Overlapping standard errors and statistical significance I have a paired data set which I have placed into $x$ and $y$ columns where $x$ are the control values and $y$ are the values following drug treatment. $N=10$ for both $x$ and $y$ columns as they are paired data. Each $x$ is the control for the corresponding $y$. 
I have seen in various texts stating that when standard error margins overlap, the data cannot significant. By standard error margin, I am referring to ($SE_\bar x = SD/\sqrt N$).
However, I have conducted two-tailed paired $t$-tests on my data set (comparing the means of all values in $x$ versus the means of all values in $y$) and my results yield statistical significance with a $p$-value $< 0.05$ (despite there being overlapping standard error margins with data in $x$ and $y$).
My question is: in a paired data set, is it possible for there to be statistical significance between the control ($x$) and drug treatment ($y$) despite having overlapping standard errors? My $t$-test was done using GraphPad prism so I'm confident there are no errors in the $t$-test. 
 A: Yes, it's quite possible for the $\pm 1$ SE error bars to overlap, but still have a significant pairwise $t$-test.  The reason is that your error bars are calculated on the between subjects data, but the test is of the within subjects data.  They are not the same thing, so they don't have to be consistent with each other.  
It is fine to calculate and present pre- and post-treatment means and error bars, but it can lead to just this confusion.  To understand what a paired $t$-test is doing, you need to think of it as a one-sample $t$-test on the differences.  That is, first subtract the pre-treatment value from the post-treatment value for each patient.  Then you have only one variable: difference scores.  Then you calculate the mean and standard error of those.  The pairwise (one-sample) $t$-test is the mean divided by the SE.  If you like, you can even make a bar plot with a single forlorn-looking bar representing the mean difference with it's corresponding error bar.  That is the actually the relevant plot for a pairwise $t$-test.  If the test was significant, the error bar will not overlap $0$.  To see this concretely, see my answer here: Is using error bars for means in a within-subjects study wrong?
