Editing Talk:2435: Geothmetic Meandian
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I do not agree with the statement that "The title text may also be a sly reference to an actual mathematical theorem, namely that if one performs this procedure only using the arithmetic mean and the harmonic mean, the result will converge to the geometric mean." Could one produce a reference to this result? A simple computer experiment does not show this "theorem" to be true, i.e. for the procedure to return the geometric mean of the original entry. [[User:Pointfivegully|Pointfivegully]] ([[User talk:Pointfivegully|talk]]) 15:04, 12 March 2021 (UTC) | I do not agree with the statement that "The title text may also be a sly reference to an actual mathematical theorem, namely that if one performs this procedure only using the arithmetic mean and the harmonic mean, the result will converge to the geometric mean." Could one produce a reference to this result? A simple computer experiment does not show this "theorem" to be true, i.e. for the procedure to return the geometric mean of the original entry. [[User:Pointfivegully|Pointfivegully]] ([[User talk:Pointfivegully|talk]]) 15:04, 12 March 2021 (UTC) | ||
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== Proof of convergence == | == Proof of convergence == | ||
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:For something as simple as this, I always find it cheating to use a package to abstract away the few actually necessary calculations. You might as well use a DWIM module and do 'result = DWIM(input)' as the sole command. But that's me for you. I'd write my own direct-to-memory screen RAM accesses, if silly things like OS HALs and GPU acceleration (once you find a way to message them as directly as possible) hadn't long since made that pretty much moot, if not actually verboten... [[Special:Contributions/141.101.99.109|141.101.99.109]] 17:53, 11 March 2021 (UTC) | :For something as simple as this, I always find it cheating to use a package to abstract away the few actually necessary calculations. You might as well use a DWIM module and do 'result = DWIM(input)' as the sole command. But that's me for you. I'd write my own direct-to-memory screen RAM accesses, if silly things like OS HALs and GPU acceleration (once you find a way to message them as directly as possible) hadn't long since made that pretty much moot, if not actually verboten... [[Special:Contributions/141.101.99.109|141.101.99.109]] 17:53, 11 March 2021 (UTC) | ||
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== Sloppy notation? == | == Sloppy notation? == | ||
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R = GMDN(a,b,c) | R = GMDN(a,b,c) | ||
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=== The RandallMunroe Set === | === The RandallMunroe Set === | ||
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result = y(2); | result = y(2); | ||
end | end | ||
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== Proof - Possibly by Induction == | == Proof - Possibly by Induction == | ||
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I believe we can produce a simpler, rigorous proof. Assuming a set of three is given, we can show that after every 2 iterations, the range is reduced by at least 1/3 of its original value, and therefore it converges exponentially to 0. We use the fact that each iteration, none of the three values will lie outside the range of the previous iteration. In addition, it can be shown that the arithmean lies at least 1/3 of the previous range away from the highest and lowest values of the previous iteration. | I believe we can produce a simpler, rigorous proof. Assuming a set of three is given, we can show that after every 2 iterations, the range is reduced by at least 1/3 of its original value, and therefore it converges exponentially to 0. We use the fact that each iteration, none of the three values will lie outside the range of the previous iteration. In addition, it can be shown that the arithmean lies at least 1/3 of the previous range away from the highest and lowest values of the previous iteration. | ||
− | If the arithmean is the highest or lowest value on the first iteration, then the range will therefore already be small enough (and won't get bigger in the second iteration.) Otherwise, the only remaining option is that it is the middle (median) value. So on the second iteration, both the median and the arithmean are within the reduced 1/3 range, and at least one of them must be the highest or lowest value. The range will always be the required size. | + | If the arithmean is the highest or lowest value on the first iteration, then the range will therefore already be small enough (and won't get bigger in the second iteration.) Otherwise, the only remaining option is that it is the middle (median) value. So on the second iteration, both the median and the arithmean are within the reduced 1/3 range, and at least one of them must be the highest or lowest value. The range will always be the required size. [[Special:Contributions/141.101.98.16|141.101.98.16]] |
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− | [[Special:Contributions/141.101.98.16|141.101.98.16]] | ||
== Why is this funny? == | == Why is this funny? == | ||
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:::'Twas not I, but note that this is 'explainxkcd' not 'explainwhyxkcdisfunny'. I think we both recognise that a cornucopia of details have been explained. It is even funnier to see someone insisting we continue to dissect the frog, but I'm not sure I need to fully explain that. ;) [[Special:Contributions/162.158.159.108|162.158.159.108]] 15:16, 12 March 2021 (UTC) | :::'Twas not I, but note that this is 'explainxkcd' not 'explainwhyxkcdisfunny'. I think we both recognise that a cornucopia of details have been explained. It is even funnier to see someone insisting we continue to dissect the frog, but I'm not sure I need to fully explain that. ;) [[Special:Contributions/162.158.159.108|162.158.159.108]] 15:16, 12 March 2021 (UTC) | ||
:Yes, here's a bit more on that.. I agree with [[User:Elektrizikekswerk|Elektrizikekswerk]] the joke is explained. The stat tip: "If you aren't sure whether to use the mean, median or geometric mean, just calculate all three, then repeat until it converges." is funny because there are many situations in the physical sciences where the arthmean, geometric mean and median for some data are different values. It is perhaps common that scientists not well versed in statistics are unsure which to use. The funny bit is imagining this less-statistically-versed-scientist throwing up their hands and just accepting the fixed constant given by iterating GMDN as the 'answer' irrelevant of any physical meaning. Also the name "geothmetic meandian" is funny because the word meandian is similar to both median, which it uses, and to ''meander'' which is indicated by the alternate assignment of the median on each iteration -- informally, this function meanders. [[User:Ramakarl|Ramakarl]] | :Yes, here's a bit more on that.. I agree with [[User:Elektrizikekswerk|Elektrizikekswerk]] the joke is explained. The stat tip: "If you aren't sure whether to use the mean, median or geometric mean, just calculate all three, then repeat until it converges." is funny because there are many situations in the physical sciences where the arthmean, geometric mean and median for some data are different values. It is perhaps common that scientists not well versed in statistics are unsure which to use. The funny bit is imagining this less-statistically-versed-scientist throwing up their hands and just accepting the fixed constant given by iterating GMDN as the 'answer' irrelevant of any physical meaning. Also the name "geothmetic meandian" is funny because the word meandian is similar to both median, which it uses, and to ''meander'' which is indicated by the alternate assignment of the median on each iteration -- informally, this function meanders. [[User:Ramakarl|Ramakarl]] | ||
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