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There are many stories about twins being able to "feel" each other even when they are apart. Typically a twin would say that they felt that an accident or some other calamity happened to the other one even though they were in different cities etc. Has this been researched?

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No, twins do not have any special magical powers, despite a number of studies investigating the claim.

A lightweight study was done in 1993 to examine if identical twins tended to have similarities in thinking, independent of any Extrasensory Perception (ESP).

The actual study that was performed is largely irrelevant here. What is interesting is the literature search in the Introduction:

Research on ESP in twins has been limited and sporadic.

It goes on to examine a fair number of studies that have been performed, and concludes:

Generally there seems to be little evidence of either a special ESP relationship between twins or a closer one between identical twins. Watson (1981) concludes that there is no evidence to support the idea of any parapsychological phenomena involved in the twin bond, and Palmer (1978), after reviewing the literature, concludes "There is no evidence that twins have any special aptitude for `telepathic' exchange

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    Just to be clear what I have done here: There is no good evidence for twins having ESP. If they did, it would be a huge sensation, and have a huge impact of physics, neuroscience and psychology. However, it isn't sufficient on Skeptics.SE to simply assert that without references, so I have found someone who has done a proper literature search and cited them as saying there is no evidence to be found. – Oddthinking Oct 15 '12 at 9:59
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    but the ministry of suppression of magic among muggles may have magically influenced the results of that test! – Chad Oct 15 '12 at 16:24
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    @Christian: (1) I included the "limited and sporadic" quote out of a sense of fairness, even though I didn't agree. The introduction did list many previous studies. – Oddthinking Oct 15 '12 at 23:01
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    (2) Yes, there is a general problem analogous to "the God of the gaps". As we eliminate possibilities, the conjectures get wilder to explain why the experiment didn't work. Even if we maimed and killed a large sample with no effect, that would still happen. "It doesn't work when disbelievers are watching." At some point, we have to say "When we have repeatedly tried to reproduce it in different ways, it has failed. Our current understanding of physics, genetics, anatomy, evolution and statistics preclude it. Let's go research malaria instead." – Oddthinking Oct 15 '12 at 23:03
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    Two words: Confirmation Bias. Case solved. :) – JasonR Oct 16 '12 at 15:33

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