Amirab 03:16, 10 November 2007 (UTC)

The latter. It is easy to give counterexample functions that – unlike the cosine function – do not have symmetries like the 2π translation; basically for any non-monotonic function f the function defined by g(x1 .. xn) = Σ f(xi) will do. Take for example f(x) = exp(x) - x.  --Lambiam 19:33, 10 November 2007 (UTC)

So, for the example f(x) = exp(x) - x, what list does not have any solution, x, of g(x1 .. xn) = Σ f(xi) = g(x .. x) between or equal to the max and min of the list? The fact that there are also solutions outside that range, as there are with the cos and other continuous functions, does not seem an adequate counterexample. Amirab 20:12, 11 November 2007 (UTC)

I made it clear that my counterexample depended on the assumption that by "not the case" you meant "not necessarily the case" – which is different from "necessarily not the case". For the latter, here is one counterexample, Define
g(x1 .. xn) = Πi≠j(xi2+(xj−1)2)·Σ(xi−2)2.
So g(x,y) = (x2+(y−1)2)(y2+(x−1)2)((x−2)2+(y−2)2). Then g(0,1) = g(m,m) has the unique solution m = 2, and it is not the case that min(0,1) ≤ m ≤ max(0,1).  --Lambiam 21:11, 11 November 2007 (UTC)

Very clever. I am impressed! The unique real solution is outside the range. However, the complexity of your example makes me doubt that just any non-monotonic function can do this trick.

Can it be done with g(x1 .. xn) = Σxi2? That is, is there a list using this simple non-monotonic function, for which its average is outside the range of the list? Perhaps simply being non-monotonic is not what allows your clever function to create your counter example. The complexity of your example makes me think it is some other property that your function possesses that allows it to come up with a counter example. Amirab 05:53, 12 November 2007 (UTC)

For g = sum of squares, there are in general two possible values for m: ±((Σxi2)/n)1/2. You can prove that at least one of these two candidates is in range, but which one depends in a non-continuous way on the xi: just consider what happens for (x1, x2) = (−1+ε, 1+ε) as ε passes through 0 – to visualize this, draw the graphs of the xi and the two m-candidates as functions of ε. When you ask for the property that my function possesses that allows a counterexample, perhaps you should be asking what property it lacks that allows this. The discontinuity for g = sum of squares suggests that this is not a mathematically trivial matter. Anyway, this is drifting outside the purpose of this talk page.  --Lambiam 08:24, 12 November 2007 (UTC)

I think that the purpose of this page is to explain what an average is and thus, at least implicitly, to make clear what is not an average. If, as seems to be agreed, an average is defined by a function of a list that generates the average (i.e. by setting the function of a list equal to the same function with the members of the list replaced by the average value sought) then it is central to this page to make precise which generating functions are allowed and which are not. For instance, I believe that functions that are not symmetric under permutation of the list should not be allowed to be called the generator of an average because I believe that this symmetry is a necessary property of an average. I do believe that the sum of squares function should be allowed. It simply gives the rms value as the average. Thus, I believe it should be accepted as an average even though this sum of squares function is not monotonic when the range can include both positive and negative values. As I understand you, you believe that the definition of average should not allow the sum of squares function to be used as the generator of what is called an average because it is not monotonic. And you think that the reason that its lack of monotonicity is a problem is because it causes the average, taken to be the solution within the range of the list, to be a non-continuous function of the list even when the generating function is a continuous function of the list. Since I do not wish the rms value to be excluded from being called an average, I would not want the definition of average to require monotonicity and thus do not see the discontinuity of the value of the average as a function of the list as excluding it from such averages.

So the question is: What is an average?

Is an average the in-range result of any symmetric generating function, or is it only the in-range result of any monotonic and symmetric generating function? Or are there other necessary restrictions that do not follow from symmetry alone in order that the definition of average properly explicates the concept of average so that it matches the general intuition of the essence of what it means to be an average? Should continuity be added and not monotonicity, excluding rms from properly being called an average? Does stability follow from the resulting definition already or does it also need to be added? I think that answering these questions is all part of defining an average and, thus, proper for this page. It might even be within the purview of this page to point out which otherwise acceptable generating functions sometimes do not provide averages because of the in-range restriction. Amirab 18:30, 13 November 2007 (UTC)

I mentioned some properties above that I'd like to see mentioned, but I added: "I have no sources I can cite". Monotonicity is a requirement stemming from my intuition of averages: if some values in the data set go up, the average won't thereby come down. While I agree that it would be nice to have a definition that matches the general intuition of what it means to be an average, such a definition should be one we can find in a reliable source and properly cite; to concoct such a definition ourselves would amount to original research that cannot be used in the article.  --Lambiam 09:43, 14 November 2007 (UTC)

I guess Wikipedia has to wait till other publications catch up to all the advances made here in order for these insights to be made available. Even though it was out of Wiki bounds, I enjoyed the discovery process. Lambiam, you really helped me advance my own understanding. The only reference I know that addresses these issues in any context at the advanced level discussed here is a chapter on annualization in the book “Advanced Portfolio Attribution Analysis, New Approaches to Return and Risk” Published by Risk Books and Edited by Carl Bacon. I do not know of any reference that addresses the problem of formulating the most general definition of an average at the advanced level discussed here. Amirab 20:37, 14 November 2007 (UTC)

That's fascinating and I thoroughly enjoyed reading these comments! But I think it might be helpful to non-technical readers of Wikipedia if we moved some of the more technical information out of this basic article. What do you think about that? --Foggy Morning (talk) 01:55, 18 November 2007 (UTC)

It seems to me that the basic article is appropriately progressive. After a brief introduction, it starts, in the section titled “Calculating Averages,” with simple examples of the most popular types of averages and presents them in the context of the general definition so that it is clear what they have in common that allows them each to correctly be designated an average. Them the formulas for various types of averages are presented in a good approximation of ascending difficulty. The technical issues on the theoretical side are only lightly broached in the subsequent section titled “Other Averages,” where these technical issues are kept to a minimum. The more technical discussion is mainly confined to this discussion thread. All this is not to say that further improvement in presentation and otherwise is not possible. Perhaps you have some suggestions. Amirab (talk) 06:15, 19 November 2007 (UTC)

[edit] Requesting consensus on simplification

Simplify I think this article should be kept very simple. I'm reasonably certain that most people looking up "average" in Wikipedia are not mathematicians. I think simple explanations of mean, median and mode would be useful, preferably with pictures and some examples that anyone could relate to (such as average height of a group of 5 men). I think greek symbols should be avoided in this article. I would move everything except the basics to the "Other averages" section or a See also list.

This would be a pretty drastic change to the article. The article is part of the mathematics project, which contributes a lot of good technical stuff to Wiki. I don't want to discourage good contributions just because I personally don't think they belong in this particular article. And I don't want to fight the math project over what should and shouldn't be in this article.

I'm not a mathematician and I'm not part of the math project. I'll leave the project to decide how to deal with this article that they're working on. --Foggy Morning (talk) 23:41, 22 November 2007 (UTC)

Like Amirab above, I think the article should be progressive in the sense that also non-mathematicians can find a basic treatment offering digestible information on what they may be looking for, but should also offer more advanced material inasmuch as it is encyclopedic and available. The Manual of style for mathematics recommends to start simple, then move toward more abstract and technical statements as the article proceeds. While there is room for improvement, the present text of the article attempts to follow that recommendation.  --Lambiam 08:34, 23 November 2007 (UTC)
I do think that this article is a bit more technical than necessary. We could reorder the material a bit. I would move the "other averages" section down, under "moving average" and perhaps even under the etymology section. The paragraph on the annualized return can go in the "other averages" section. The paragraph on the Heronian mean can be deleted as that's a very uncommon one, as far as I know. The table can then go at the end of the article. I think that this would put the simpler stuff higher up. Additionally, the text in the first paragraph does more or less assume that you already know about the (arithmetic) mean, mode and median. I think there should be a bit more explanation on what they are, though of course we shouldn't duplicate the existing articles. Before explaining what unifies the mean, mode and median, perhaps we should first explain how they differ. -- Jitse Niesen (talk) 14:08, 23 November 2007 (UTC)
In my opinion every mathematical article that gets a significant percentage of stray hits from non-mathematicians should give them something, because the absolute numbers will be so high. In many cases a short introduction will do: While it's clear they needn't read on, it can provide them with some vague imagery to take away. But this article has the potential for much more. E.g.:
"The easiest way to take the average of some values is the arithmetic mean: Their sum divided by how many they are, e.g. (7+3+8+4+4)/5=5.2. Even easier to compute is the median: After sorting the values by size: 3,4,4,7,8, one or two (depending on whether it's an odd number of values or an even one) will be in the middle of the list. If it's one, that's already the median. If it's two, their arithmetic mean is the median.
In some cases the arithmetic mean is not adequate. E.g. if you want to determine the average body length of the vertebrae in a forest, a single elephant can make a bigger difference than it should: (7+3+8+4+4+601)/6=104.5. The simplest solution is to take the median. (In the example it increases from 4 to (4+7)/2=5.5, which isn't so bad.) Or you can throw away the maverick values before taking the arithmetic mean. Yet another possibility is to take the geometric mean, e.g. the fifth root of 7×3×8×4×4."
It should be possible to express this in appropriate language without making it too long. With the right kind of examples (to keep it concrete for the non-mathematicians) even mathematicians will enjoy this. --Hans Adler (talk) 20:42, 23 November 2007 (UTC)

Comment Hans Adler, I like your approach. For those of you wondering what to include here, try imagining that your young son asks you, "Dad, what does average mean?" So you and he decide look up "average" together in Wikipedia. What you read together at the very beginning is

"In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. The most common method is the arithmetic mean, but there are many other types of averages."

By the time you've tried to explain "central tendency", "data set", and "descriptive statistics" to your son, you've both agreed to go toss a ball in the back yard rather than try to decipher this maze of mathematical lingo that's completely beyond your son's comprehension. PLEASE try to put yourselves in the readers' shoes! --Foggy Morning (talk) 01:02, 24 November 2007 (UTC)

Comment We have Mean already, so making "Average" elementary, with a pointer to "Mean" for further study, makes perfect sense to me. The topic is obviously worth broad development. Pete St.John (talk) 15:58, 24 November 2007 (UTC)

I also believe that any article on average should start with the easiest case. The paragraph on the arithmetic mean does so with the example 2 + 8 = A + A. I believe that the article should make clear at each step why the particular average being considered is subsumed under the general name of an average. Otherwise this page on average should not exist and there should just be a separate article for each kind of average instead of this general article explaining the general concept that covers them all. The calculation section makes clear the connection to the general idea of average by explaining how each of the most common types of averages can be understood in its simplest form as an instance of the general concept. The section goes on to the next steps in understanding the concept of an average by explaining how to: Expand the arithmetic average from two to more terms, Calculate a geometric average, Calculate a mode. Calculate a median. It does this in a way that is not only clear, easy and accessible (and should be made even more so), but also does so in a way that makes it clear why each calculation is an example of the basic concept. Too often math is taught as a litany of algorithms without an explanation. That helps no one at any level. That is why it is important in an article that explains “average” that each kind of average be clearly seen as an instance of the essential concept, a function of a list replicated by the same function of a constant list, that makes something an average.

The “function of a list” definition of average includes all legitimate averages and is the clearest general definition, simplest general definition, best motivated by our intuition and best gets to the heart of the matter of what is the essence of an average. The Heronian mean and the annualization of returns, which are not all of a single year in duration, are examples that cannot be subsumed under the generalized f-mean. These two examples are the only ones presented in the article that show that the “function of a list” approach is technically necessary and superior to the f-mean definition. Amirab (talk) 19:52, 24 November 2007 (UTC)

As to the relationship of “average” to “mean,” to my ear average is more general since it sounds right to call a median a kind of average but it sounds awkward to call a median a kind of mean. Amirab (talk) 20:00, 24 November 2007 (UTC)

Do you have children?--Foggy Morning (talk) 01:43, 26 November 2007 (UTC)
What I was trying to get at is that "average" is a common English word, most children experience averaged grades pretty early, while "mean" (and, re the generalization Amirab gives, "moment") are technical terms. It's reasonable to assume some amount of numeracy for articles on technical subjects, and to try and reach a broader audience (with links to more technical articles) for items of broader interest. An article about chess should be approachable by schoolchildren, an article about the Najdorf Poisoned Pawn need not be. Pete St.John (talk) 18:12, 26 November 2007 (UTC)
After looking at the article Mean, I agree that most of the "more advanced stuff" here is more in place there, in the article Mean, than here. This article could just confine itself to three notions of average: (1) mean in the usual meaning of arithmetic mean, while pointing out that there are other means and referring for that to Mean; (2) mode; and (3) median (not necessarily presented in that order). Furthermore, I think all mention of Heronian mean should be removed as being OR.  --Lambiam 21:10, 26 November 2007 (UTC)
Sounds good, only I don't understand why you want to include the mode. I wouldn't know what to do with it and when it's safe to use it for anything. In fact, I had never heard of it before! Or is this standard school stuff in English-speaking countries? (As to the Heronian mean, yes I agree for the generalized version presented in the article. But the original one for just two values could provide just the kind of historical details that many non-mathematicians will like.) --Hans Adler (talk) 22:34, 26 November 2007 (UTC)
Just do a Google search on ["measures of central tendency"]. The first hit: "This section defines the three most common measures of central tendency: the mean, the median, and the mode."[2] The next: "Measures of central tendency—mean, median, and mode—can help you capture, with a single number, what is typical of the data."[3] And so on. The search term ["measures of average"] gives similar results:  --Lambiam 06:26, 27 November 2007 (UTC)
P.S. And here is a quote from the intro paragraph of our own article Mean: "It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. For instance, average house prices almost always use the median value for the average."  --Lambiam 06:31, 27 November 2007 (UTC)
Sorry, I am not questioning that it is a valid concept, and I see that I probably misunderstood your intentions, as I was thinking of a longish article discussing a few concepts that everybody should know in great detail. My point is that we probably can't explain the mode to laypeople in a safe way. If Wikipedia tells all the world in simple words that the mode is as valid as an average as are the mean and the median, this could make it simple for a company to make crappy claims such as: "Value X for our product is about 50. The approximate values of X for the competition's products are 10, 10, 70, 80, 90, 110, 232, 234 and 300, as is proved by these independent sources. Therefore the average is 10, and our product is five times better than average." Perhaps we can say something like: "The term 'average' can refer to arithmetic mean, median or one of many similar notions (see mean). As a technical term in statistics it also refers to the mode." And then go on to explain arithmetic mean and median with lots of examples of when and how to use them (or not). --Hans Adler (talk) 08:56, 27 November 2007 (UTC)
Explaining and illustrating possible pitfalls of the use of the mode of a data set as being representative for the data set does not seem that daunting a task to me. In any case, if the mode is commonly understood to be one of the common measures of central tendency, we as encyclopedians should not hide the information from our readers for fear of possible abuse, which is clearly still equally possible if we censure the information here.  --Lambiam 13:27, 27 November 2007 (UTC)

Something that Lambiam wrote just made me check whether there is a redirect from measure of central tendency to average. There is, and there are also redirects from central tendency, measure of central tendency, measures of central tendency, statistical average, average value and mean value. So here is a suggestion:

  1. The current technical content of average is copied or moved to measures of central tendency.
  2. The article average gets a warning like, e.g., "This article is an introduction to averages in the usual sense of the word. For the technical term in statistics, see under the synonym central tendency." And the page then proceeds to treat the subject in a way that laypeople can understand, that is consistent with dictionary definitions of the word, and which is technically sound apart from using the "wrong" definition.
  3. statistical average, average value and mean value, as the only synonyms with potential for confusion, continue to redirect to average. All the others redirect to measures of central tendency.

--Hans Adler (talk) 21:23, 27 November 2007 (UTC)

A problem I have observed with assigning different roles to separate articles about what are essentially synonyms for the same topic, is that editors, unaware of the intention behind the division of content (or reaching different conclusions about the fuzzy criteria as to what belongs where), typically do not respect the intended respective characters of the several articles, leading in the long run to randomly different parallel articles on the same topic. Is it possible to give clear sharp criteria about what kind of treatment (none, basic, advanced) about which aspects is to be appropriate in articles on, respectively, Average, Central tendency, and Mean? I don't think a formulation like "averages in the usual sense of the word" cuts it. The dictionary definition "a typical amount, rate, degree, etc.; norm", the second sense for the lemma "average" on dictionary.com,[4], as well as the later "A number that typifies a set of numbers of which it is a function" from the American Heritage Dictionary, could also have been used as the definition of "measure of central tendency".  --Lambiam 07:52, 28 November 2007 (UTC)
How about articles "Average (common usage)" and "Average (mathematics)"? Then an editor who wants to link to average would either choose explicitly, or get a disambiguation page (at "Average"). Either way would simplify and clarify, for readers of any level. Pete St.John (talk) 18:37, 28 November 2007 (UTC)

As Lambiam said, I am already completely confused as to what this discussion intends will be on the "common usage" page and what will be on the "mathematics" page. Could it be that the "common usage" page will just be a list of a few (criteria?) ideas related to the concept of average without giving any indication of what they have in common or what the general concept is supposed to be. If so, I repeat, that it would be much better for each element of the list to have its own page and there be no general page at all, so people do not get confused into thinking they are supposed to learn anything about the general concept when they go to a page about averages in general. Or maybe the "common" page should just be a list of buttons to other pages without any explanation or comments at all; at least that will not be misleading. However, I still think it is possible and am holding out hope for a page to be properly structured so that it starts easy, explains the essence of the concept, and then moves on to technical examples and comments. What would an article be for if it purposely avoids explaining the essence of its concept? Amirab 68.175.110.10 (talk) 08:22, 29 November 2007 (UTC)

I don't know what "the essence of the concept" is (uniquely). I believe that a lay exposition (aimed mostly at schoolchildren) would read differently from a formal exposition of statistical moments (aimed mostly at college students). The page "Average (common usage)" would teach people how to divide the sum of a series of numbers by the number of numbers. It would have links to more technical pages (such as "Average (mathematics)" for those who want to learn more, and more technical articles could have links back to lay expositions, for people who bit off more than they can chew, or who are interested in teaching children, etc. I don't know why the "common usage" page would have to be "just a list" or why it would have to lack "any indication of what they have in common..." etc. Pete St.John (talk) 17:32, 29 November 2007 (UTC)

It seems that Lambian has edited the article to delete all mention of the Heronian mean. The Heronian mean has even been deleted from the list of equations of examples so that its very existence is censored from the article. Now the article no longer provides any example of why the definition of average that it offers is superior to the generalized f-mean. Amirab 68.175.110.10 (talk) 18:07, 12 December 2007 (UTC)

Well, as I wrote, this meaning of "Heronian mean" as operating on more than 2 values is original research, which should have no place in Wikipedia. If you can find a source for this meaning, I'll gladly reinstate it. However, to be quite honest, as far as I can see the "superior" definition of average is also OR. While it makes sense, it does not work for everything called "average" (weighted average, running average), and personally I think "my" criteria, which I did not put in the article because that would have been OR, are even more superiorer.  --Lambiam 21:28, 12 December 2007 (UTC)

[edit] Is there a nifty name for this average?

What about

e^{E(log(x_i)}

Is there a nice name for this average? If I for instance deal with log-normal distributed things it would be handy (now, in the log-normal case it happens to coincide with the median, but anyway it would be useful). —Preceding unsigned comment added by 80.252.189.234 (talk) 16:10, 2 January 2008 (UTC)

This is the geometric average. Our article only gives the formula for a data set, but looking at the formula in the section Geometric mean#Relationship with arithmetic mean of logarithms it should be obvious this is the same.  --Lambiam 17:26, 2 January 2008 (UTC)

[edit] Get real, please

Average is a word more or less understood in a general way when a kid is 8-10 years old. But not from this article, that's for certain. Please simplify this article and put more complex stuff in linked articles. Your help is greatly appreciated! --Foggy Morning (talk) 01:51, 22 January 2008 (UTC)

fwiw, I had advocated technical thoroughness under mean and simplicity here, for just that reason, but it didn't fly. I have trouble imagining someone looking under "average" to learn about geometric mean, which can merely be pointed to from "average" for further technical material. Unfortunately, I don't know of any WP:GETREAL :-( Pete St.John (talk) 19:30, 22 January 2008 (UTC)

I agree completely. Even as a mathematician, I am myself totally confused by the current mess that results from having separate pages on average and on mean, which seem to be contradicting each other as far as the uses of these two terms are concerned, and which are covering more or less the same material. I am too far removed from the subject to comment on the correct technical uses of these terms, and I don't want to edit in this article before that is settled. We are currently trying to heptagonate the moon by aiming at 10-year-olds and experts in the same asterisking article just because a technical term happens to coincide with a loosely related word of natural language. I will support every reasonable solution to this problem, including Peter's (if it turns out that the term "mean" is sufficiently general). Btw, this is of course not the only instance of this general problem. E.g. we have length vs. distance, weight vs. mass, and just one article on number (completely biased towards mathematics and its history in a narrow sense, ignoring other cultural aspects altogether). What I would really like to see is a solution like tree vs. tree (graph theory) that gives both aspects the weight that they deserve. --Hans Adler (talk) 20:55, 22 January 2008 (UTC)

Where are the high-school math teachers when you need one? :-) Pete St.John (talk) 21:08, 22 January 2008 (UTC)