Why does the p-value double when using two-tailed test compared to one-tailed one? Since this has been marked as a duplicate, I want to clarify that it's not about critical values of one- vs two-tailed tests, but the calculation of the p-Value in this case. 
I get that in a two-tailed test, you look at both sides of the distribution and therefore you split alpha in half and you need a more extreme test statistic to get a significant result (at the same alpha level). 
In my understanding, the p-value is the probability to get this or a more extreme test statistics if the H0 is true. This is easily calculated for a one-tailed test: I just calculate the integral right of the empirical test statistic

For a normal distribution, a z-Value of 1.645 gives me a p-Value of ~0.05. But if I do the same procedure with a two-tailed test, SPSS/Excel doubles the p-value. I get that you now look on the other side of the distribution as well, and therefore I think it's the integral of z(test statistic) + z(-test statistic), but why? I did not get the result -1.645, so I don't understand why I should add the 2.5% of the other side. 
 A: A p-value is the probability of obtaining a result at least as extreme as the one observed.
In the case of a two-tailed z-test, "more extreme" means having a z-value at least as great in magnitude (at least as far from zero) as the observed z-value.
So if your sample gives a z-value of say 1.3 (just for an example), then the p-value will be the area to the right of 1.3 plus the area to the left of -1.3. 

Similarly if your sample gives a z-value of -2.1, then the p-value will be the area to the left of -2.1 plus the area to the right of 2.1. 
A: When you do a two-tailed test you are in fact obtaining both the positive and the negative of the statistic. Remember a two-tailed test means that your are testing whether your alternative hypothesis is different from the null, which could mean either greater or less than. The "greater than" part gives you the critical region on the positive side of the curve, whereas the "less than" part gives you other critical region on the negative side of the curve.
Since both regions are the same, you will usually only use either of the statistics (positive or negative) and divide your level of significance in half to account for the other side.
