There is a general rule in experimental physics that "more data means less uncertainty." This is because the estimate of the uncertainty is usually assumed to be
$$
\sigma = \sqrt{ \frac{1}{n-1} \sum_{i = 1}^n (x_i - \bar{x})^2 },
$$
and a higher $n$ leads to a smaller $\sigma$.
However, the uncertainty in your overall measurement will not decrease if you simply count the data points twice. This is because the standard deviation of the data is only a good estimate of the uncertainty in $\bar{x}$ if the individual measurements $x_i$ are uncorrelated with each other. This basically means that the value of $x_i$ is independent of the value of $x_j$ for all $i$ and $j$. In your experiment, the values of the "duplicated" data are highly (in fact, perfectly) correlated with the original data points, so the equation doesn't allow you to reliably estimate the uncertainty in your measurement of $\bar{x}$.
In fact, it's not too hard to show that "duplicating" the measurements like this doesn't change your uncertainty at all. To get into this, you have to generalize the ideas of error propagation to the cases where errors are correlated with each other. This isn't usually done in intro physics courses, but it's not that much of a stretch. To get a flavor of it, take a look at the Wikipedia page on error propagation; the $\sigma_{AB}$ terms are the ones you usually neglect in intro physics labs, but if the measurements are correlated with each other, you need to consider them.
