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grade math text that elicits answers regarding these three particular concepts. The exercise in her homework does not discuss the concept of "average" or use "mean" interchangeably with the former term. Additionally, from a layman's perspective, the concepts of "mean" or "average", "mode," and "median" are distinct concepts, not just different flavors of "average," and the most common use of "average" does not encompass either "mode" or "median". Otherwise, there would be confusion as to what, for example, the "average age" of a given population group was (i.e., is it the "mode," "median," or "mean"?). A more in-depth treatment of the subtle differences between various "tendencies" is best addressed elsewhere; the topic's breath is somewhat greater than what most people will be searching for.
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somehow, and one imaginable way is a random-value producing apparatus that keeps producing a stream of digits making up the decimal expansion of the values for the random variables. So imagine that after several eons of steady production we have that one stream is "0.7785459(to be continued)" and so is another. Are the two values duplicates of each other? Perhaps, but most likely not. They may diverge at every next digit. But even if they are, we'll never know, since the streams will never be complete. So actually, the probability of any duplicate elements is 0, and, moreover, even if duplicacy (is that an
English word?) occurs, it remains forever unknowable. Might as well say: can't happen. --
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1011:: imagine a random variable X on with p(x)=2x. X obviously has a mode at x=1. Then define a random variable Y determined by X by (a) writing X as a binary fraction , (b) changing all the '1' digits to '2' digits, and (c) reading the result as a ternary fraction. This is a monotonic transformation so the mode of Y should be at 0.2222...(base 3), i.e. at 1. I have not convinced myself and I won't press the point. --
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we look at this from a practical point of view, wouldn't it be more helpful to talk about the practical situation where you get, say, temperature readings with +/- .01 degrees accuracy and it's quite possible, but unlikely, that you'll get the same reading twice, rendering the mode meaningless and requiring the technique described? -- In any case, thanks for explaining!
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Quite a reasonable article, though the language could perhaps be made simpler and flow better in places (i've just had a go at the first sentence). This isn't a complex concept and not as important as the mean or median in practice so i don't think the article needs to be any longer – if anything, it
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agreed, giving "Mode: n/a". For the
Rumping distribution, the answer 1 seems rather reasonable; in an experimental process, the experimentator would definitely be led to hypothesize this, and in the limit, using appropriately finer and finer discretizations as the number of trials grows, the modes of
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observe the following data 1,3,4,5,7,9,12,13,15,15,15,15,15,16,17,19,22,24,24,25,25,26,26,27,30,30,31,32,32,33,33,34,35,35,36,36,37,,39,40,40,45,45,46,47,48,49,49. class intervals freq. 0-10 6 10-20 10 20-30 8 30-40 14 40-50
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Huh. Seems like I understood alright, but you guys have a seriously different way of looking at these things. :-) (I was assuming that we were using the language of theory, since continuous distributions are a theoretical/modelling tool rather than something you'd deal with in practice.) Although, if
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Also, I should mention that anybody searching for "central tendency" is likely a technical person not interested in most of the average article. They are interested in what all the different measures are to apply in some application they are building. I will in the next week or so break the redirect
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for this (and you cannot know the required size in advance). Unless you have a set of library functions for this, that is not an easy programming exercise. If all values are different, the size grows to be as large as the data sample. It is almost always easier and more efficient to sort the values,
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Thank you so much!!! I appreciate your taking the time to help me. I didn't see that example in the article even though it was staring me in the face. Oops! :) In this case I think I'll write that the mode is "44 and 93" and see how that works. Again, thank you!!! (Thanks as well to whoever moved my
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It is indeed an interesting point, and it crossed my mind that the Cantor set could be said to be the mode. On the other hand, the somewhat loose definition given for mode ("the most frequent value assumed by a random variable") is not designed for such pathological cases, and it would be stressing
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The article gives the example of a data sample and states: "the mode is not unique". That is about all that can be said about it. Depending on your needs, predilections, and local customs, you can pick your choice between: (a) for this sample the mode is undefined; (b) this sample actually has two
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If the population distribution is continuous, the chance of observing two or more identical values is 0. If you discretize the data, then the result may (of course) depend on the details. If you have a distribution whose density function has a thin spike, it gets diluted when the intervals are much
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Indeed, more precisely formulated, the probability of any duplicate elements is 0. So if you were to repeat this for the rest of the life of the universe, you'd never expect a duplicate. Isn't that good enough for "will occur precisely once"? More strongly, though, the values have to be represented
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Probably because there is no promotion section. The article, as far as i checked, provides factual and unbiased information on this statistic, and it is up to the reader to criticize it or not. If you feel some properties which make mode a bad statistic to use are missing you can probably edit the
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It assumes at the beginning that we know the mode, in order to form a confidence interval for the mode. Likewise, later in discussing symmetric distributions, the confidence interval for the mean/median/mode depends on the true value of the mean/median/mode. This renders it not useful in practice
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In the
Knowledge article we read "For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode." I propose to change to: "For samples, if it is known that they are drawn from a symmetric unimodal distribution, the
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On a more practical bent, elementary school texts are discussing the three concepts, "mode," "median," and "mean" together without even mentioning the "average" concept. What drew me to the mode article (and to the section on mode, mean, and median) was a basic exercise from my daughter's sixth
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I find it unfortunate that this table uses a different data set for each example, and that the clarity and usefulness would be greatly improved if the same data set was used for each entry. I would like to edit it with the following data set: 1, 2, 2, 3, 4, 7, 9. This would give a mean of 4, a
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I don't understand why in "a sample from a continuous distribution each value will occur precisely once." I'm not particularly familiar with statistics (I didn't know what a 'mode' was before reading this article), so this may just reflect confusion on my part, but I don't see why sample from a
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Before computing the mode of a series of values it is probable wise to see if the distribution is unimodal. The dip test can be used for this purpose. If it is not unimodal this may complicate the programming somewhat. Determining the number of modes in an arbitrary list of numbers if it is not
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wider. The sample data show a thin spike at 15 and a robust mound around 35. The choice of intervals does actually not conform to the advice given in the article, but it is also possible to construct an example that does conform but still displays this phenomenon. --
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you don't use intervals but just tally the values: 3× a 0, 3× a 1, 4× a 2, 9× a 3, 5× a 4, 1× a 5, 2× a 6. If the frequencies are very low, you could lump groups of adjacent values together, making sure the groups have equal sizes. If the underlying distribution is
844:(lookup table). For example, if the data is something like {2404030, 712, 20574, 2404030, 657032180, 20574, 712, 2404030, 712, 20574, 2404030, 657032180, 657032180, 2404030, 31710510, 657032180, 31710510, 20574, 712, 712}, you don't want to allocate a normal
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It is fine if arithmetic mean gets more attention than other kinds of mean and than median and mode, because it is the most common kind of average, but it would be a mistake to leave the others out. But since each already has an article, the treatment in
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I think it belongs in a section on a central tenancy article. Central tendency currently redirects to average, which I think is a mistake because average is just one example of a measure of central tendency. Comparison logically belongs there.
624:; actual random-value producing apparatus doesn't last that long. For practical application, the article tells you to discretize the data, as for a histogram, so that you will actually get (still in practice) frequencies greather than 1. --
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modes: 1 and 4; and (c) the mode is indeterminate (whatever that means). I would not take the average except when you have a histogram peaking in two adjacent slots, because that could be severely misleading. Now
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occurrences. If the data is unimodal, these two will be the same. If it is possible to append an "infinite" value to the data (or any value exceeding all original data values), the last block can (and should) be
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sample mean can be used as an estimate of the population mode." Otherwise, it fails for example for values 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, having mean 3.5 which is not a good estimate of the two modes 2 and 5.
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9 according to the defination of the mode, 30-40 is the modal class. and the mode will lie in that interval. But if we look at the raw data then 15 is the mode. how do we resolve this contradiction ?
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maxCnt: (maxCnt, loMode, hiMode) := (cnt, last, last); else if cnt = maxCnt: if last < loMode: loMode := last; end if if last : -->
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mode" when no uniqueness assumption is in effect ? I find the current presentation misleading and i don't think there would be any problem with using the indefinite article throughout. Thank you.
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I brought this to question since the definition uses "frequency," and it would appear that they are all equally frequent and thus all modes? I'm not sure of the official accepted interpretation.
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hiMode: hiMode := last; end if else: skip; end if (cnt, last) := (1, x); else: cnt := cnt+1; end if end for
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1235:: "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,
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Rather strange. A multimodal continuous case: several local maxima, possibly of different "heights". But multimodal discrete case: several local maxima of the same "height". Really so?
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My question is I have a set of 10 numbers with 44 and 93 appearing twice. Is the mode "44 and 93" or is it the average of the two, as calculated with "median" involving even numbers?
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continuous distribution will contain every element only once (although it seems to me that it would do so with probabililty 1, i.e., almost certainly). Could someone explain?
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maxCnt: (maxCnt, loMode, hiMode) := (cnt, last, last); else if cnt = maxCnt: if last < loMode: loMode := last; end if if last : -->
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when discussing the linearity of mean, mode, and median? It does mean what we want to say, but it will convey it only to readers who don't need this article.
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Just do a Google search on . The first hit: "This section defines the three most common measures of central tendency: the mean, the median, and the mode."
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At the end of Pass 3 the last remembered (i.e. stored) value is the mode. (With a little more thought, Pass 1 and 2 can be achieved in a single pass.)
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620:"You guys"? I'm just one editor with one viewpoint (who happened to have written the relevant paragraph). The above was all very theoretical and only
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should really be rewritten to cover mainly the three well-known measures of central tendency, arithmetic mean, median and mode (with "See also"s for
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I got a proublem from my math class which implies there is a mode for a set of 5 numbers. (It's a hard proublem to explain so I won't right it out.)
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1112:, where it was even argued that mentioning the concept of statistical mode might harm laypeople who would not know how to use it in a safe way. --
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This requires that you maintain a data structure for counting, for each possible sample value, its frequency. In general, that has to be an
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Thanks for that! another hiccup! what happens if the data is given as a discontinuous data. do we make the intervals continuous, if so why?
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Strange, yes. But this seems to be how statisticians use these terms. References would be worthwhile. I'll add one for the discrete case.
1708:"The mode of a continuous probability distribution is the value x at which its probability density function has a locally maximum value";
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Can someone give an example of "Furthermore, like the mean, the mode of a probability distribution can be (plus or minus) infinity"? --
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is supposed to be a value that is different from all possible data values (and always compares as different). At the end, the variable
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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In both places where it is used, it is immediately explained. Doesn't that work? I can't judge, because I don't need this article.
856:(cnt, last) := (0, NAN); for x ← sorted(data): if x ≠ last: if cnt = 0: skip; else if cnt : -->
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1631:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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1674:. It comprises well over 1/3 of the article, despite being about a very esoteric point (samples with only a single data point).
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https://web.archive.org/web/20071030070638/http://www.stats4students.com/Essentials/Measures-Central-Tendency/Overview_2.php
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Hi. I would like to see a section on how to effectively compute the mode of a list of numbers. That would be useful. --
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its applicability beyond its scope. I see that the anonymous editor who set up the "Probability distribution" box at
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into equi-sized parts. These intervals must be contiguous: together they must cover the whole space without gaps. --
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Not at all - just that I originally doubted whether there was a dististibution on the reals with an infinite mode.
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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Are you trying to say that the distribution associated with the Cantor function does have a defined mode? --
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As to your question, it is an interesting point; I think it might be possible to say that the modes of the
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after which you can find the mode in a single pass. Here is a way of doing that, presented in pseudocode:
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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question to the bottom -- I wasn't aware of the wiki custom of oldest on the top/newest at the bottom!)
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are synonyms. The mean, the median and the mode are all different kinds of averages. As I wrote on the
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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intervals? And isn't the data (if finite in size) always "discontinuous"? If the data comes from a
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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1219:—mean, median, and mode—can help you capture, with a single number, what is typical of the data."
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contains the highest frequency observed. The result is delivered in a pair of program variables:
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This was a multiple choice question on a standardized test, so would I say 4 modes or no modes?--
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before doing mass systematic removals. This message is updated dynamically through the template
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I expect that attempts to unify the two notions will produced strained and non-natural results.
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At least part of it doesn't make sense (the first part of the subsection "Statistical tests").
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1243:. For instance, average house prices almost always use the median value for the average." --
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Is the mode all of them, since the highest frequency is one, or is there no mode whatsoever?
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Rather than try to explain the usage, I kept the idea. See what you think; I wouldn't put
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The mode cannot be calculated in such a varied set of data and thus should not be used. -
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unimodal is difficult. It may be impossible but I am not sure if this is a known fact.
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If you found an error with any archives or the URLs themselves, you can fix them with
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article at all. I think it should focus much more on just the arithmetic mean and the
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Last edited at 15:00, 22 February 2008 (UTC). Substituted at 20:06, 1 May 2016 (UTC)
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1069:? I propose moving it there with a link to it from here. Let me know what you think.
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A problem with including this algorithm in the text of the article is that this is "
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http://www.stats4students.com/Essentials/Measures-Central-Tendency/Overview_2.php
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unless I hear an objection. There are at least three problems with this section:
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The correct statement is that the difference between the mean and the MEDIAN is
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aka
Weighted mean), and then this section can be moved to the rewritten article
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1793:. No special action is required regarding these talk page notices, other than
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There is no universal quantification here, and the commuting operations are a
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under the See also, but go ahead if you like. Perhaps a general article on
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is more to the point, but even I :) can see that that is a bridge too far.
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hiMode: hiMode := last; end if else: skip; end if
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Compute the mode by three successive passes through the data list thus:
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Let me know if there is any objection to removing the entire section.
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P.S. And here is a quote from the intro paragraph of our own article
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No modes, according to my programming teacher who worked for NASA.
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The way I see it is as follows. For "commutative" in the sense of
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Start a separate count variable = 0 for each sample value and do
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because it was merged with that article. As used in statistics,
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An editor has identified a potential problem with the redirect
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Here the universal quantification is essential, and there is
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operations (although generalizations are possible and usual).
1108:. Unfortunately, we did not reach consensus on this over at
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Knowledge:Redirects for discussion/Log/2022 May 3#Math mode
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I've replaced the sentence by one that seems more true. --
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for additional information. I made the following changes:
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and start a new article if nobody persuades me otherwise.
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Confidence interval for the mode with a single data point
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and that the difference between the mean and the MODE is
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because it does not really cover the concept meant here.
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Does anyone else think this section would fit better in
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That's the problem. Nobody who is editing it does. (And
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Comparison of mean, median and mode: move to "average"?
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I texified the "Unimodal distributions" section under
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And so on. The search term gives similar results: --
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and central tendency pages should discuss the rest.
201:, a collaborative effort to improve the coverage of
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1797:using the archive tool instructions below. Editors
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1104:– inasmuch as it is not OR – should be merged into
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1042:the discretized distributions will tend to 1. --
815:: Increment the appropriate count for each sample
731:I have a similar question to the first one above
1952:Knowledge level-4 vital articles in Mathematics
1700:Different treatment of discrete and continuous?
1627:, and are posted here for posterity. Following
1783:This message was posted before February 2018.
407:I have reverted the addition of a wikilink to
1621:The comment(s) below were originally left at
804:: Count the number of different sample values
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853:(maxCnt, loMode, hiMode) := (0, NAN, NAN);
1753:I have just modified one external link on
1681:even if we did have a single-value sample.
1601:. This is what is written in the articles.
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1324:Why isn't there a criticism section???
262:Question: Let's say the set is 1,2,3,4.
1942:Knowledge vital articles in Mathematics
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1007:. See what you think of the following
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819:Start a frequency variable = 0 and do
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1967:Top-importance Statistics articles
1556:{\displaystyle \sigma (3/5)^{1/2}}
1443:{\displaystyle \sigma (3/5)^{1/2}}
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1982:Mid-priority mathematics articles
1757:. Please take a moment to review
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1305:median of 3, and a mode of 2. --
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1594:{\displaystyle \sigma 3^{1/2}}
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271:DOES ANYONE KNOW THE ANSWER?
209:and see a list of open tasks.
104:and see a list of open tasks.
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1261:can be relatively brief. --
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1266:21:24, 26 July 2008 (UTC)
1175:01:42, 21 July 2008 (UTC)
1157:I don't like the current
1149:01:38, 21 July 2008 (UTC)
1133:01:34, 21 July 2008 (UTC)
1100:. The current content of
715:12:27, 8 April 2007 (UTC)
671:06:30, 8 April 2007 (UTC)
428:Well it would be nice if
361:Do we really need to say
234:
167:
129:
78:
57:
1858:Redirects for discussion
1638:should be more concise.
1611:13:31, 7 July 2012 (UTC)
1294:Better Data Set for the
925:13:35, 7 July 2012 (UTC)
616:14:30, 15 May 2006 (UTC)
606:13:50, 15 May 2006 (UTC)
591:13:23, 15 May 2006 (UTC)
293:21:02, 16 May 2006 (UTC)
241:project's priority scale
1891:05:20, 3 May 2022 (UTC)
1872:and has thus listed it
1213:The next: "Measures of
1092:aka Moving average and
1003:are the members of the
958:Thought not. Thanks --
908:15:38, 8 May 2007 (UTC)
835:07:48, 8 May 2007 (UTC)
573:01:58, 7 May 2006 (UTC)
556:18:51, 6 May 2006 (UTC)
441:03:25, 6 May 2006 (UTC)
424:02:13, 6 May 2006 (UTC)
401:01:51, 6 May 2006 (UTC)
396:might be just as bad.)
384:01:28, 6 May 2006 (UTC)
370:21:23, 5 May 2006 (UTC)
198:WikiProject Mathematics
1932:B-Class vital articles
1856:"Math mode" listed at
1595:
1557:
1491:Which one is correct?
1482:
1444:
1390:Unimodal distributions
93:WikiProject Statistics
1596:
1558:
1483:
1445:
1320:Criticism of the mode
1009:singular distribution
502:commutative operation
430:commutative operation
409:Commutative operation
311:comment was added by
36:level-4 vital article
1795:regular verification
1567:
1515:
1454:
1402:
1201:talk page of Average
687:Poisson distribution
622:thought experimental
434:commutative diagrams
221:mathematics articles
1785:After February 2018
1039:Cantor distribution
1001:Cantor distribution
562:Commutative diagram
540:operation which is
450:commutative diagram
413:Commutative diagram
116:Statistics articles
1839:InternetArchiveBot
1790:InternetArchiveBot
1617:Assessment comment
1591:
1553:
1478:
1440:
784:Computing the mode
700:gamma distribution
190:Mathematics portal
45:content assessment
1904:Why not talk of "
1815:
1755:Mode (statistics)
1653:
1652:
1374:comment added by
1363:Probably an error
1330:comment added by
1290:
1278:comment added by
905:
897:original research
883:occurrences, and
842:associative array
763:
712:
658:
630:
603:
553:
479:
469:
421:
381:
324:
255:
254:
251:
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146:
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1989:
1867:
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1840:
1813:
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1487:
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1449:
1447:
1446:
1441:
1439:
1438:
1434:
1418:
1386:
1342:
1273:
1216:central tendency
1185:Central tendency
1094:Weighted average
904:
762:
720:thanks for that!
711:
657:
629:
602:
552:
477:
467:
420:
380:
306:
223:
222:
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136:importance scale
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42:
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24:
16:
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1987:
1986:
1922:
1921:
1902:
1861:
1843:
1838:
1806:
1799:have permission
1789:
1763:this simple FaQ
1748:
1717:Boris Tsirelson
1704:From the lead:
1702:
1661:
1622:
1619:
1573:
1565:
1564:
1535:
1513:
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1460:
1452:
1451:
1422:
1400:
1399:
1392:
1369:
1365:
1325:
1322:
1302:
1090:Running average
1084:In my opinion,
1063:
933:
864:
859:
854:
786:
757:data sample? --
733:
698:, like for the
583:
570:Septentrionalis
438:Septentrionalis
398:Septentrionalis
367:Septentrionalis
359:
307:—The preceding
260:
220:
217:
214:
211:
210:
188:
181:
161:
115:
112:
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106:
105:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
1995:
1993:
1985:
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1969:
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1959:
1954:
1949:
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1924:
1923:
1901:
1894:
1874:for discussion
1860:
1854:
1833:
1832:
1825:
1778:
1777:
1769:Added archive
1747:
1744:
1743:
1742:
1713:
1712:
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1683:
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1678:
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1437:
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1425:
1421:
1417:
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1407:
1391:
1388:
1376:140.105.52.132
1364:
1361:
1360:
1359:
1332:129.98.192.217
1321:
1318:
1301:
1292:
1269:
1268:
1254:
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1229:
1180:
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732:
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718:
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685:, such as the
664:
663:
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635:
634:
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582:
579:
578:
577:
576:
575:
545:
534:
533:
532:
531:in the domain.
497:
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207:the discussion
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132:Top-importance
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102:the discussion
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73:Top‑importance
67:
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26:
13:
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1983:
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1187:redirects to
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961:
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948:
944:
943:
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938:
931:Infinite mode
930:
926:
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918:
913:
912:
909:
906:
902:
898:
894:
888:
884:
880:
876:
872:
868:
867:The constant
866:
865:
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843:
839:
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730:
728:
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723:
722:59.180.55.151
716:
713:
709:
705:
701:
697:
692:
688:
684:
680:
676:
675:
674:
672:
669:
668:59.180.85.146
662:
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631:
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567:
566:Commutativity
563:
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18:
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1837:
1834:
1809:source check
1788:
1782:
1779:
1752:
1749:
1714:
1703:
1684:
1672:Undue weight
1662:
1654:
1620:
1500:
1499:
1493:
1490:
1393:
1370:— Preceding
1366:
1323:
1303:
1296:
1295:
1280:71.34.165.94
1270:
1215:
1214:
1196:
1192:
1183:The article
1121:
1110:Talk:Average
1064:
934:
829:
823:
818:
812:
807:
801:
796:
787:
773:LandOfIsrael
754:
750:
740:LandOfIsrael
737:
734:
719:
704:sample space
690:
678:
677:Do you mean
665:
638:
584:
541:
537:
528:
524:
520:
516:
512:
508:
500:
493:
489:
480:
474:
470:
464:
461:commute if:
458:
454:
448:
393:
389:
363:commute with
362:
360:
357:Commute with
270:
267:
264:
261:
237:Mid-priority
236:
196:
162:Mid‑priority
131:
91:
51:WikiProjects
34:
1326:—Preceding
1274:—Preceding
1167:Jason Quinn
1141:Jason Quinn
1125:Jason Quinn
832:Cuddlyable3
394:independent
212:Mathematics
203:mathematics
159:Mathematics
1926:Categories
1846:Report bug
1005:Cantor set
696:continuous
679:contiguous
107:Statistics
98:statistics
70:Statistics
1870:Math mode
1829:this tool
1822:this tool
673:reeded13
313:75.8.96.7
39:is rated
1885:1234qwer
1882:1234qwer
1835:Cheers.—
1501:WHAT!!??
1372:unsigned
1328:unsigned
1276:unsigned
892:omitted.
738:Thanks!
523:for all
346:Lophoole
341:Lophoole
309:unsigned
1900:mode" ?
1759:my edit
1603:DrMicro
1495:Tebello
1307:Pdcurry
1263:Lambiam
1259:Average
1245:Lambiam
1224:Lambiam
1193:average
1189:Average
1159:average
1114:Lambiam
1102:Average
1098:Average
1086:Average
1067:average
1044:Lambiam
1013:Rumping
977:Lambiam
960:Rumping
947:Lambiam
937:Rumping
917:DrMicro
901:Lambiam
759:Lambiam
727:reeded
708:Lambiam
691:usually
689:, then
654:Lambiam
647:reeded
626:Lambiam
599:Lambiam
549:Lambiam
417:Lambiam
377:Lambiam
239:on the
134:on the
41:B-class
1910:Plm203
1732:Loraof
1687:Loraof
1396:median
1349:Plm203
1239:, and
1237:median
899:". --
889:maxCnt
885:hiMode
881:maxCnt
877:loMode
873:maxCnt
824:Pass 3
813:Pass 2
802:Pass 1
790:Spoon!
642:Reeded
542:binary
390:linear
290:Dch111
277:Ferret
47:scale.
1300:table
846:array
755:empty
613:Benja
588:Benja
494:unary
28:This
1914:talk
1736:talk
1721:talk
1691:talk
1644:talk
1640:Qwfp
1607:talk
1380:talk
1353:talk
1336:talk
1311:talk
1284:talk
1241:mode
1233:Mean
1195:and
1171:talk
1163:mean
1145:talk
1129:talk
1106:Mean
1075:talk
1071:Qwfp
921:talk
751:here
527:and
490:pair
457:and
317:talk
1898:the
1803:RfC
1773:to
1498:The
869:NAN
538:one
492:of
392:or
231:Mid
126:Top
1928::
1916:)
1816:.
1811:}}
1807:{{
1738:)
1723:)
1693:)
1646:)
1609:)
1571:σ
1519:σ
1488:.
1458:σ
1406:σ
1382:)
1355:)
1338:)
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1286:)
1173:)
1147:)
1131:)
1077:)
923:)
568:?
519:⊕
515:=
511:⊕
473:=
319:)
1912:(
1906:a
1888:4
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1831:.
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1169:(
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529:y
525:x
521:x
517:y
513:y
509:x
483:.
481:f
478:o
475:g
471:g
468:o
465:f
459:g
455:f
323:.
315:(
243:.
138:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
