How can I divert down spout water above ground to a different location?
These are the things you need to consider:


the pipe diameter would need to be at least the size of the downspout or larger. So we are talking three to four inch diameter.  Two possibilities that come to mind are:


four inch diameter drain pipe with sleeve, not perforated
PVC pipe in four inch diameter

both are not pretty to look at and detract from the appearance of your house
in both cases you must ensure that there is a grade or drop of at least one inch over four feet. See here for details. A rough calculation for the twenty foot length indicates that the end of the drain would have to be five inches lower than the start of the drain.  If you fail to have a consistent grade water will pool in the pipe and freeze in the winter. This could deform or break PVC pipe.
there is a risk that the pipe will not be able to accommodate the quantity of water a "once in a hundred year storm" could supply.  Weather is unpredictable, if your pipe fails, then it fails next to the house.  The idea of drains is to get the water away from the house as fast as possible.


So, you could do what you describe and it might work fine, or you get a bad rain storm a few years from now and have the water dumped in a window well and into your basement. Who can say?

Here are similar questions with some excellent answers.  As well consider a french drain, this is an ideal solution to your problem.