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What is relationship between "significant correlation" and "significant difference" between two variables?

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If you have two different variables, you can calculate how they are correlated. Does height correlate with blood pressure? Does age correlate with income? You can also compute a 95% confidence interval for the correlation coefficient.

If you have one variable measured in two groups, you compute the difference between the two means. You can also compute a 95% confidence interval for the difference.

The concept of "significance" is so often misunderstood, I think it is best to just stop with the confidence intervals. But you can ask if the correlation coefficient is far enough away from zero (considering the sample size) to be statistically significant, and if the difference between the two means is far enough from zero (considering the sizes of the two samples, and the variability within the two samples) to be statistically significant.

The key distinction is whether you have two variables, or one variable in two groups.

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  • $\begingroup$ My question is that what is relationship between "correlation" and "difference" between two variables? X and Y are correlated. Do they must have difference? X and Y are different. Do they must have correlation? $\endgroup$
    – Arash
    Apr 3, 2014 at 2:02
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Your comment is clearer than your original question – enough so that I may be able to clear things up for you.

My question is that what is relationship between "correlation" and "difference" between two variables? X and Y are correlated. Do they must have difference? X and Y are different. Do they must have correlation?

The short answer to both is no.

If $X$ and $Y$ correlate, any less-than-perfect correlation $(|\rho|<0)$ implies differences in individual values, but not necessarily in summary statistics like means, medians, variances, etc.
For example, with $X=\{0,0,1\},Y=\{0,1,0\},\mu_x=\mu_y,\rho=-.5$. With $X=Y,\rho=1$. However, with with $X=-Y,\mu_x=-\mu_y,\rho=-1$.
Thus only in the case of a perfect correlation can two variables exhibit no differences whatsoever, but the possibility remains that they are exact opposites. Correlation implies nothing about differences in means, medians, or variances.

If $X$ and $Y$ differ, they may or may not correlate; there is no "must" here. For example, with $X=\{0,0,0\},Y=\{0,1,0\},\mu_x<\mu_y$, but since $X$ has $\sigma=0$, $\rho$ is undefined. Conversely, with $X=\{0,0,1\},Y=\{1,1,0\},\mu_x<\mu_y,\rho=-1$. These examples are the simplest I can think of.

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  • $\begingroup$ Dear Nick, What is your interpretation about undefined ρ? $\endgroup$
    – Arash
    Apr 3, 2014 at 3:49
  • $\begingroup$ There is no information to interpret about correlations involving one variable with completely invariant data. $\endgroup$ Apr 3, 2014 at 3:52
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Testing the statistical significance of correlation coefficients involves the same logic as testing the statistical significance of differences between group means or proportions. Whereas the null hypothesis in the latter tests states that there is no real difference between groups in the population and that any observed differece in our sample is due to sampling error, the null hypothesis with correlation coefficients states that there is no real correlation in the population (i.e., $r= 0.00$) and that any observed correlation in our sample is due to sampling error. Just as in other tests of significance, sampling error when testing the statistical significance of correlation coefficients involves the use of theoretical sampling distributions with an assumed mean correlation of $0$.

From :Statistics for Evidence-Based Practice and Evaluation By Allen Rubin

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    $\begingroup$ I'm sorry I can't even read that. You're welcome to type in a short quote if you want your answer to be legible, but it would also be a good idea to add some of your own thoughts, e.g. interpreting the material in the light of the question. $\endgroup$
    – TooTone
    Apr 7, 2014 at 13:45

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