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June 28

Citation for Pooled_variance#Sample-based_statistics

I have read many relevant sections from textbooks on statistics, but none of them mentioned the formula in the linked section. I hope someone will provide a reference for that. Thanks.

${\begin{aligned}\mu _{X}&={\frac {1}{\sum _{i}{N_{X_{i}}}}}\left(\sum _{i}{N_{X_{i}}\mu _{X_{i}}}\right)\\[3pt]\sigma _{X}&={\sqrt {{\frac {1}{\sum _{i}{N_{X_{i}}-1}}}\left(\sum _{i}{\left[(N_{X_{i}}-1)\sigma _{X_{i}}^{2}+N_{X_{i}}\mu _{X_{i}}^{2}\right]}-\left[\sum _{i}{N_{X_{i}}}\right]\mu _{X}^{2}\right)}}\end{aligned}}$

--117.136.2.134 (talk) 00:26, 28 June 2017 (UTC)[reply]

The whole article is poorly referenced and it's not entirely clear that the topic is notable (in the WP sense). I'm not an expert in statistics though, so maybe it's more to it. In any case, the formula doesn't seem that difficult to derive. If M_0,i, M_1,i and M_2,i denote the sums over X_i of 1, x, and x², then you can determine each of these values from the given (N_i, μ_i, σ_i), add them to get the overall sums M₀, M₁ and M₂, and use these to get the to get overall (N, μ, σ).

It seems to me though that the formula isn't solving the problem stated in the article lead. The formula computes the population σ over the union of the different populations, but the idea of the pooled variance was you could combine statistics for populations with different means under the assumption that they had the same theoretical variance. You're not interested in the variance in the union but in the common variance within each population. The article doesn't give a scenario where the means of different populations are different but their variances are expected to be the same, so I guess my question is when would situation actually arise? --RDBury (talk) 08:54, 29 June 2017 (UTC)[reply]

I've never used pooled variance, but I think it's a standard topic in stat. The see-also section has links to several other things that come up in the same context.

If I understand it correctly, the context is very broad. First, suppose you have a true relationship y = a + bx + e. At every different x value there's a different mean of y; but the x-contingent variance of y is always the same—namely, the variance of e. The variance of e can be estimated using OLS. But now, suppose the functional form of the effect of x on y is unknown but may be nonlinear. That's where pooled variance comes in. So it's actually more general than linear regression. Loraof (talk) 14:31, 29 June 2017 (UTC)[reply]

As RDBury pointed out, the formula cited by the OP is irrelevant to the article. So I've removed it. Loraof (talk) 21:06, 29 June 2017 (UTC)[reply]

I want to compute the population σ over the union of the different populations. Is there a reference for that? Thanks.--222.129.63.181 (talk) 00:20, 30 June 2017 (UTC)[reply]

Hilbert

I encountered the concept of a Hilbert space a couple of years ago as the result of a categorization and Wikidata-related problem. Today, I encountered the concept of a Hilbert transform. Apparently they're both related to waves, and they're named for the same guy, but beyond that I'm not clear: are they at all related? My mathematical education stopped with an unsuccessful attempt at introductory calculus. Nyttend backup (talk) 13:14, 28 June 2017 (UTC)[reply]

There is no any direct relation between them. Ruslik_Zero 20:24, 28 June 2017 (UTC)[reply]

Not that it answers your question, but it's worth noting that Hilbert was prolific enough that his profligacy itself warrants a wikipedia page: List_of_things_named_after_David_Hilbert. 92.29.145.27 (talk) 20:51, 28 June 2017 (UTC)[reply]

David Hilbert was not a profligate. He was a modest man. What you said is insulting. --AboutFace 22 (talk) 22:44, 29 June 2017 (UTC)[reply]

Not sure how seriously you meant your comment, but anon has obviously simply used the wrong word where he meant "prolificacy" (maybe it was even autocomplete or something). A typo is not the same as an insult. -- Meni Rosenfeld (talk) 00:20, 30 June 2017 (UTC)[reply]

Profligacy is fine in the context as in meaning 1 of [1] 'wildly extravagant' or wiktionary 'Careless wastefulness'. A billionaire throwing away money on his wedding day for instance is prolifgate but it certainly is no insult to say that, he has lots more where that came from. Dmcq (talk) 12:36, 30 June 2017 (UTC)[reply]

Profligacy has a strong negative connotation no matter how you approach it. Seldom used too. --AboutFace 22 (talk) 22:11, 30 June 2017 (UTC)[reply]

I feel like I must stress again that this was obviously a typo. "Prolificacy" is the attribute of being prolific, and Hilbert definitely has plenty of that. Anon clearly meant to write "...Hilbert was prolific enough that his prolificacy itself...". Either the OP had a temporary mixup between words, or there was some autocorrect at play. It seems "prolificacy" is not in Firefox's dictionary, and anon might have used a feature to replace the word with the closest word in the dictionary, which happened to be "profligacy", even though it makes no sense in this context. -- Meni Rosenfeld (talk) 09:01, 3 July 2017 (UTC)[reply]

I see nothing wrong with the word in the context. If you have a look at a recent discussion at Wikipedia_talk:WikiProject_Mathematics#Aggressive_spamming_of_a_recent_arXiv_posting.2C_.22decision_streams.22 the difference might become more obvious. I do not know if the person pushing their paper there is prolific or not, but they are most certainly not profligate with their production. They did not produce it and leave it with careless abandon and just go on to their next idea of many. Using words that for most people would indicate they are wastrels is common for emphasizing the ability of someone who has the means to overcome any such potential disadvantage because they have so much.. Dmcq (talk) 09:30, 5 July 2017 (UTC)[reply]

Coming back to the main point, although there is no prior relation between these two things named after Hilbert, it is not difficult to relate them. The Hilbert transform defines the unique unitary translation-invariant linear complex structure on the Hilbert space of square-integrable (real) functions of a real variable. Thus the Hilbert transform is what mediates between a real Hilbert space and a complex Hilbert space. Sławomir Biały (talk) 12:40, 30 June 2017 (UTC)[reply]