We were not the first or last people to conduct such tests however.
What Matt and I determined was:
#1. Stabilizers really only helped a little bit at short distances; Its primary purpose was for shooting long distances.
#2. Absorbing vibrations from the bow was not the primary function of a stabilizer, making the bow bottom heavy was the primary function.
Our testing wasn't very scientific, but fortunately Field and Stream (a fishing/hunting magazine based in the USA) did their own series of tests in 2013 and theirs was much more scientific than ours - and yet concluded the same things.
Manufacturers tell us that the primary purpose of stabilizers is to reduce vibrations in the bow, and while that may end up having a minor effect, it is so minor it may not even be worth mentioning.
During the tests Matt and I did we determined that the biggest effect on accuracy was simply the addition of weight - making the bow bottom heavy and then easier to make sure it was perfectly balanced (left to right) and was not canting to the left or right by accident. A loose and relaxed grip plus extra weight made all the difference to the accuracy.
The Field and Stream tests kept careful track of what they were doing:
They had three different people shoot three different bows, each with and without a small hunting-style stabilizer and then measured the size of the arrow clusters and got an average number. They also varied the distances by 30, 40 and 50 yards.
Field and Stream Test Results
Total average group sizes for 6-inch hunting stabilizer, no stabilizer:
30 Yards: 2.59 with stabilizer, 2.75 without stabilizer (a 6% increase in cluster accuracy)
40 Yards: 3.66 with stabilizer, 3.61 without stabilizer (cluster accuracy was actually 2% worse)
60 Yards: 5.07 with stabilizer, 5.23 without stabilizer (a 3% increase in cluster accuracy)
So their conclusions was that a small stabilizer really had very little difference in terms of cluster accuracy, and they felt the test was a bit inconclusive because the averages might have been effected by fatigue, wind conditions and physical differences between the three shooters.
During the tests Matt and I conducted we saw virtually no difference when shooting at approx. 20 yards. Our conclusions was that stabilizers were basically unnecessary at close range. The difference in cluster quality was more noticeable when we were shooting at 30 yards or more.
Field and Streams Test #2
After the somewhat inconclusive results of the first test above, Field and Stream conducted a 2nd test using longer and heavier 10-inch or 12-inch stabilizers.
Due to space restrictions and the fact that they only had two people available for the 2nd test they didn't publish their full results of the 2nd test, but what they did note was that: "We both shot markedly better with them."
So the extra weight was certainly a benefit. A big benefit. "[T]he full story is that these results were even more surprising than those of the original test. Damned-near astonishing, in fact."
They didn't notice any major differences at 40 yards or less, but at 60 yards there was a big difference.
"Shooting a Bowtech Insanity CPXL with your basic 5-inch hunting stabilizer, I shot 10 three-shot groups that averaged 4.82 inches—which was on par with most of my other 60-yard groups. Then I screwed on a 10-3/8-inch, 10.5-ounce, Doinker EFDF (doinker.com), and shot 10 more groups at 60. Average size: 2.97 inches. I figured that had to be a fluke, so I shot another 10 groups. They averaged 3.21 inches. I later got similar results with a 12-inch Doinker D.I.S.H. and a 10-inch Bee Stinger Pro Hunter, too (beestinger.com)."
"Meanwhile, Brantley cut his 60-yard groups nearly in half (from 6.12 to 3.50) when he went from no stabilizer to a 12-inch, 17-ounce Fuse Carbon Bowhunter Freestyle (fusearchery.com) on a Hoyt Spyder Turbo. He called the result “amazing.” In the end, we agreed that it makes no sense to hang such a big, heavy model on a short-range woods bow, but on long-range western or 3D bow, we’d definitely carry the extra weight to get all that extra accuracy."
So the bigger heavier stabilizers really do help - but only at long distances and only a small bit at close distances. At close range or point blank range they are pretty much unnecessary.
One last note...
Years ago I witnessed a fellow archer who prefers to shoot Olympic style go from a mid-weight stabilizer to a very expensive / much heavier stabilizer. Prior to switch he was shooting doughnut sized clusters at 30 yards. After the switch his clusters became the size of footlong subs, going up and down. Now you might wonder what was he doing wrong? If the only thing that had changed was the stabilizer, why would it be causing his clusters to go up and down so dramatically?
The answer is that his bow arm/shoulder was too weak for the huge / heavy stabilizer. He had spent $400 on a very expensive stabilizer that was actually too big for him to handle properly.
And to top it off, according to the test results by both myself, Matt and the good folks at Field and Stream - he really did not need a stabilizer to be shooting at 30 yards. It was an unnecessary crutch and in his case he was actually getting worse.
Now you might think "Won't he get stronger if he keeps using it?" Maybe. However from past conversations with him I know that he is against exercising and argues that "You don't need to exercise to do archery."
Which is true, you do not need to exercise to do archery. But it certainly helps your accuracy if your muscles are stronger in all the right places.
In the same vein, you don't need a stabilizer to do long distance archery. But it does help improve your accuracy a little bit.
Ultimately it is up to personal choice. Some people prefer to shoot with stabilizers, some prefer to shoot without them. I personally only put stabilizers on when shooting my compound bow. I rarely use them on recurve bows.