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:
This answer elaborates a bit on the answer above, pulling from some YouTube videos done by Specialized Bicycles in their own wind tunnel as well as a few other sources. It goes on a (hopefully acceptable) tangent by discussing an alternative hand position for road cyclists, some of whom may not be able (due to racing regulations) or willing (due to vanity) to install aero bars. It also discusses the potential use of aero bars in group rides - there are some situations when this is acceptable!
Aerodynamic drag produces most of the resistance you need to overcome when in motion, unless you are riding up a very steep hill. Anecdotally, I believe I've heard aerodynamic experts (Josh Poertner of Zipp, the folks at Flo Cycling) discuss on podcasts that the average cyclist needs to be on about a 6% climb before the effect of lighter weight overcomes that of better aerodynamics.
Quoting @RChung (and simplifying a bit), the power needed to overcome aerodynamic drag is:
0.5 * drag coefficient * frontal area * air density * speed ^ 3
In many contexts, you'll hear the term CdA, or coefficient of drag area - this is basically the drag coefficient * drag area. Aero bars and getting in the drops reduce your CdA. (Note: corrected some text thanks to @RChung's comment)
The original post didn't mention the aero hoods position. You have your hands on your hoods, but you crouch so that your forearms are parallel to the ground. In the image below, originally by Getty and linked from an article by Peloton magazine, the lead rider (Jolien D'Hoore) is in the aero hoods position, and the second rider (Anna van der Breggen) is in the drops.
Aero hoods is actually more aero than riding in the drops, as shown subsequently. Consider that in the drops, your forearms are actually exposed to the wind, unless you are crouching extremely (and unsustainably) low in the drops. Aero hoods shields your forearms from the wind. Moreover, the wind 'sees' your forearms roughly as cylinders, and cylindrical shapes have a high drag coefficient. Further, torso positions are comparable between aero hoods and drops. Thus, even though the aero hoods position looks only a bit different from the drops, the difference in drag is measurable. Even though clip on aero bars on a standard drop bar bike may have the pads above the handlebar, they also shield your forearms, and they probably also streamline your body's leading edge to the wind, reducing your drag coefficient more than aero hoods.
Otherwise, the usual positions for road cyclists without aero bars include, from most to least drag, hands on the tops, hands on the brake hoods, and hands in the drops. Previously, some road racers put their wrists on the handlebars close to the stem, as if they were holding invisible aero bars (which is one name for this position, another being the puppy paws position). This does reduce your control over the bike, although no known crashes have been attributed to it. The UCI (International Cycling Union) has banned this position in road races it sanctions.
Much of this section draws on work by Specialized. For each position or upgrade, Specialized prefer to present the number of seconds saved per 40 kilometers in a constant solo effort. They argue, in this article by Bikerumor, that a change in position or in equipment tends to produce about the same amount of total time saved between 20 and 50 km/hr (about 12.4 to 31.1 mph). That is, no matter what your own speed, you should save about equal time. The amount of power saved, naturally, varies significantly. (Image from Specialized presented in the BikeRumour article)
In this YouTube video, Specialized demonstrated that for the test rider:
A different video shows that for a different rider on a different bike:
Notably, in the last clip, when the rider was in her aero bar position, she sitting noticeably higher up than when she was on the drops. (See 0:58 for a frontal view in the aero bars, 0:50 for frontal view in drops.) This was without changing the rest of her bike position. I believe that one of the commenters on the original question may have found it counterintuitive that a relatively high aero bar position could still reduce drag. It can.
This article by road.cc discusses an academic article comparing the regular hoods position, the drops, aero hoods, and short reach aero bars. This is the same rider on the same bike. I'm unable to access the article, but drawing from the text, this is the power required by the studied rider to maintain 28 mph (45 km/h) in a wind tunnel, at 0 degrees yaw:
The test discussed above shows a bit over 5% reduction in required power at their testing speed when going from standard drop position to clip on aero bars, or a bit over 4% reduction from drops to aero hoods. The early sections of this GCN video corroborate that the aero hoods is a significant power savings over the drops (33.7W at 30km/h; they discussed that the rider's CdA on his road bike in aero hoods was comparable to some riders on their time trial bikes).
One issue to consider is that if you have aero bars installed but you are not using them, this seems like it should cause some additional drag, since you are adding frontal area (recall the formula cited above) to one of the leading edges of your bicycle. If you are in a triathlon or time trial and you were going to be in your aero bars almost all the time, this is obviously moot. It could possibly be something to consider if you are not in a triathlon or time trial, as discussed later. However, I'm not aware of testing on the additional drag imposed by unused aero bars.
In any case, versus the standard upright position on the hoods and the drops, aero bars are a significant reduction in drag. Roadies who can't or don't want to use aero bars may want to be aware of the aero hoods position, because it comes very close to clip on aero bars in drag. This does strain your triceps a bit, so it isn't typically used over very long distances. Naturally, one could train for this position specifically if one were interested.
Never mind just drag: in very long rides, aero bars do offer an additional riding position that you can cycle through. This can reduce strain on your neck, shoulders, and arms over time, as articulated in this Cycling About article on bikepacking. However, many long rides involve cycling in a group at least some of the time. This leads to the next section.
The following is irrelevant to the OP's, but it's worth discussing how or if aero bars should be used in groups. You lack easy access to your brakes and your steering control is reduced when on aerobars. This was alluded to in the other answer. In a paceline, you may have to brake or maneuver on short notice. Thus, aero bars are outright banned in road races. (The Cinelli Spinacis were small clip on aero bars that were UCI legal for a short time, but were banned in 1997.)
However, many longer rides, like gravel races, audaxes, or road gran fondos or century rides don't explicitly ban aero bars. Anyone considering using aero bars in these events should exercise caution and use some common sense. In particular, I would recommend staying out of your aero bars if you are in a paceline behind other riders. I would actively object to riders who use their aero bars in a paceline, although I would probably accept it if they were taking a turn at the front and they demonstrated sufficient handling skill. I would take additional caution on gravel, and I would take additional caution if there were a lot of other riders around (e.g. my group catches another group in front, or just after starting the race when most riders are together).
Geoff Kabush, a professional mountain bike and gravel racer, vehemently opposes aero bars in gravel rides due to safety issues, and he thinks they should be banned outright. The lead groups often are riding fast and in close proximity, and obviously they are mostly on loose terrain. A counterpoint by Mat Stephens, another competitive gravel rider, is available here. He essentially says that people need to get over it, and that they can save the rider a considerable amount of power expended in a long race with many windy stretches. Recall that if you save power, you also have to eat less. Of course, Stephens has presumably has good bike handling skills. And because he has outright won a major gravel race, he could probably drop most people objecting.
See Chris H's comment below on aero bars being widely accepted in audax rides. These tend to be very long, multiple day rides. I would expect the groups to be very stretched out. This particular road Gran Fondo (Italian for "big ride", generally means a big organized single day ride that isn't a sanctioned race) strongly discourages aero bars but does not ban them. This likely does not represent all Gran Fondos, but is an example of a possible step a ride could take.
In group rides organized by clubs, my experience is that I have rarely seen aero bars as road cyclists tend to prefer not to use them. I think this is a combination of fashion, plus aero bars being banned in sanctioned road races. I suspect that many clubs will lack specific rules because they encounter the situation rarely. However, a ride leader might exercise their discretion to ask you to leave. Triathlete group rides are likely to be different because aero bars are de rigueur in triathlons, but I suspect triathletes also ride further away from each other than road-oriented groups. See Vladimir F's comment about his own cycling club.
In a group situation, remember that your actions affect the people riding close to you. Because of group cycling dynamics, this can include people who did not make a conscious choice to ride with you!