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I'm trying to answer the question: if twins are raised in separate environments, what is the standard deviation in their heights? Just as some example data, let's say the As are twins, the Bs are twins and so on. Let's say A1 is 170 cm and A2 is 171 cm, B1 is 155 cm and B2 is 157 cm and so on.

I don't have access to any raw data about this.

However, the Minnesota Study of Twins Reared Apart reports that in their sample of 56 sets of twins, the correlation coefficient $r$ between twins was 0.86 (in contrast, for their 274 twins raised together, $r$ was 0.93)

Is there any way to get from this $r$ value back to the expected standard deviation in the normal distribution of which a pair of twins are two sample points? Intuitively I would have thought that if you repeatedly take 2 samples from different normal distributions, and then calculate the correlation of those samples, it should be possible to calculate the average standard deviation of the original normal distributions?

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  • $\begingroup$ This looks as if it might be an exercise, Hint: Your last paragraph seems to ignore the fact that even if distributions are normal, the problem concerns a bivariate normal. Otherwise the definition of correlation is instructive. The correlation depends on covariance as well as means and SDs. $\endgroup$
    – Nick Cox
    Commented Oct 3, 2021 at 8:34
  • $\begingroup$ Thanks @NickCox , no, just a layperson trying to understand something. To be honest, I'm not sure how their 0.86 figure was arrived at, but looking at the paper, it says that they used an interclass correlation. $\endgroup$
    – Bruce
    Commented Oct 3, 2021 at 8:42
  • $\begingroup$ I think you mean intraclass $\endgroup$
    – Nick Cox
    Commented Oct 3, 2021 at 8:47
  • $\begingroup$ Thanks @NickCox, I've checked and the paper definitely says 'interclass'. But using Pearson R, the numbers seem sensible, so I guess it's just a typo in the paper. $\endgroup$
    – Bruce
    Commented Oct 3, 2021 at 10:52

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It's not possible. If you have two normal random variables $X$ and $Y$ with a certain correlation $\rho$ (as estimated by $r$), and you multiply them both by constants $c_1, c_2>0$, their standard deviations will change but their correlation will remain the same. This shows that the correlation doesn't hold enough information to identify the standard deviations.

Note that there is an additional complication in your question because you're not talking about the two separate distributions of the first and the second twin, but rather about "the normal distribution of which a pair of twins are two sample points". Unless the distribution for first and second twin is identical, this will be a mixture of normals rather than a single normal, and even if they are identical, it's not a standard i.i.d. sample because of the dependence between the two twins. Anyway the argument in the first paragraph still applies - the correlation does not hold information about the standard deviation of that distribution, be it a normal or a mixture, for the reason given above.

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  • $\begingroup$ Ah great - thank you! $\endgroup$
    – Bruce
    Commented Oct 3, 2021 at 15:14

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