Normal distribution. Hmm, that would seem to imply that the game determines metal and crystal counts independently.
Anyways, actually, 1d6 has an expected value of 3.5 ((1+2+3+4+5+6)/6 = 21/6 = 3.5) As two dice rolls are independent, it follows that E(2d6) = E(1d6 + 1d6) = E(1d6) + E(1d6) = 3.5 + 3.5 = 7
For 1d12, E(1d12) = (1+2+3+4+5+6+7+8+9+10+11+12)/12 = 78/12 = 6.5. So actually, 2d6 has a slightly higher expected value than 1d12. Interesting.
So on a normal planet, the range for a single resource type is 0 to 4 (both bounds inclusive). What is the distribution of that? (again, on a SINGLE TYPE of resource)
Well those numbers are correct, I normally like to drop the .5 as you can't roll a .5 on a die, even if it not perfectly correct.
And the fact that 2d6 has a slightly higher expected value is not only well know, but if you read any D&D optimization guide you notice that they always go for weapons that roll 2d6 or 2d4 over 1d12 or 1d8. Even more often is that they also look for brutal weapons as well, even not brutal, rolling 2d6 over 1d12 means a range of 2-12 versus 1-12, with the first averaging higher. Throw in brutal 1 and you get a range of 4-12, Brutal 2 becomes 6-12.
On the graph, it shows that 3,4 and 5 resources types make up the bulk of the galaxy, so balancing for 4 was not quite wrong. The greater issue likely is as it was pointed out, planets like the ones with ARS tend to have 5+ rather then finding an ARS on a planet with only 1 or 2. So maybe how those are being seeded should be adjusted rather then income from a harvester.
After all, you should need to take planets for resources, planets for ARS and if you are lucky they are the same planet, but it sounds like that is not the case.
Given that I personally only play Fallen Spire, I don't often look at how much each planet makes, as I am going to have to take them all anyway.