Scatter plot: calculate box where 80 % of the points are I've a scatter plot with x- and y-axis.

Now I want to calculate a rectangle which displays where 80% of the points in this plot are.

I'm not very familiar with statistics, so I don't know the method I have to use. Read something about percentiles. Is that correct?

I know I have to calculate where 80% are on the x-axis and also where 80% are on the y-axis and then I got my rectangle corners, but how do I calculate the 80%?

Sample data:
Your question seems to imply that there will always be just a single cluster of points.  If this is the case, I would go with a fairly simple heuristic like the following-

Calculate the mean and variance of the point positions in both the x and y directions.  The mean is the center point of your box, and you can use the ratio of the x/y variances to determine in which directions the box should grow faster.  For instance, if the x variance is "1" and the y variance is "2", I would have the box expand twice as fast in the y direction.

Now simply expand the box until you have at least 80% of the points contained.

If there is ever more than one cluster of points you may need to use a more sophisticated solution that uses clustering algorithms to identify the clusters and a box for each cluster.