Aero Bars vs The Drops On my road bike I am debating installing aero bars. 

Will I notice a significant measurable improvement in speed using aero bars vs being in the drops?

I am looking for specific numbers here with resources, not just opinions.

The case I am really curious about is riding solo, on flat ground around 20 mph where there is no wind.
Yes, typically you will obtain an improvement in speed by adding clip-on aero bars to your bike. 

How much extra speed is dependent on many variables though (mostly how it changes your aerodynamics) but a couple of km/h faster is certainly possible. 

One of the several compromises of adding aero bars to a road bike versus having a dedicated time trial bike is that the clip-on bars are set up fairly high when clamped onto the tops of the road bike bars. They also take up space where you might wish to rest your hands at times and of course unlike a TT bike you need to move hand position to access gear shifters. It's not a big deal if the road is of a very consistent gradient, but will become a bit of a pain if the road is a little more variable.

Another factor to consider with such bars is how they affect your saddle position as sometimes it helps to bring the saddle forward a little when stretched out more on bar extensions.

Be careful with the clamps and be sure not to over tighten and damage the handlebars, but also don't have them too loose so that they slip when you are riding / go over bumps!

If you do try them then take some time to practice riding with them and stay safe!

Edit: Since the OP wants some specifics, I'll provide an example. 

I did some analysis of my own TT performance over a course I raced twice in 2009. It turns out the wind and air density were very similar on both days which meant a direct comparison was reasonably possible. I use various analysis methods to account for variations in environmental conditions, power output and pacing (some of which are derivatives of original work of R. Chung).

The first race I rode on my standard road race bike, the second on a borrowed TT bike and using the same clothing and helmet for each ride. I accounted for differences in pacing and power (only 1W difference) and calculated the difference in coefficient of drag area (CdA) which is the primary measure of an object's "aeroness". My CdA reduced from 0.334m^2 on the road bike to 0.286m^2 on the borrowed TT bike, a 14% reduction.

Now a 14% reduction in CdA does not mean a 14% improvement in speed. In this case it meant an improvement of 3% in speed or a little over 1km/h. Keep in mind that this wasn't a totally flat course, so on flat ground that would equate to a speed gain of ~1.9km/h. Had I simply added TT bars to my road bike, the speed gains on flat road would have been less.

Here is the link detailing the impromptu assessment and data:
http://alex-cycle.blogspot.com.au/2009/04/new-skool.html

Now these things are individual, so YMMV, and what sort of speed gain is possible for anyone depends a lot on the aero characteristics of the starting position, the ending position and what if any power change is experienced when riding with the TT bars (often riders experience a loss of power when riding in a more aggressively tucked or stretched out position).

Further edit:

Here's some more sample data, this time showing the difference in CdA for four continental professional riders between their road bike (hands in the drops) and their TT bike. Again it's not clip on aero bars, however it's a good example of the variance in aero improvement between riders when position changes from road position to TT position: