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We have a thread, Real life examples of difference between independence and correlation, which currently reads:

It is well known that independence implies zero correlation but zero correlation need not imply independence.

I came across plenty of examples using random variables supporting the latter result. Are there any real life examples to support this fact?

[Edit: @amoeba, @gung and I have performed some editing, so the second paragraph now reads:

I came across plenty of mathematical examples demonstrating dependence despite zero correlation. Are there any real life examples to support this fact?

But this does not seem to have been deemed a sufficient change for a successful reopen review.]

The thread is currently closed as a duplicate of Simplest example of uncorrelated but not independent X and Y?, which is itself closed as a duplicate of How to show operations on two random variables (each Bernoulli) are dependent but not correlated? I'm not sure how much sense it makes to close a question as a duplicate of a duplicate in general, but can see some logic behind doing so in this case.

However, I think it's clear that, at its heart, the "real life examples" thread is getting at a different point to the other threads, which are rather more oriented towards mathematical statistics. Moreover it has two nice "real-life" examples posted in the answers, which wouldn't quite seem to fit as answers in the "simplest example of uncorrelated but not independent X and Y" that it's closed as duplicate of, and certainly wouldn't fit in the Bernoulli thread that the "simplest example" thread was closed as a duplicate of.

I seem to have been the only person who wanted to reopen in the thread's last reopen review, which suggests in its current state it is not viewed as sufficiently distinctive. But it does seem clear, at least to me, that the original poster (OP) was hoping for a thread quite different in character to our existing ones. Could someone suggest to what extent the wording of the question would need to be clarified/strengthened to emphasise the question's distinctive character, so that it could return as an open thread?

Alternatively, if this question is to be viewed as essentially indistinguishable from the "simplest example" question, then perhaps this thread should be merged into that one (and presumably that one would be reopened, since if "real life example" answers were deemed acceptable there, then that thread is not redundant to the Bernoulli variable one). Personally, I don't find this alternative particularly attractive, because it strikes me there is worthwhile distinction between a minimalist mathematically simple example, and a more heuristic/intuitive real life example, but I think it would be more consistent than the current situation.

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    $\begingroup$ I think both threads should be re-opened and I have now voted for that. Real life examples is one thing, simple examples is another thing, and a question clarifying some technical confusion about one particular case is yet another thing. There is probably a good case for making the real-life and the simple-examples questions community wiki (my recent Meta question about CW notwithstanding). I have now edited the simple-examples Q to make it more explicitly CW-like (so that it asks about "some simple examples" and not about "the simplest example"). $\endgroup$
    – amoeba
    Commented Mar 5, 2016 at 0:46
  • $\begingroup$ As a general remark, whenever I see some question closed as a duplicate of another question that is closed as a duplicate, I try to sort this situation out (usually by flagging for moderator attention and suggesting some more reasonable solution as a flag comment). I think it should not happen. $\endgroup$
    – amoeba
    Commented Mar 5, 2016 at 0:47
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    $\begingroup$ Real life examples are much harder to find than mathematical justifications and for many people are convincing and interesting, so, if I knew how, I would vote for the real life one to be reopened. $\endgroup$
    – mdewey
    Commented Mar 5, 2016 at 16:06
  • $\begingroup$ @mdewey: You cannot vote because you don't have enough reputation for that (yet). I think you need 2k rep to get access to voting. $\endgroup$
    – amoeba
    Commented Mar 5, 2016 at 22:52
  • $\begingroup$ @mdewey, actually, it is 3k. stats.stackexchange.com/help/privileges/close-questions $\endgroup$
    – Andre Silva
    Commented Mar 7, 2016 at 20:37
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    $\begingroup$ Update: simple examples thread got reopened & made CW (by whuber). $\endgroup$
    – amoeba
    Commented Mar 17, 2016 at 10:33
  • $\begingroup$ @amoeba Thanks. This rather changes things; in particular, it raises the question of whether the "real life examples" be merged with the "simple examples" one? $\endgroup$
    – Silverfish
    Commented Mar 17, 2016 at 11:09
  • $\begingroup$ I still think that the threads are quite clearly distinct enough to remain open. I don't really care about this specific case but it saddens me that decisions like that are so unbelievably difficult to discuss here. We are very fast to close but to reopen anything is a nightmare. $\endgroup$
    – amoeba
    Commented Mar 21, 2016 at 10:07
  • $\begingroup$ @amoeba I feel you. I think that CV is just slightly not big enough to get that kind of responsiveness - if there were about 20-30 regular Meta participants, then I think proposals would gain traction faster (in particular, it'd be easier to reach 5 reopen votes). A question I don't know the answer to, is whether a Mod would be open to implementing a "decision" reached through Meta discussion on this issue, rather than the reopen vote procedure. $\endgroup$
    – Silverfish
    Commented Mar 21, 2016 at 11:43
  • $\begingroup$ Have you seen a recent similar thread, by the way? meta.stats.stackexchange.com/questions/3017 $\endgroup$
    – amoeba
    Commented Mar 21, 2016 at 11:48
  • $\begingroup$ @amoeba I did and I note it came to a similar lack of resolution! (In that case, my Bayesian knowledge is too rusty for me to see how well the proposed duplicate worked. Perhaps it was a good choice but if so, it was not entirely obvious to me. This thread here is rather different, in that it seems so much more a matter of judgment but the issues involved are not very technical. $\endgroup$
    – Silverfish
    Commented Mar 21, 2016 at 15:36
  • $\begingroup$ I have difficulty understanding how "real life examples" can demonstrate any mathematical assertion, much less this one. I am reminded of Hotelling's criticism of a similar effort: "To "prove" such a mathematical result by [a large number of real-life examples] ... is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoology or to mathematics." $\endgroup$
    – whuber Mod
    Commented Mar 26, 2016 at 23:05
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    $\begingroup$ @whuber Thanks for the comment. I don't think the question was demanding a mathematical proof: at least, I didn't interpret it as such. Practical examples have more and better uses than providing technically incomplete proofs. "Real world" examples are pedagogically valuable because they are motivating (why do or should we care about the distinction between "independent" and "uncorrelated"?) and grant us intuition (eg by demonstrating some of the "failure modes" of the incorrect statement uncorrelated implies independent, which purpose the answers in that thread serve admirably) $\endgroup$
    – Silverfish
    Commented Mar 26, 2016 at 23:45
  • $\begingroup$ Okay, I can accept that. What still isn't clear is why the criteria of motivation and intuition distinguish "real-world" from "theoretical" examples. It also seems to me that existing answers miss that mark because they fall short of showing what errors ensue from failing to distinguish the appearance of correlation from the appearance of independence in datasets. I'm not suggesting these are bad answers, only that they don't seem to accomplish any of the "pedogogically valuable" things you list. That in turn makes me wonder whether the question itself might have multiple interpretations. $\endgroup$
    – whuber Mod
    Commented Mar 26, 2016 at 23:51
  • $\begingroup$ @whuber It's a common misconception that if X and Y are uncorrelated, their scatter plot is the classic "blob" (essentially a joint normal distribution with $\rho=0$). Since that's also the scatter plot that they'd draw if asked to produce a plot showing independence, it's no wonder people conflate the two. Peter Flom's "U shaped curve" knocks that misconception squarely on the head. I think Adrian's answer is also nice: it illustrates a different (perhaps less obvious) "failure mode" of uncorrelated implies independent, with very clear obvious "real world" ("why should I care") implications $\endgroup$
    – Silverfish
    Commented Mar 26, 2016 at 23:59

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I think both threads should be re-opened and I have voted for that.

Real life examples is one thing, simple examples is another thing, and a question clarifying some technical confusion about one particular case is yet another thing. They are all sufficiently different to remain non-duplicate.

There is probably a good case for making the real-life and the simple-examples questions community wiki (my recent Meta question about CW notwithstanding). I have now edited the simple-examples Q to make it more explicitly CW-like (so that it asks about "some simple examples" and not about "the simplest example").

Update (Mar 28): Both threads are re-opened now. Both were reopened via moderator action; for one of them the reopening vote failed and it only got reopened after I raised a corresponding flag.

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Please upvote this answer if you could not care less.

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Please upvote this answer if, despite what @Silverfish and myself wrote, you think the threads are really duplicates and should stay closed as such.

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