Talk:Median - Knowledge

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edition of Numerical Recipes versus the latest edition. I have a quibble about the definition in the Preamble where it says "If there are an even number of observations, one often takes the mean of the two middle values.” – but this last is ambiguous: Is the median of {0,1,1,2,2,2} 1.5, the average of numbers 1 & 2, or, rather and perhaps better, 1.6, the weighted mean of the sub-multiset formed of the two middle values {1,1,2,2,2}? The Preamble perhaps means (sorry!) to say "one often takes the average of the values of the two middle observations". But I like the subtlety of the other definition a little better, somehow; and it ought not to be too hard to construct an example of a discrete distribution limiting to a continuous one, in which the first definition behaves, in the limit, worse than the second.

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distribution? Even if section 2 of the article does follow logically, I have never seen anyone defy convention and state that there are two medians and to not interpolate. Can you please provide some sort of citaion where this nonuniqueness of the median idea is presented? If not, it might be good to pull it infavour of the interpolating definition to avoid confusing students. If you are correct and there are solid sources then we should add a bit on the interpolating convention in section 2 as a warning.

1199:. A sample has observations and a population has members. Starting at the second sentence thru the end of the second paragraph, the article needs to have the word distribution changed to sample and population changed to observations. None of it is referring to a distribution (population). The above definitions need to be included in the "Medians in descriptive statistics" with the correct definition of a unique median. I will make the appropriate changes after after discussion and concurrence. 254: 233: 200: 2390:

this accounts for the word ‘unique’ creeping into their definition; but it unnecessarily limits the scope of their concept. A better reference, in many respects, is a paper written before Davis’s work by Charles Plott in 1967 (C. R. Plott, “A Notion of Equilibrium and its Possibility Under Majority Rule”). He uses a similar concept and requires no assumption of uniqueness, but his own term is ‘majority rule equilibrium’ which is hardly suitable for a general-purpose location parameter.

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for example a list of species that have sequentially evolved from one another. You can order them, so you can find the "middle" one(s). For even cardinality, there are two candidates. You cannot compute the average of those. You can, however, employ a _single_ definition for the median that allows for both candidates to qualify as the median: The median is the object in the sorted list whose distance from the center is minimal.

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giving it its own section. Then moved most of the more technical material into a "discussion" section. In places where it makes sense to introduce technical material early I've cordoned it off at the end of a paragraph by using words like "in more technical terms" at the start of a sentence. And everywhere tried to deduplicate and reduce sentence and paragraph length. Let me know if you have any thoughts on this.

431: 191: 2008:, and that here has now been improved. I haven't looked to see what was copied where, but I would suggest that that other aricle is the better place to have a longer outline of median-unbiasedness, with perhaps a reduction in the present article concentrating on whether/why the sample median is median-unbiased rather than median-unbiasedness itself. 2422:

Recent changes to this article suggested that reordering data elements changes the median - that is simply incorrect. To arrive at the median, one finds the value such that at least half the set is less than or equal to the proposed value and half is greater than or equal (this is a paraphrase of the

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I've was looking at this article and wondering if there's any sense in adding a basic illustration of the median concept at the top of the article, and if so how people think this could be done - any thoughts? I feel it would be nice to have a basic illustration of how you take the median (maybe from

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is presently a redirect, somehow using the anchor presently within this article. On more general "multivariate median" changes, do note that there is an early mention of the multivariate case in the section on "An inequality relating means and medians" that may need to be put in a more logical place,

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it is noted that there are number of ways in existing software packages (R, SAS) for computing quantiles. (Disclaimer: I have contributed to that page.) These apply for the first 2-quantile, which we call the median. In particular, the median in your first example need not be 1.5, but could be 1 or

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to the discussion. The median can be defined for any list of objects that reside on an ordinal scale, which means that they can be ordered. An ordinal scale does not necessarily lend itself for computing an arithmetic mean. For that, the object would need to reside on at least an interval scale. Take

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is quite right that the reference (and several others) uses the expression “unique median in all directions” but the term is highly misleading. If an even number points lie along a straight line, then there exist an infinite number of omnidirectional medians – are these “non-unique unique medians in

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I vaguely recall that a running median or median filter was included by Tukey et al. as part of "exploratory data analysis". Can anyone contribute text and reference for this? The present content covers image processing only, but this would have been for a time-series context. I see that there is an

1674:- which basically is the set member closest to the median - this median definition suggested here is not the same. For example, given the vector set {(2,3),(3,1),(1,2)} the x-median is 2, the y-median is 2, resulting in a combined median of (2,2); this is not the medoid (either (1,2) or (2,3) is). -- 1346:

The fourth "citation needed" probably is a reminder that you can do this on an arbitrary ordering of the values. I could order numbers by their prime divisor representation, resulting in an order such as, {1, 2, 4, 8, 3, 6, 12, 9, 5, 10, 7, 11, 13} (based on {1, 2, 2*2, 2*2*2, 3, 3*2, 3*2*2, 3*3, 5,

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You are correct that I am an instructor telling students how to do exercises and it is very confusing when the posting emphasizes this nonstandard, and as far as I can tell unaccepted interpretation of the definition. Are there two definitions of median?? The middle variable(s) and the middle of the

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P.S. it also works for booleans. The boolean median is basically 0 when there are more 0s and 1 when there are more 1s in the set. Also relevant for computing the mean over computer data to stay in the right domain. Yet another use is data anonymization. When the values don't belong to the original

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One more thing: no, having an integer median for a set of integers CAN be quite important. But in just as many cases, you will require that the median actually is an element of the set itself. In this setting, the set {1, 2, 4, 5} would have two medians, 2 and 4. This happens a lot when the objects

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I should clarify that the original section was technically correct, except for the use of "underflow." However, the editor took an article about index size overflow when finding the "median" index during a binary search, and applied the same concept to his own median calculation using actual values

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The best precise definition of "the median" of an even-cardinality multiset is not easy to discern. I have seen several good discussions and I'll try to find some references. The bottom line is that it seems not many practitioners care; but it can matter in automated software -- see the penultimate

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Since computer integers are complete within their range, and the median by definition is neither larger than the largest element nor smaller than the smallest element, an "integer median" of computer integers, is a computer integer. (e.g. in a byte context, the integer median will never be outside

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The omnidirectional median as I presented it is a hybrid of the ideas in the two papers, abstracting out Plott’s domain specificity and Davis’s implied assumption of uniqueness. Strict Wikipedians rules might reject it as ‘original research’, in which case it would be better to delete my addition

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I don’t have a reference for the omnidirectional median under any non-misleading name. The Davis paper concerns multivariate generalisations of the ‘median voter theorem’ which, in a single dimension, is conventionally limited to odd numbers of voters to avoid problems of non-uniqueness. I assume

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I wonder if it would be appropriate to add a couple ways of determining the median (in nonproblematical cases). For example, in an unordered and uncounted list, start with the largest and smallest numbers, and then the next pairs etc. OR, in a spread sheet, take the number of entries, divide by

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Sorry, you fail to make your case. That particular external link is not cogent unless perhaps it's an instructor telling students how he wished exercises done, or otherwise there for some extraneous reason. The basic definition of "median" given in that same external link logically entails that

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I've never heard of the term before, so apologies to not knowing the full context. I'm just here to verify the source information and complete the reference. Please feel free to correct or reword the content. A better source that uses the preferred term "omnidirectional median" would be welcome.

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What I've decided to do instead is give the article a reorganisation. I've written an introduction that's a lot more non-technical and really aimed for people with no more than basic maths if that. Then I've emphasised the basic procedure (which most people will be coming to this article for) by

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This is a nice illustration of the 2-dimensional marginal median, in which the median is taken by components. (The marginal median should generally be avoided when the number of observations exceeds the dimension squared, as a rule of thumb). This application is discussed in recent papers on the

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has written several articles in recent years on median selection (and then with reference to resource-allocation problems, etc.), which are extremely detailed and point out errors in the literature (even by outstanding computer scientists). He is a meticulous programmer and mathematician, and I

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Although there are indubitably circumstances in which the median of some integers must be an integer, I suspect that in scientific practice in general, with an even number of values, the mean of the two middle values is used whether or not it is an integer. (But I have no citations to back me

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A discussion of multivariate extensions of the univariate median is needed. Since this is a statistics article (rather than a computer-science algorithmics article), this should have priority over the extensive discussion of sorting. (I agree that it should come after a discussion of a simple

1218:). First, "distribution" should be written as "sample" and the sample median is unique by definition. Secondly, the median will be unique when there are observations of the same value, for example {1, 4, 4, 5} yields a sample median of 4. Maybe I don't know what is being said here. 949:

REPLY: I would suggest looking at the textbook by Casella and Berger, for some discussion of results on the mean-median-mode inequality and the 6 sigma inequality, with references to the literature (e.g. the article by Pukelsheim et alia); see also Joag-Dev and Dharmadhikari's

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But here's my guess as to the general meaning of "weighted median". Assign weights, i.e., non-negative numbers whose sum is 1, to all the numbers in your list; different numbers may carry different weights. Take the median of the resulting discrete probablity distribution.

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Note that the mathematical median should satisfy this condition. However there can be more than one "median" according to this definition (indeed, in the {1,2,3,4} case any of 2, 2.5, 2.75 3 should satisfy this condition), while in the mathematical definition it is uniquely

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Look in 2004(?) Statistical Science, which had a special issue on nonparametric statistics. The last article was on nonparametric multivariate time series. There are some nice articles using spatial medians in chemometrics and image processing, for example.

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The other medians should probably go in another article. I am glad that we agree about Oja, etc. Oja's median is too complicated, to sketch here. You should look at Oja's homepage, which used to have a hilarious, self-deprecating quotation of a slam from a

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Corrected terminology in the "Easy explanation of the sample median" section. The is no "distribution" given nor are we working with numbers. We are working with (sample) observations with measured values, i.e. length of growth, scores for a quiz, etc.

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I recently rewrote the Computer Science median section. The original contribution strayed from the referenced article. The referenced article did not specifically deal with the "even case," and the original editor incorrectly used the term underflow.

1849:(emphasizing the Euclidean norm: other norms could be mentioned). The proper definition of the spatial median should be given for the population distribution. The current heuristic definition in terms of absolute loss is limited to L1, as noted. 1387:

I like your clarification that the median depends upon the ordering, not only the elements. It especially well highlights the fact that taking the average of the two middle elements is not always sensible, even when the numbers are real

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Do you mean that although sorting time is O(nlogn) the median can be found in O(1) time if the list is sorted? Or does it mean that for an unsorted list the median can be found in O(n) time? I feel this needs clarification in the article

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There are a few citation needed markers there, where I'm not sure if they are really needed. I hadn't read the article before, but I find these parts rather obvious. Albeit they could probably be rewritten to be easier to understand.

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Wouldn't the proposed alternative median calculation of using A + ((B − A)/2) also overflow, if for instance A is MAXINT? In any case I don't see why the implementation or the possible overflows should be mentioned in this article.

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I think the page currently incorrectly says that median is 50% percentile, rather than 50th percentile. I'm not an expert in statistics, so this is really just a question: Isn't this equivalent to 0.5-quantile (50%-quantile)? -- JKľ

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Completed the example for an even number of observations as defined for a sample median. If there is a question regarding the uniqueness of a sample median for an even number of observations, please discuss first before changing the

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what this article says is right. So do innumerable books. Certainly in some situations it makes sense to interpolate like that, but that doesn't mean the article is wrong to point out non-uniqueness of the median in certain cases.

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Since in the example above 1.5 is not integer, it's not useful. But you can use a "lower median" or "upper median" then, by rounding appropriately (i.e. the "lower integer median" would be 1, the "upper integer median" would be

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I think that the marginal median and the spatial median should be mentioned, and their properties briefly noted, in a WP:Summary fashion. I trust that the arithmetic mean notes that it is defined and used for multivariate

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are complex and you cannot "interpolate" between objects. Say, we have four players, each with scores and points. We can sort them by their score, then determine the "median" players points. Sounds too constructed? Well,

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Part 2 about the nonuniqueness is directly contradicted by the first external link at the bottom. Unless someone can find some documentation supporting not interpolating the median then this section needs to be removed.

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The page says, "In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half." Numerical value is

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due to precision limitations. Again there will probably some rounding happen, and you could probably again define a lower and upper median (although I bet most people will just use whatever the FPU computes for

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unique (by definition). Also, the statement, "...The median may not be unique, as there may be a number of observations with the same value occupying the middle range of a distribution" mixes terminology of

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as the absolute value of a number, and as far as I know, the median need not be above zero, so I've changed numerical value to number. I'm not an expert, so make sure to change it back if I'm wrong.

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I think the relation between the median and mode is missing. The 'popular explanation' might actually be misunderstood to describe the mode, and the difference between the two concept is not given.

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5*2, 7, 11, 13}, which is in fact a well-defined order of the positive integers derived from their ordered prime divisor sequences). This ordered sequence obviously has the median 12, while in the

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centered at each one. Take a weighted average of those bell-shaped functions, getting a probability density. Finally, take the median of that distribution. The bell curves are the "smoothing".

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So much wrong, I got distracted. The statement, "...The median may not be unique, as there may be a number of observations with the same value occupying the middle range of a "distribution" (

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should be a separate article. Just the Oja et al. literature alone would fill more than one wikipedia article. In fact, I almost started one but did not have the time to work on it.

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I'd call this a "set theoretic median definition", but I currently do not have a citation for it, it's solely based on my intuition and probably on things I've learned some years ago.

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While it is true that floating point numbers are not as precise as the mathematical real numbers, I offer that this distinction is not sufficiently relevant for the present article.

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is desribed in a linked, separate article. Note that Knowledge does not mean to include pseudocode for all possible algorithms, this is still a dictionary, not a programming book --

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the same as the median of a sample. The median of a population is given in the section "Medians of probability distributions" and the median need not be unique. However, given an

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Why hasn't this issue been addressed yet? Something must be very wrong if the introduction of the section promises an O(n) algorithm but the following algorithm is O(n log n). --

2105:"Numerical value", according to the source linked by you, means not always "absolute value"; sometimes also "definite quantity". But anyway, I agree: just "number" is better. 1637:

or in words: A median (note that there could be more than one) is an element such that less than half of the elements are larger and less than half of the element are smaller.

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For some discussions of this issue you might want to read: Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, _American Statistician_, *50*, 361-365.

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an odd-numbered dataset, then an even one), and a graphic showing how the mean can be skewed by outliers - essentially a representation of the median as it is taught on

2456: 1061:"Median is the middle value after arranging data by any order." This obviously follows from the definition, does a definitive source have to be cited in this case? -- 1015:

or deleting this section, since I added a short discussion of median-unbiased estimators and the absolute-value loss function (again following the ideas of Laplace).

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on seismology which at least confirms Tukey as a basic origin for running medians and it has a range of interesting results. But I expect there are better sources.

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2. Also, I don't think that this is an integer vs. real number issue; For example, if the set were {1.1, 2.2, 3.3, 4.4} then the median could be 2.75, 2.2, or 3.3.

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I just wanted to check, the median's equal to the 50th percentile, right? I think this would be helpful to have in the defintion of median (assuming it's right).

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Thanks, I understand now. I think it might preferable to move the "Efficient computation" part below the "Easy explanation" part. The way it is now is confusing.

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Thanks for your helpful edits, here, as elsewhere, Mathstat. I shall have to look at your draft. (Epstein also has an interest in zonotopes, I'd bet! C.f.

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This is not better, and has weird consequences. For example, the median of {0,1,1,2,2,2} (1.6) is then larger than the median of {0,1,1,2,3,3} (1.333...).

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REPLY: This is a typical homework problem, whose solution can be found by visiting standard intermediate textbooks (only some of which have the solution).

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says "Note, however, that c is not always unique, and therefore not well defined in general." Can someone give me an example where c is not unique? -

2496: 1134:(aka sample median). This is similar to knowing the difference between the mean of a population and the mean of a sample . Median of a population 2461: 2205: 876:

I agree. This section of the article is so trivial, and the result it gives so unimportant and obvious, that I think it should be cut entirely.

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The overall implementation of a median in Computer Science still needs a good source. Much appreciation to anyone who can find a good link.

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computer floats are usually just a subset of real numbers (well, usually plus infinity, plus NaN, and with two zeros), with tons of gaps

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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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can be generalized to non-metric-spaces by using this kind of median instead of the mean. This is then called "k-median clustering"

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Isn't it part of Knowledge's mission to make information more accessible? An easier to read introduction would help a lot here.

2446: 833:(not indexes). An additional section about calculating an actual median needs to be added (hopefully with a proper citation :) 99: 30: 1810: 1856:

could be discussed. A discussion of data depth functionals, etc., would probably exceed the depth desirable for this article!

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Should mention that U.S. census reports make very extensive use of medians where people might expect averages to be used...

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I’m not going to revert Mwtoews’s edit, but I suspect he’s susbstituted palpable nonsense for non-standard terminology. The

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I was somehow logged out, the above IP is mine. My last edit to the Median page was also under this IP. Sorry for the mess.

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The first and second paragraphs are mostly incorrect. First, one must understand the difference between the median of a

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Also, the even/odd cases are not currently dealt with (they weren't explained properly in the original section either).

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For continuous probability distributions, the difference between the median and the mean is less than or equal to one

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https://web.archive.org/web/20110310043642/http://www.universityofcalifornia.edu/senate/inmemoriam/georgewbrown.htm

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Laplace was older then Fechner and used the median frequently. Therefore, I suggest using the alternative wording

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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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https://web.archive.org/web/20100730032416/http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm

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computer floating point numbers have precision issues. Depending on where the floating point values lie, the

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example). I would suggest that a discussion of multivariate medians should discuss the following concepts:

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Just looked at the entry to learn more but found the piece a bit hard going with a liberal arts education.

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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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The paragraph packed with "citation needed" markers probably tries to provide solutions for all these.

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formal definition section) - that condition is clearly independent of the order of the data elements.

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multivariate median. Just search for "spatial median" and "United States" and "population center".

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all directions”? The word “unique” is already defined in all contexts: you can’t define a “unique

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I've heard of a "smoothed weighted median", defined something like this. Given a list of numbers

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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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before doing mass systematic removals. This message is updated dynamically through the template

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completely until a reference comes along which presents the concept in suitable generality.

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can you please include and intermediate definition of this. from and inteermidiate student!

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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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Note, however, that c is not always unique, and therefore not well defined in general.

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Let's establish a few things first, to make sure we are at least agreeing on these:

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computer integers are in a finite range, but the range is complete without gaps

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domain, they can easily be identified as anonymized/fake/modified/whatever. --

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Biometry by Sokal and Rolf, 3rd edition, pages 44 to 46, specifically box 4.1

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However, I think this holds for any distribution, not just continuous one.

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Okay, I will show you mine, and you show me yours for better understanding.

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Creating a section that is more readable for the mathematically challenged

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Articles created or improved during WikiProject Europe's 10,000 Challenge

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Until recent edits, the text in this article was very similar to that in

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in a population of integers, the median may eventually not be an integer:

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http://www.universityofcalifornia.edu/senate/inmemoriam/georgewbrown.htm

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Maybe someone else can make more sense of their definition than I can.

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Anyone up for rewriting this section to make it more understandable? --

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is well understood and fairly widely used; the terminology is a mess.

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Thank you for the helpful comment, and for catching my stray anchor.

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http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm

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On "spatial median", in any changes/splitting please recall that

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Does someone know how to demonstrate this optimality property ?

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is even, then the sample median is the arithmetic mean of the

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Hmm... I'd say "no". To me, the concept of the median is an

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used the sample median (to estimate the population median).

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for additional information. I made the following changes:

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Statisitics Explained by Mckillup 2006, page 74 and 75.

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http://scholar.google.de/scholar?q=k-median+clustering

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could donate lecture notes or text from an article.

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http://en.wikipedia.org/Median#Theoretical_properties

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would recommend his articles to interested readers.

370:, a collaborative effort to improve the coverage of 281:, a collaborative effort to improve the coverage of 2242:using the archive tool instructions below. Editors 1627: 1540: 1454: 1434: 1266:Melcombe did a fantastic job of cleaning this up. 2332:Davis et al also define their median in terms of 1100:Who can tell me ? i do not know how to find it. 438:This article was created or improved during the " 33:for general discussion of the article's subject. 2467:Knowledge level-4 vital articles in Mathematics 2006:Bias_of_an_estimator#Median_unbiased_estimators 2228:This message was posted before February 2018. 616:Does anyone know what a "weighted median" is? 174: 8: 1670:While closely related to the concept of the 1337:median may not be representable in computer 1040: 1038: 1298:For example, the median of {0, 1, 2, 3} on 332: 227: 2188:I have just modified 2 external links on 1614: 1606: 1603: 1595: 1576: 1557: 1555: 1527: 1519: 1516: 1508: 1489: 1470: 1468: 1447: 1427: 1548:m\right\}|\leq {\frac {|S|}{2}}}" /: --> 910:An inequality relating means and medians 2457:Knowledge vital articles in Mathematics 1698: 1034: 1007:further popularized the median for the 952:Unimodality, Convexity and Applications 485:The posted definition is partly wrong!! 334: 229: 188: 2181:External links modified (January 2018) 1466:m\right\}|\leq {\frac {|S|}{2}}}": --> 2472:B-Class vital articles in Mathematics 2121:"The median is of central importance" 1304:However, if you restrict this to the 1170:observations. The median of a sample 1096:Find Mean,Median,Mode in Grouped data 583:2003:C9:8715:7B00:484A:84B6:CE19:C97E 7: 1541:{\displaystyle |\left\{s\in S|s: --> 576:I would like to add the variable of 440:The 20,000 Challenge: UK and Ireland 364:This article is within the scope of 275:This article is within the scope of 218:It is of interest to the following 23:for discussing improvements to the 2492:Top-importance Statistics articles 2482:High-priority mathematics articles 1896:If anyone wants to work on it ... 14: 2192:. Please take a moment to review 1048:(1921), Pt II Ch XVII §5 (p 201). 295:Knowledge:WikiProject Mathematics 2452:Knowledge level-4 vital articles 1542:m\right\}|\leq {\frac {|S|}{2}}} 1057:Citation Needed for obvious fact 969:History?? Laplace versus Fechner 384:Knowledge:WikiProject Statistics 357: 336: 298:Template:WikiProject Mathematics 262: 252: 231: 198: 189: 45:Click here to start a new topic. 2497:WikiProject Statistics articles 1759:Running median or median filter 1318:Are we good on these problems? 404:This article has been rated as 387:Template:WikiProject Statistics 315:This article has been rated as 2462:B-Class level-4 vital articles 1959:or given a forward-reference. 1615: 1607: 1596: 1577: 1558: 1528: 1520: 1509: 1490: 1471: 1195:is to sample as members is to 767:does somebody have a picture? 1: 2147:an average high-school course 1276:21:37, 24 November 2009 (UTC) 1256:00:35, 22 November 2009 (UTC) 1228:11:42, 22 November 2009 (UTC) 1209:21:24, 24 November 2009 (UTC) 1150:is odd, the sample median is 857:02:59, 29 November 2008 (UTC) 843:02:47, 29 November 2008 (UTC) 828:23:24, 28 November 2008 (UTC) 611:01:55, 22 November 2009 (UTC) 591:14:00, 21 February 2024 (UTC) 546:11:19, 11 December 2006 (UTC) 378:and see a list of open tasks. 289:and see a list of open tasks. 42:Put new text under old text. 2477:B-Class mathematics articles 2433:03:03, 10 January 2024 (UTC) 2296:06:24, 24 January 2018 (UTC) 2125:I see what you did there... 2038:01:42, 16 January 2014 (UTC) 1730:20:18, 1 November 2010 (UTC) 1116:09:15, 3 November 2009 (UTC) 1090:09:37, 14 January 2010 (UTC) 1071:23:07, 25 October 2009 (UTC) 943:14:22, 19 January 2009 (UTC) 777:19:13, 9 December 2007 (UTC) 762:03:35, 1 November 2006 (UTC) 745:23:50, 22 January 2011 (UTC) 727:08:22, 21 January 2011 (UTC) 708:19:19, 20 January 2011 (UTC) 521:19:43, 17 October 2006 (UTC) 2487:B-Class Statistics articles 2321:” and allow that a “unique 2068:Number, not numerical value 1284:Computer median, revisited. 805:Medians in Computer Science 563:11:37, 27 August 2012 (UTC) 50:New to Knowledge? Welcome! 2518: 2317:”, you have to define an “ 2259:(last update: 5 June 2024) 2185:Hello fellow Wikipedians, 1809:I have found (via google) 872:10:25, 13 April 2009 (UTC) 2410:07:26, 21 July 2021 (UTC) 2381:21:13, 20 July 2021 (UTC) 2360:07:03, 20 July 2021 (UTC) 2175:21:09, 18 July 2016 (UTC) 2159:04:23, 16 June 2016 (UTC) 2115:05:48, 6 March 2015 (UTC) 2098:23:37, 5 March 2015 (UTC) 2063:20:45, 5 April 2014 (UTC) 1046:A Treatise on Probability 800:10:04, 15 June 2008 (UTC) 694:02:26, 2 April 2006 (UTC) 572:03:30, 6 March 2007 (UTC) 403: 352: 314: 247: 226: 80:Be welcoming to newcomers 2301:Median in all directions 2140:Graphical representation 2135:19:19, 27 May 2015 (UTC) 2023:Calculating by hand etc. 1684:21:14, 17 May 2010 (UTC) 1666:20:48, 17 May 2010 (UTC) 1403:19:30, 17 May 2010 (UTC) 1364:17:02, 17 May 2010 (UTC) 1025:18:22, 3 June 2009 (UTC) 964:18:21, 3 June 2009 (UTC) 905:18:27, 3 June 2009 (UTC) 886:13:14, 12 May 2009 (UTC) 669:23:04, 20 Apr 2005 (UTC) 657:Median = 50th percentile 480:18:17, 3 June 2009 (UTC) 321:project's priority scale 2018:09:06, 9 May 2012 (UTC) 1990:21:45, 7 May 2012 (UTC) 1969:21:32, 7 May 2012 (UTC) 1942:19:09, 7 May 2012 (UTC) 1906:17:30, 7 May 2012 (UTC) 1884:16:50, 7 May 2012 (UTC) 1823:23:22, 7 May 2012 (UTC) 1801:21:52, 7 May 2012 (UTC) 1778:13:03, 7 May 2012 (UTC) 1753:21:48, 7 May 2012 (UTC) 713:This is Knowledge. The 652:21:52, 5 Jan 2005 (UTC) 278:WikiProject Mathematics 2447:B-Class vital articles 1894:User:Mathstat/MVMedian 1629: 1543: 1456: 1436: 1369:A couple of thoughts: 1044:Keynes, John Maynard; 1013: 994: 455: 434: 367:WikiProject Statistics 75:avoid personal attacks 2418:Order of the data set 1912:Shapley-Folkman lemma 1630: 1544: 1457: 1437: 1351:order it would be 7. 1330:the byte value range) 1302:is by definition 1.5. 1122:Definition (Nov 2009) 998: 975: 684:Efficient Computation 454: 433: 205:level-4 vital article 100:Neutral point of view 2240:regular verification 1554: 1467: 1446: 1426: 1130:and the median of a 988:the median into the 578:Level_of_measurement 301:mathematics articles 105:No original research 2329:which is unique. 2230:After February 2018 1890:Multivariate median 1862:User:David Eppstein 1860:I would guess that 1831:Multivariate median 1372:In the article for 1146:observations where 1132:set of observations 715:selection algorithm 638:, first consider a 463:Optimality property 390:Statistics articles 2284:InternetArchiveBot 2235:InternetArchiveBot 1650:k-means clustering 1625: 1538: 1452: 1432: 1422:A median of a set 1191:. In other words, 1011:analysis of data. 992:analysis of data. 918:standard deviation 456: 435: 270:Mathematics portal 214:content assessment 86:dispute resolution 47: 2260: 2101: 2084:comment added by 1988: 1940: 1882: 1799: 1751: 1623: 1536: 1455:{\displaystyle m} 1435:{\displaystyle S} 1308:, there is no 1.5 1259: 1242:comment added by 1106:comment added by 946: 929:comment added by 914:The article says 640:bell-shaped curve 596:The sample median 511: 497:comment added by 460: 459: 424: 423: 420: 419: 416: 415: 331: 330: 327: 326: 183: 182: 66:Assume good faith 43: 2509: 2379: 2377: 2372: 2311: 2294: 2285: 2258: 2257: 2236: 2100: 2078: 1987: 1981: 1974: 1939: 1933: 1926: 1881: 1875: 1868: 1811:Evans pdf (1981) 1798: 1792: 1785: 1750: 1744: 1737: 1708: 1703: 1634: 1632: 1631: 1626: 1624: 1619: 1618: 1610: 1604: 1599: 1594: 1590: 1580: 1561: 1549: 1546: 1545: 1539: 1537: 1532: 1531: 1523: 1517: 1512: 1507: 1503: 1493: 1474: 1461: 1459: 1458: 1453: 1441: 1439: 1438: 1433: 1258: 1236: 1158:observation. If 1118: 1049: 1042: 1017:Kiefer.Wolfowitz 956:Kiefer.Wolfowitz 945: 923: 897:Kiefer.Wolfowitz 510: 491: 472:Kiefer.Wolfowitz 426: 410:importance scale 392: 391: 388: 385: 382: 361: 354: 353: 348: 340: 333: 303: 302: 299: 296: 293: 272: 267: 266: 256: 249: 248: 243: 235: 228: 211: 202: 201: 194: 193: 185: 179: 178: 164: 95:Article policies 16: 2517: 2516: 2512: 2511: 2510: 2508: 2507: 2506: 2437: 2436: 2420: 2375: 2370: 2367: 2305: 2303: 2288: 2283: 2251: 2244:have permission 2234: 2198:this simple FaQ 2183: 2142: 2123: 2107:Boris Tsirelson 2079: 2070: 2045: 2025: 2002: 2000:Median-unbiased 1977: 1975: 1929: 1927: 1871: 1869: 1841:Marginal median 1833: 1788: 1786: 1761: 1740: 1738: 1718: 1716:U.S. Census use 1713: 1712: 1711: 1704: 1700: 1605: 1566: 1562: 1552: 1551: 1518: 1479: 1475: 1464: 1463: 1444: 1443: 1424: 1423: 1286: 1237: 1124: 1101: 1098: 1059: 1054: 1053: 1052: 1043: 1036: 980: 971: 924: 912: 849:JoshuaSchaeffer 820:JoshuaSchaeffer 807: 785: 755: 686: 659: 629: 618: 598: 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1602: 1598: 1593: 1589: 1586: 1583: 1579: 1575: 1572: 1569: 1565: 1560: 1535: 1530: 1526: 1522: 1515: 1511: 1506: 1502: 1499: 1496: 1492: 1488: 1485: 1482: 1478: 1473: 1451: 1442:is an element 1431: 1420: 1415:(or lets say, 1406: 1405: 1391: 1390: 1389: 1385: 1382: 1378: 1344: 1343: 1339:floating point 1331: 1327: 1316: 1315: 1312: 1309: 1306:integer domain 1303: 1297: 1285: 1282: 1281: 1280: 1279: 1278: 1261: 1260: 1231: 1230: 1140:ordered sample 1123: 1120: 1097: 1094: 1093: 1092: 1058: 1055: 1051: 1050: 1033: 1032: 1028: 1005:Gustav Fechner 983:Gustav Fechner 979: 976: 970: 967: 911: 908: 864:190.245.13.228 806: 803: 792:72.221.120.191 784: 781: 766: 754: 751: 750: 749: 748: 747: 730: 729: 685: 682: 681: 680: 679: 678: 671: 670: 658: 655: 654: 653: 644: 643: 625: 617: 614: 597: 594: 538: 524: 523: 486: 483: 464: 461: 458: 457: 448: 436: 422: 421: 418: 417: 414: 413: 406:Top-importance 402: 396: 395: 393: 376:the discussion 362: 350: 349: 347:Top‑importance 341: 329: 328: 325: 324: 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696: 695: 692: 691:203.33.164.42 683: 675: 674: 673: 672: 668: 667:Michael Hardy 664: 663: 662: 656: 651: 650:Michael Hardy 646: 645: 641: 637: 633: 628: 624: 620: 619: 615: 613: 612: 608: 604: 601:information. 595: 593: 592: 588: 584: 579: 574: 573: 570: 569:Hadleywickham 565: 564: 560: 556: 551: 548: 547: 544: 537: 533: 530: 527: 522: 519: 518:Michael Hardy 514: 513: 512: 508: 504: 500: 496: 484: 482: 481: 477: 473: 468: 462: 453: 449: 447: 446: 445:You can help! 441: 437: 432: 428: 427: 411: 407: 401: 398: 397: 394: 377: 373: 369: 368: 363: 360: 356: 355: 351: 345: 342: 339: 335: 322: 318: 317:High-priority 312: 309: 308: 305: 288: 284: 280: 279: 271: 265: 260: 258: 255: 251: 250: 246: 242:High‑priority 240: 237: 234: 230: 225: 221: 215: 207: 206: 196: 192: 187: 186: 177: 173: 170: 167: 163: 159: 155: 152: 149: 146: 143: 140: 137: 134: 131: 127: 124: 123:Find sources: 120: 119: 111: 110:Verifiability 108: 106: 103: 101: 98: 97: 96: 87: 83: 81: 78: 76: 72: 69: 67: 64: 63: 57: 53: 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Kiwiel 889: 875: 860: 846: 831: 818: 815: 812: 808: 786: 765: 756: 719:87.174.120.8 697: 687: 660: 635: 631: 626: 622: 599: 575: 566: 555:132.230.10.6 552: 549: 543:75.36.232.78 539: 534: 531: 528: 525: 488: 469: 466: 444: 405: 365: 316: 276: 220:WikiProjects 203: 171: 165: 157: 150: 144: 138: 132: 122: 94: 19:This is the 2055:Petercascio 1922:referee! ;) 1764:article on 1644:determined. 1238:—Preceding 1102:—Preceding 1009:statistical 925:—Preceding 835:68.0.255.35 493:—Preceding 292:Mathematics 283:mathematics 239:Mathematics 148:free images 31:not a forum 2441:Categories 2291:Report bug 2086:Y2N1-09631 1854:Oja median 1694:References 1462:such that 1411:well, the 1128:population 1030:References 986:introduced 677:2006-04-24 634:= 1, ..., 499:Jdparker28 381:Statistics 372:statistics 344:Statistics 2274:this tool 2267:this tool 2167:Blythwood 2151:Blythwood 1985:Wolfowitz 1937:Wolfowitz 1879:Wolfowitz 1796:Wolfowitz 1748:Wolfowitz 1417:numerical 1395:Quantling 1374:quantiles 973:I quote 665:Correct. 208:is rated 88:if needed 71:Be polite 21:talk page 2325:” is an 2280:Cheers.— 2094:contribs 2082:unsigned 2028:two.... 2010:Melcombe 1961:Melcombe 1898:Mathstat 1867:Thanks! 1815:Melcombe 1770:Melcombe 1722:AnonMoos 1388:numbers. 1268:JackOL31 1252:contribs 1244:JackOL31 1240:unsigned 1220:JackOL31 1201:JackOL31 1187:used in 1179:used in 1104:unsigned 1082:Nwinther 939:contribs 927:unsigned 878:Ben Finn 603:JackOL31 507:contribs 495:unsigned 56:get help 29:This is 27:article. 2425:MrOllie 2348:concept 2308:Mwtoews 2194:my edit 2075:defined 1349:natural 1185:members 1181:samples 1164:(n/2)th 1154:as the 1152:defined 1001:Laplace 978:History 931:Matumba 769:Andries 408:on the 319:on the 210:B-class 154:WP refs 142:scholar 2190:Median 1979:Kiefer 1931:Kiefer 1873:Kiefer 1790:Kiefer 1742:Kiefer 1672:medoid 1658:Chire2 1356:Chire2 1342:them). 1136:is not 990:formal 759:Junuxx 216:scale. 126:Google 25:Median 1498:: --> 1183:with 1078:axiom 737:Roerd 700:Roerd 197:This 169:JSTOR 130:books 84:Seek 2429:talk 2406:talk 2356:talk 2171:talk 2155:talk 2131:talk 2111:talk 2090:talk 2059:talk 2034:talk 2014:talk 1965:talk 1902:talk 1819:talk 1774:talk 1726:talk 1680:talk 1662:talk 1585:< 1550:and 1399:talk 1381:up!) 1360:talk 1272:talk 1248:talk 1224:talk 1205:talk 1166:and 1112:talk 1086:talk 1067:talk 1021:talk 960:talk 935:talk 901:talk 882:talk 868:talk 853:talk 839:talk 824:talk 796:talk 773:talk 753:Mode 741:talk 723:talk 704:talk 630:for 607:talk 587:talk 559:talk 503:talk 476:talk 311:High 162:FENS 136:news 73:and 2248:RfC 2218:to 2208:to 1216:sic 1142:of 400:Top 176:TWL 2443:: 2431:) 2408:) 2358:) 2261:. 2256:}} 2252:{{ 2173:) 2157:) 2133:) 2113:) 2096:) 2092:• 2061:) 2036:) 2016:) 1967:) 1914:). 1904:) 1821:) 1776:) 1728:) 1682:) 1664:) 1601:≤ 1571:∈ 1514:≤ 1484:∈ 1401:) 1362:) 1326:2. 1274:) 1254:) 1250:• 1226:) 1207:) 1172:is 1168:th 1156:th 1114:) 1088:) 1069:) 1037:^ 1023:) 962:) 954:. 941:) 937:• 903:) 884:) 870:) 855:) 841:) 826:) 798:) 775:) 743:) 725:) 706:) 609:) 589:) 561:) 509:) 505:• 478:) 156:) 54:; 2427:( 2404:( 2376:t 2371:m 2368:+ 2354:( 2338:X 2334:X 2327:M 2323:M 2319:M 2315:M 2310:: 2306:@ 2293:) 2289:( 2276:. 2269:. 2169:( 2153:( 2129:( 2109:( 2088:( 2057:( 2032:( 2012:( 1983:. 1963:( 1935:. 1900:( 1877:. 1817:( 1794:. 1772:( 1746:. 1724:( 1678:( 1660:( 1621:2 1616:| 1612:S 1608:| 1597:| 1592:} 1588:m 1582:s 1578:| 1574:S 1568:s 1564:{ 1559:| 1534:2 1529:| 1525:S 1521:| 1510:| 1505:} 1501:m 1495:s 1491:| 1487:S 1481:s 1477:{ 1472:| 1450:m 1430:S 1397:( 1358:( 1270:( 1246:( 1222:( 1203:( 1160:n 1148:n 1144:n 1110:( 1084:( 1065:( 1019:( 958:( 933:( 920:. 899:( 880:( 866:( 851:( 837:( 822:( 794:( 771:( 739:( 721:( 702:( 636:n 632:i 627:i 623:x 605:( 585:( 557:( 501:( 474:( 412:. 323:. 222:: 172:· 166:· 158:· 151:· 145:· 139:· 133:· 128:( 58:.

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