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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
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
"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
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 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.
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)
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? [Reaction interpolated by 69.223.83.131 22:42, 9 May 2007]
The above means you can calculate different kinds of average even when you do not have access to the original data.
[Above enquiry presented by
81.104.12.82 -
19:46, 30 June 2006 +
19:46, 30 June 2006 +
20:04, 30 June 2006]
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)
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)
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)
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)
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 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)
Thank you for improving. A few comments:
Bo Jacoby 11:53, 6 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)
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)
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)
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)
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 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)
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)
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)
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
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)
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:
-- Hans Adler ( talk) 21:23, 27 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)
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)
What about
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)
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)
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)
Will someone with an understanding of finance please correct the first example of Annualized return?
First, (1 − 10%) × (1 + 60%) = (1 + R) × (1 + R) is not a valid equation. The author is mixing percentages and real numbers. It should be:
.
Second, the solutions to that equation are -0.0513 and -1.9487, which can by no stretch of the imagination be transformed to 20% as stated in the article. In other words, the passage:
is gobbledygook.
Cheers Io ( talk) 21:29, 27 July 2008 (UTC)
Not sure if this is helpful or not, but in finance average annual returns are an average of annual percentage yields. The 10% loss and 60% gain in that example are based on different starting numbers. Like if you started with $100, lost 10%, you'd be down to $90. The 60% gain in the second period is based on $90, not $100. So the 0.6 is 60% of $90, which is 54% of the original $100. -- 64.181.88.31 ( talk) 11:11, 24 September 2008 (UTC)
I'm sorry but i don't understand the formula shown here already for this at all. I understand Summation notation but I belive that this is more right:
where the mean = the sum of all the numbers from the 1st through to the nth (amount of number that you have) this is then divided by the amount of number you have. I just don't get the 1 over n in the one already there, as far as I an see it doesn't work. Could whoever answers my quiery please explain it to me simply, without being rude. After all i'm intrigued in things like this. 95jb14 ( talk) 19:54, 12 December 2008 (UTC) Oooooops my mistake - sorry, I was wrong! —Preceding unsigned comment added by 95jb14 ( talk • contribs) 19:59, 12 December 2008 (UTC)
The article says in the " arithmetic mean" section:
Is there anything simple about that equation when you consider that the readers are total lay people (even in terms of arithmetic)? Unfortunately, most people on the internet can barely multiply numbers over 3. And even if one considers the readership to be more mathematically inclined than that, this equation is still written out in a wildly convoluted way.
Where I'm from, "simple" would be as follows:
The word simple appeared over three times in this section. To begin with, that word should be eliminated. Phrases like "It is simple to [then] calculate..." should not appear in this article. Three out of the four times the word appeared, it was next to some incomprehensible calculation or a calculation that the reader could hardly understand why s/he was doing it. Simple to whom? Often, what's written as simple is far from it. And considering the article's basic grammar mistakes, it's a bit ludicrous to banty that word around.
If you want to talk about "simple," look at the basic grammatical mistakes I uncovered in this article. A comma and a conjunction separates two independent clauses not a dependent and independent one. All of this may be perceived as just plain insulting and certainly prohibitive to non-mathemically inclined people. I know we can do a bit better here. 68.161.240.184 ( talk) 22:03, 14 January 2009 (UTC)
I'm glad that you agree. But there is still another issue. I personally believe that the section is written in a far too technical manner. An arithmetic mean or average is a concept that readers should be able to understand with no algebra background. There should be zero prerequisites, except for arithmetic of course, to understanding this concept via Wikipedia's article on it. Take the following paragraph:
I'm no expert on the subject, but it seems that this explanation is too algebraic. Most readers will not understand or gain insight from the sentence, The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. This may not seem so to some editors here, but it is actually confusing to someone without knowledge of algebra. Drop the variables (or even just the second variable) and it becomes intelligible to your audience. I am not suggesting that this entire paragraph be eliminated, but, rather, that it be fleshed out in lay language as well. Cheers, 68.161.240.184 ( talk) 16:17, 15 January 2009 (UTC)
Is there any different between "average" and "mean"? As I know is the same thing, so maybe these entries should be merged. Could anyone tell me know if I'm wrong? In Hebrew the meaning of "mean" and "average" is the same, but there is no link to the hebrew entry for "average" although there is a link from the "mean" page. Deltafunction ( talk) 13:32, 18 February 2009 (UTC)
I'd like to revisit this question. There is currently so much duplication in the two articles that they are virtually identical at the moment. I do appreciate that there is a difference between "mean" and "average", but I think that the content pertaining specifically to means should probably be merged to the mean article. Sławomir Biały ( talk) 03:28, 26 June 2009 (UTC)
Justin Peterson---- I think that mean should not be merged with average instead it should be the other way. I mean come on "average" is just like slang to Mean. Don't delete the "Mean" article delete the "average" article. Who agrees?
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JS - They shouldn't be merged.
Average defined as central tendency specifically refers to arithmetic mean in terms of sampling. Although the central tendency(population mean) of a sample may change as the population changes. E.g. calculating the average age within a population over time, i.e. as the population itself ages. While at any one time, the population does have an average age which is given by the arithmetic mean at that time, in the future the average age within the population will no longer be the arithmetic mean calculated in at present. It may be the geometric or harmonic mean of the current age distribution possibly. Average specifically refers to arithmetic mean, while the term "mean" almost always refers to the same thing, it however has other technical uses.
The Average article should have everything referencing the more complicated aspects of mean removed and only discuss central tendency and arithmetic mean(the words "arithmetic mean" should point to the appropriate section in the "mean" article). The "mean" article should have a subheading discussing average and central tendency though with links pointing back to the "average" article. I have never(or at least rarely... as I can't recall any) heard someone say "geometric average" or "harmonic average". Although saying it that way wouldn't necessarily be wrong. It just goes against convention. The point is that an average is a specific type of mean, and the word "mean" usually refers to expected value, but can refer to some other generic type of mean such as the harmonic mean.
The harmonic mean is a type of mean. So the set of all types of means contains the class of harmonic means.
The word "mean" in a mathematical context doesn't necessarily refer to "arithmetic mean"(maybe it does 99.99% or more of the time). I'd like to see a reference where the word "average" in a mathematical context refers to something other than the arithmetic mean. The true way to determine the dispute would be to find the average use of the word "average" and compare it to the average use of the word "mean"(in mathematical/statistical contexts). Then compare those two averages, or are they then proportions? Whichever one points to "arithmetic mean" more often loses out, i.e. gets its article cut down.
I agree with Derek Ross' last post.
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No it should not be merged. Median and mode are averages, but not a mean. It would be wrong to merge.
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There seems to be no overall consensus for merging, more against in fact, so I'm removing the templates. Dmcq ( talk) 11:17, 19 June 2010 (UTC)
Here is a step by step way to work out the mean median and modal of a frquency table: step 1: look at your table - Number of Hours TV | Students (frequency)
0 | 1 1 | 5 2 | 4 3 | 8 4 | 3 5 or more | 1
step 2: we need to work out how many students took part in this survey
1+5+4+8+3+1= 22
step 3: to work out the mean we need to divide the total number of students by the number of numbers which in this case is 6 so we: 22/6= 3.666667 which when simplified is =3.7 step 4: the median is a mathematical term for the middle number. to work this out you divide the total number of students by 2 in this case it is 11 and 12, so we need to find the 11th and 12th students and the number of hours of TV which is they are both in the 3 hours catergory. step 5: the modal is a mathematcal tern for the most common number, to work this out we look at the frequency coloumn first and the highest number here is 8. after finding this we look acroos to see where abouts that is on the other column in this case it is 3 hours. step 5: now we need to make our answers clear
Mean: 3.7 Median: 3 hours Mode: 3 hours
That was a simple question on a frequency table they can be much harder but once you understand the mathematcal terms u will be able to tackle any question. —Preceding unsigned comment added by Anfran94 ( talk • contribs) 17:52, 16 September 2009 (UTC)
In External links, add the following link: Averages: A New Approach.
This book provides additional information about averages and means, and it "contains neutral and accurate material that cannot be integrated into the Wikipedia article due to ... amount of detail ... ." (See Wikipedia's "External links": http://en.wikipedia.org/wiki/Wikipedia:External_links#What_should_be_linked). Smithpith ( talk) 19:00, 6 October 2009 (UTC)
Isn't there a sign for the average that is widely accepted and understood. This one:
Or is it just my perception, that this is common? It is not in the article and neither listed here: http://en.wikipedia.org/wiki/Table_of_mathematical_symbols —Preceding unsigned comment added by Kalyxo ( talk • contribs) 15:38, 10 January 2010 (UTC)
I put in my two cents in Talk:Mean#Disapproval and would also like to touch on something disturbing. While reading the talk pages I ran across this;
I also read this;
I will return to the regular broadcast and the subject at hand but first a disclaimer;
By the time I wrote this the template had been removed. This is one thing that gives Wikipedia an advantage over any other mainstream encyclopidia. The commercial was not over and the problem was solved. Thanks, Otr500 ( talk) 12:55, 19 June 2010 (UTC)
I don't believe the definition of mode (or the listed procedure for calculating it) listed in this article is adequate. For example, what is the mode of this set: {1, 1, 1, 2, 2, 2, 4, 4, 4} ? The article seems to assume that "most frequent elements" either come singly, or in pairs. 70.247.169.94 ( talk) 04:07, 4 July 2010 (UTC)
The Miscellaneous types section notes trimean and trimedian (and normalized mean). In an internet search I found definitions for trimean agreeing wth what is in the present article for trimean but almost nothing for trimedian, one of which was the same as what is in trimean and another which gave the following definitions:
so that here trimean disgrees with trimean.
So does anyone have a decent source defining these things in some well established way (so that the redlinks can be solved)? The citation at the end of the sentence does not seem to mention them. Similarly for "normalized mean", for which what I found seems to relate to means of subgroups compared to the overall mean, which seems out of place at that point in the article. Melcombe ( talk) 01:48, 23 May 2012 (UTC)
Does anyone think this image at the top of the "Calculations" section adds anything helpful to the article? I had to keep staring at it just to figure out what's going on in it, and once I did I still can't seem to gain any insight from it.
Can we remove it? Duoduoduo ( talk) 23:02, 19 April 2013 (UTC)
Agree with moves. I agree with all the sectional move tags. I propose that these moves be made, and that this article should end up as (1) the current lede, (2) the etymology section, and (3) a list of links preceded by something like "Average can refer in various contexts to any of the following." Duoduoduo ( talk) 14:19, 28 April 2013 (UTC)
I have moved and re-written this section.
1. This section was originally inserted into the middle of the Pythagorean means sections. It clearly didn't belong there.
2. The difference between an antiderivative and an integral is rather technical, but antiderivative certainly shouldn't be used here.
3. The average may still be finite even if the function tends to infinity. What matters is that the integral is finite.
4. I have removed some nonsense, for example “The prove for this equation lies in the equation to calculate an approximation of an area under a curve without using integration, where we would multiply the of a curve by we would be using."
5. If you read earlier entries in this talk page you will see that there are several complaints that this article is too mathematical. Someone else tagged sections to be merged elsewhere. We have very slowly been cleaning up the article and so I have tagged this section similarly. I still consider that this section is better covered in Mean with perhaps a very short note here. Dingo1729 ( talk) 05:19, 6 October 2013 (UTC)
Splitting the article would negate the benefit of focusing, in one place, on the word "average". Use of the word "average" can be a source of confusion due to author lack of clear delineation of the particular mean (arithmetic, geometric, harmonic) being reported on and inconsistency of verbiage within the authors report (e.g., after stating the particular mean being used authors fall into the confusion trap of using the word "average" subsequently). Such inconsistency leads to miscommunication and confusion. The Wikipedia article helps address these and related literature short comings. It also ties into one neat article the basic 'measures of central tendency' parameters.
Expansion of the basic parameter information of this article by other articles is a useful way to go. Furthermore. in a more general context, some repetition of information in various articles is beneficial. How to distribute related information across articles and manage repetition is an important topic unto its self. Managing repetition with an eye to erring on the side of more rather than less in a reasonable manner is the way to go.
Tegangwer ( talk) 04:10, 28 November 2013 (UTC)
Is this still a "start class" article? Seems pretty decent to me. -- PeterLFlomPhD ( talk) 22:10, 23 July 2015 (UTC)
Hello,
I've added a new "history" section, with a "origin" sub topic. I didn't figure out how to make the citations work for this section. Please help me clean it up. Thanks. Tal Galili ( talk) 18:59, 22 October 2015 (UTC)
The comment(s) below were originally left at Talk:Average/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Consider adding more elementary/introductory material. References would also help.
Geometry guy
14:45, 21 May 2007 (UTC)
"Measures of central tendency" section needs to be rewritten to be less technical. Kaldari 19:44, 29 October 2007 (UTC) This article is much too technical for an introductory article, and it's also lacking inline citations. -- Foggy Morning 02:25, 8 November 2007 (UTC) |
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Currently the lead says
The first and third sentences says that "average" means the arithmetic mean, contradicting the rest of the article, which gives a variety of alternative uses.
The second sentence is vacuous and was probably a soft vandalism.
The last sentence introduces two other concepts without saying whether they are averages.
So I'm going to rewrite the lead consistently with the rest of the article. Loraof ( talk) 16:12, 6 July 2017 (UTC)
The last sentence from the lead is "in colloquial usage any of these [mean, median, mode] might be called an average value. " is utter rubbish. Only an idiot would use the term average to refer to median or mode. There is no way that this is colloquial usage of the word "average." — Preceding unsigned comment added by 135.23.120.120 ( talk) 20:42, 8 October 2021 (UTC)
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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
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
"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
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 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.
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)
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? [Reaction interpolated by 69.223.83.131 22:42, 9 May 2007]
The above means you can calculate different kinds of average even when you do not have access to the original data.
[Above enquiry presented by
81.104.12.82 -
19:46, 30 June 2006 +
19:46, 30 June 2006 +
20:04, 30 June 2006]
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)
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)
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)
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)
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 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)
Thank you for improving. A few comments:
Bo Jacoby 11:53, 6 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)
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)
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)
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)
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 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)
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)
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)
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
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)
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:
-- Hans Adler ( talk) 21:23, 27 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)
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)
What about
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)
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)
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)
Will someone with an understanding of finance please correct the first example of Annualized return?
First, (1 − 10%) × (1 + 60%) = (1 + R) × (1 + R) is not a valid equation. The author is mixing percentages and real numbers. It should be:
.
Second, the solutions to that equation are -0.0513 and -1.9487, which can by no stretch of the imagination be transformed to 20% as stated in the article. In other words, the passage:
is gobbledygook.
Cheers Io ( talk) 21:29, 27 July 2008 (UTC)
Not sure if this is helpful or not, but in finance average annual returns are an average of annual percentage yields. The 10% loss and 60% gain in that example are based on different starting numbers. Like if you started with $100, lost 10%, you'd be down to $90. The 60% gain in the second period is based on $90, not $100. So the 0.6 is 60% of $90, which is 54% of the original $100. -- 64.181.88.31 ( talk) 11:11, 24 September 2008 (UTC)
I'm sorry but i don't understand the formula shown here already for this at all. I understand Summation notation but I belive that this is more right:
where the mean = the sum of all the numbers from the 1st through to the nth (amount of number that you have) this is then divided by the amount of number you have. I just don't get the 1 over n in the one already there, as far as I an see it doesn't work. Could whoever answers my quiery please explain it to me simply, without being rude. After all i'm intrigued in things like this. 95jb14 ( talk) 19:54, 12 December 2008 (UTC) Oooooops my mistake - sorry, I was wrong! —Preceding unsigned comment added by 95jb14 ( talk • contribs) 19:59, 12 December 2008 (UTC)
The article says in the " arithmetic mean" section:
Is there anything simple about that equation when you consider that the readers are total lay people (even in terms of arithmetic)? Unfortunately, most people on the internet can barely multiply numbers over 3. And even if one considers the readership to be more mathematically inclined than that, this equation is still written out in a wildly convoluted way.
Where I'm from, "simple" would be as follows:
The word simple appeared over three times in this section. To begin with, that word should be eliminated. Phrases like "It is simple to [then] calculate..." should not appear in this article. Three out of the four times the word appeared, it was next to some incomprehensible calculation or a calculation that the reader could hardly understand why s/he was doing it. Simple to whom? Often, what's written as simple is far from it. And considering the article's basic grammar mistakes, it's a bit ludicrous to banty that word around.
If you want to talk about "simple," look at the basic grammatical mistakes I uncovered in this article. A comma and a conjunction separates two independent clauses not a dependent and independent one. All of this may be perceived as just plain insulting and certainly prohibitive to non-mathemically inclined people. I know we can do a bit better here. 68.161.240.184 ( talk) 22:03, 14 January 2009 (UTC)
I'm glad that you agree. But there is still another issue. I personally believe that the section is written in a far too technical manner. An arithmetic mean or average is a concept that readers should be able to understand with no algebra background. There should be zero prerequisites, except for arithmetic of course, to understanding this concept via Wikipedia's article on it. Take the following paragraph:
I'm no expert on the subject, but it seems that this explanation is too algebraic. Most readers will not understand or gain insight from the sentence, The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. This may not seem so to some editors here, but it is actually confusing to someone without knowledge of algebra. Drop the variables (or even just the second variable) and it becomes intelligible to your audience. I am not suggesting that this entire paragraph be eliminated, but, rather, that it be fleshed out in lay language as well. Cheers, 68.161.240.184 ( talk) 16:17, 15 January 2009 (UTC)
Is there any different between "average" and "mean"? As I know is the same thing, so maybe these entries should be merged. Could anyone tell me know if I'm wrong? In Hebrew the meaning of "mean" and "average" is the same, but there is no link to the hebrew entry for "average" although there is a link from the "mean" page. Deltafunction ( talk) 13:32, 18 February 2009 (UTC)
I'd like to revisit this question. There is currently so much duplication in the two articles that they are virtually identical at the moment. I do appreciate that there is a difference between "mean" and "average", but I think that the content pertaining specifically to means should probably be merged to the mean article. Sławomir Biały ( talk) 03:28, 26 June 2009 (UTC)
Justin Peterson---- I think that mean should not be merged with average instead it should be the other way. I mean come on "average" is just like slang to Mean. Don't delete the "Mean" article delete the "average" article. Who agrees?
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JS - They shouldn't be merged.
Average defined as central tendency specifically refers to arithmetic mean in terms of sampling. Although the central tendency(population mean) of a sample may change as the population changes. E.g. calculating the average age within a population over time, i.e. as the population itself ages. While at any one time, the population does have an average age which is given by the arithmetic mean at that time, in the future the average age within the population will no longer be the arithmetic mean calculated in at present. It may be the geometric or harmonic mean of the current age distribution possibly. Average specifically refers to arithmetic mean, while the term "mean" almost always refers to the same thing, it however has other technical uses.
The Average article should have everything referencing the more complicated aspects of mean removed and only discuss central tendency and arithmetic mean(the words "arithmetic mean" should point to the appropriate section in the "mean" article). The "mean" article should have a subheading discussing average and central tendency though with links pointing back to the "average" article. I have never(or at least rarely... as I can't recall any) heard someone say "geometric average" or "harmonic average". Although saying it that way wouldn't necessarily be wrong. It just goes against convention. The point is that an average is a specific type of mean, and the word "mean" usually refers to expected value, but can refer to some other generic type of mean such as the harmonic mean.
The harmonic mean is a type of mean. So the set of all types of means contains the class of harmonic means.
The word "mean" in a mathematical context doesn't necessarily refer to "arithmetic mean"(maybe it does 99.99% or more of the time). I'd like to see a reference where the word "average" in a mathematical context refers to something other than the arithmetic mean. The true way to determine the dispute would be to find the average use of the word "average" and compare it to the average use of the word "mean"(in mathematical/statistical contexts). Then compare those two averages, or are they then proportions? Whichever one points to "arithmetic mean" more often loses out, i.e. gets its article cut down.
I agree with Derek Ross' last post.
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No it should not be merged. Median and mode are averages, but not a mean. It would be wrong to merge.
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There seems to be no overall consensus for merging, more against in fact, so I'm removing the templates. Dmcq ( talk) 11:17, 19 June 2010 (UTC)
Here is a step by step way to work out the mean median and modal of a frquency table: step 1: look at your table - Number of Hours TV | Students (frequency)
0 | 1 1 | 5 2 | 4 3 | 8 4 | 3 5 or more | 1
step 2: we need to work out how many students took part in this survey
1+5+4+8+3+1= 22
step 3: to work out the mean we need to divide the total number of students by the number of numbers which in this case is 6 so we: 22/6= 3.666667 which when simplified is =3.7 step 4: the median is a mathematical term for the middle number. to work this out you divide the total number of students by 2 in this case it is 11 and 12, so we need to find the 11th and 12th students and the number of hours of TV which is they are both in the 3 hours catergory. step 5: the modal is a mathematcal tern for the most common number, to work this out we look at the frequency coloumn first and the highest number here is 8. after finding this we look acroos to see where abouts that is on the other column in this case it is 3 hours. step 5: now we need to make our answers clear
Mean: 3.7 Median: 3 hours Mode: 3 hours
That was a simple question on a frequency table they can be much harder but once you understand the mathematcal terms u will be able to tackle any question. —Preceding unsigned comment added by Anfran94 ( talk • contribs) 17:52, 16 September 2009 (UTC)
In External links, add the following link: Averages: A New Approach.
This book provides additional information about averages and means, and it "contains neutral and accurate material that cannot be integrated into the Wikipedia article due to ... amount of detail ... ." (See Wikipedia's "External links": http://en.wikipedia.org/wiki/Wikipedia:External_links#What_should_be_linked). Smithpith ( talk) 19:00, 6 October 2009 (UTC)
Isn't there a sign for the average that is widely accepted and understood. This one:
Or is it just my perception, that this is common? It is not in the article and neither listed here: http://en.wikipedia.org/wiki/Table_of_mathematical_symbols —Preceding unsigned comment added by Kalyxo ( talk • contribs) 15:38, 10 January 2010 (UTC)
I put in my two cents in Talk:Mean#Disapproval and would also like to touch on something disturbing. While reading the talk pages I ran across this;
I also read this;
I will return to the regular broadcast and the subject at hand but first a disclaimer;
By the time I wrote this the template had been removed. This is one thing that gives Wikipedia an advantage over any other mainstream encyclopidia. The commercial was not over and the problem was solved. Thanks, Otr500 ( talk) 12:55, 19 June 2010 (UTC)
I don't believe the definition of mode (or the listed procedure for calculating it) listed in this article is adequate. For example, what is the mode of this set: {1, 1, 1, 2, 2, 2, 4, 4, 4} ? The article seems to assume that "most frequent elements" either come singly, or in pairs. 70.247.169.94 ( talk) 04:07, 4 July 2010 (UTC)
The Miscellaneous types section notes trimean and trimedian (and normalized mean). In an internet search I found definitions for trimean agreeing wth what is in the present article for trimean but almost nothing for trimedian, one of which was the same as what is in trimean and another which gave the following definitions:
so that here trimean disgrees with trimean.
So does anyone have a decent source defining these things in some well established way (so that the redlinks can be solved)? The citation at the end of the sentence does not seem to mention them. Similarly for "normalized mean", for which what I found seems to relate to means of subgroups compared to the overall mean, which seems out of place at that point in the article. Melcombe ( talk) 01:48, 23 May 2012 (UTC)
Does anyone think this image at the top of the "Calculations" section adds anything helpful to the article? I had to keep staring at it just to figure out what's going on in it, and once I did I still can't seem to gain any insight from it.
Can we remove it? Duoduoduo ( talk) 23:02, 19 April 2013 (UTC)
Agree with moves. I agree with all the sectional move tags. I propose that these moves be made, and that this article should end up as (1) the current lede, (2) the etymology section, and (3) a list of links preceded by something like "Average can refer in various contexts to any of the following." Duoduoduo ( talk) 14:19, 28 April 2013 (UTC)
I have moved and re-written this section.
1. This section was originally inserted into the middle of the Pythagorean means sections. It clearly didn't belong there.
2. The difference between an antiderivative and an integral is rather technical, but antiderivative certainly shouldn't be used here.
3. The average may still be finite even if the function tends to infinity. What matters is that the integral is finite.
4. I have removed some nonsense, for example “The prove for this equation lies in the equation to calculate an approximation of an area under a curve without using integration, where we would multiply the of a curve by we would be using."
5. If you read earlier entries in this talk page you will see that there are several complaints that this article is too mathematical. Someone else tagged sections to be merged elsewhere. We have very slowly been cleaning up the article and so I have tagged this section similarly. I still consider that this section is better covered in Mean with perhaps a very short note here. Dingo1729 ( talk) 05:19, 6 October 2013 (UTC)
Splitting the article would negate the benefit of focusing, in one place, on the word "average". Use of the word "average" can be a source of confusion due to author lack of clear delineation of the particular mean (arithmetic, geometric, harmonic) being reported on and inconsistency of verbiage within the authors report (e.g., after stating the particular mean being used authors fall into the confusion trap of using the word "average" subsequently). Such inconsistency leads to miscommunication and confusion. The Wikipedia article helps address these and related literature short comings. It also ties into one neat article the basic 'measures of central tendency' parameters.
Expansion of the basic parameter information of this article by other articles is a useful way to go. Furthermore. in a more general context, some repetition of information in various articles is beneficial. How to distribute related information across articles and manage repetition is an important topic unto its self. Managing repetition with an eye to erring on the side of more rather than less in a reasonable manner is the way to go.
Tegangwer ( talk) 04:10, 28 November 2013 (UTC)
Is this still a "start class" article? Seems pretty decent to me. -- PeterLFlomPhD ( talk) 22:10, 23 July 2015 (UTC)
Hello,
I've added a new "history" section, with a "origin" sub topic. I didn't figure out how to make the citations work for this section. Please help me clean it up. Thanks. Tal Galili ( talk) 18:59, 22 October 2015 (UTC)
The comment(s) below were originally left at Talk:Average/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Consider adding more elementary/introductory material. References would also help.
Geometry guy
14:45, 21 May 2007 (UTC)
"Measures of central tendency" section needs to be rewritten to be less technical. Kaldari 19:44, 29 October 2007 (UTC) This article is much too technical for an introductory article, and it's also lacking inline citations. -- Foggy Morning 02:25, 8 November 2007 (UTC) |
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Currently the lead says
The first and third sentences says that "average" means the arithmetic mean, contradicting the rest of the article, which gives a variety of alternative uses.
The second sentence is vacuous and was probably a soft vandalism.
The last sentence introduces two other concepts without saying whether they are averages.
So I'm going to rewrite the lead consistently with the rest of the article. Loraof ( talk) 16:12, 6 July 2017 (UTC)
The last sentence from the lead is "in colloquial usage any of these [mean, median, mode] might be called an average value. " is utter rubbish. Only an idiot would use the term average to refer to median or mode. There is no way that this is colloquial usage of the word "average." — Preceding unsigned comment added by 135.23.120.120 ( talk) 20:42, 8 October 2021 (UTC)