How big a sample size do I need to test my hypotheses?

The four determinants of statistical power are related. If you know three of them, you can figure out the fourth. A prospective power analysis can thus be used to determine the minimum sample size (N) given prior expectations regarding the effect size, the alpha significance criterion, and the desired level of statistical power.

For example, if you hope to detect an effect of size r = .40 using a two-tailed test, you can look up a table to learn that you will need a sample size of at least N = 46 given conventional alpha and power levels.

To detect a smaller effect of r = .20 under the same circumstances, you will need a sample of at least N = 193.

The only tricky part in this exercise is estimating the size of the effect that you hope to find. If you overestimate the expected effect size, your minimum sample size will be underestimated and your study will be underpowered. In other words, you will have a lower probability of obtaining a statistically significant result. If statistical significance is important to you (e.g., because it pleases reviewers or PhD supervisors), then you might want to look for ways to boost statistical power.

More here.