So, I need to adjust that 95% down to a more reasonable average then look again. Thanks!
The other important thing you are forgetting is the advantage you get from using Reckless attack, which increases both your hit chance and your crit chance . For a character who is not a half-orc, maul and greatsword win out during the earlier tiers of play, but greataxe starts to shine at around 13th level when you get Brutal Critical (2 dice). For a half-orc with Savage Attacks, the break-even point is even earlier--I believe level 9. I did the full math at one point, factoring in advantage and a base 60% chance to hit, but I don't have it handy. I think I even factored in the -5/+10 from Great Weapon Master.
I have quickly done the math and here are my findings without Great Weapon Master.
TLDR: the average difference for 2d6 and 1d12 seems vary comparable, until you get to +2d critical dice, with advantage on the attack roll. Or you need to expand your critical range to see 1d12 dealing out a little more damage on average. Yet on average is the key word here as 1d12 has a large variance.
So my results show that for damage consistency you are best of with 2d6, Great Weapon Fighting style (+1,33 average damage) and Great Weapon Master. Because this gives you a very noticeable +10 damage boost which is ~3d6 on average, the penalty to hit can be off set by obtaining advantage on the roll to hit.
With normal Criticals, 2d6 has 7% more damage output on average with 2d6 as the base for the relative damage difference, regardless of the critical range or advantage (ADV) to hit.
If you add +1d to criticals, 1d12 breaks even at 50% hit chance (or lower) with expanded critical range 19-20. With simple ADV the difference between 2d6 and 1d12 begins to diminish, with 1% at 40% hit chance. With expanded Crit Range and ADV, 1d12 breaks even from 80-65% hit chance and lower. Still the maximum difference in favor of 1d12 is only -3% at 40% hit rate. I didn't calculate below 40% hit rate because I didn't think it remains relevant there but it will only increase as the relative impact of critical hits become greater.
At +2d to criticals, 1d12 breaks even at 55% hit chance (or lower) for normal 20 only crits. Expanded 19-20 crits on all to hit chances, 1d12 pulls ahead with -1% on 95% hit rate. On ADV on normal crits 1d12 breaks even around 80-95% chance to hit. On ADV and expanded crits it is better, beginning with -5% on 95% hit rate.
At +3d to criticals, 1d12 begins to pull ahead from 80% or lower chance to hit for normal 20 crits. So from here on on, 1d12 is the weapon of choice for barbarians.
So like leugren pointed out: a Half-Orc barbarian from level 13 and on will naturally pick the Great-Axe as his weapon of choice for maximum damage output. Yet other races would have to wait until level 17.
I started a half-orc barbarian recently. Starting 17 Str and at L4 I'll take Orcish Fury to get 18 Str and add 1 die roll to any attack. This, plus the Savage Attack ability, made me go with the greataxe. Sure, the 2d6 greatsword is better than a 1d12 greataxe, but there's fun to be had in getting to roll 5d12+6 at level 4. :-D
And as others have noted, 2d6 is a bell curve on damage. That means you're more likely to do average damage, and less likely to do extremely high and low damage. So the greataxe is a "high risk, high reward" weapon. If you like that style of play, go with the greataxe. If you're more of a "Slow and steady wins the race"-type of person, go with the greatsword/maul. :-)
I would take the Greataxe. the focus of barbarians is doing a lot of critical hit damage, that is why (a) Barbarians get a lot of brutal criticals , and (b) people say Half-Orcs are good Barbarians. they get a brutal critical at level 1. if you were playing a fighter or paladin, I would take either the maul or Greatsword. I would lean towards the maul because there are more monsters resistant and immune to slashing than with bludgeoning.
I have quickly done the math and here are my findings without Great Weapon Master.
TLDR: the average difference for 2d6 and 1d12 seems vary comparable, until you get to +2d critical dice, with advantage on the attack roll. Or you need to expand your critical range to see 1d12 dealing out a little more damage on average. Yet on average is the key word here as 1d12 has a large variance.
So my results show that for damage consistency you are best of with 2d6, Great Weapon Fighting style (+1,33 average damage) and Great Weapon Master. Because this gives you a very noticeable +10 damage boost which is ~3d6 on average, the penalty to hit can be off set by obtaining advantage on the roll to hit.
With normal Criticals, 2d6 has 7% more damage output on average with 2d6 as the base for the relative damage difference, regardless of the critical range or advantage (ADV) to hit.
If you add +1d to criticals, 1d12 breaks even at 50% hit chance (or lower) with expanded critical range 19-20. With simple ADV the difference between 2d6 and 1d12 begins to diminish, with 1% at 40% hit chance.
With expanded Crit Range and ADV, 1d12 breaks even from 80-65% hit chance and lower. Still the maximum difference in favor of 1d12 is only -3% at 40% hit rate. I didn't calculate below 40% hit rate because I didn't think it remains relevant there but it will only increase as the relative impact of critical hits become greater.
At +2d to criticals, 1d12 breaks even at 55% hit chance (or lower) for normal 20 only crits.
Expanded 19-20 crits on all to hit chances, 1d12 pulls ahead with -1% on 95% hit rate.
On ADV on normal crits 1d12 breaks even around 80-95% chance to hit.
On ADV and expanded crits it is better, beginning with -5% on 95% hit rate.
At +3d to criticals, 1d12 begins to pull ahead from 80% or lower chance to hit for normal 20 crits. So from here on on, 1d12 is the weapon of choice for barbarians.
So like leugren pointed out: a Half-Orc barbarian from level 13 and on will naturally pick the Great-Axe as his weapon of choice for maximum damage output. Yet other races would have to wait until level 17.
great sword or maul would be the way to go, however it makes more sense for a maul since it is something you can see a half orc weilding
I started a half-orc barbarian recently. Starting 17 Str and at L4 I'll take Orcish Fury to get 18 Str and add 1 die roll to any attack. This, plus the Savage Attack ability, made me go with the greataxe. Sure, the 2d6 greatsword is better than a 1d12 greataxe, but there's fun to be had in getting to roll 5d12+6 at level 4. :-D
And as others have noted, 2d6 is a bell curve on damage. That means you're more likely to do average damage, and less likely to do extremely high and low damage. So the greataxe is a "high risk, high reward" weapon. If you like that style of play, go with the greataxe. If you're more of a "Slow and steady wins the race"-type of person, go with the greatsword/maul. :-)
I would take the Greataxe. the focus of barbarians is doing a lot of critical hit damage, that is why (a) Barbarians get a lot of brutal criticals , and (b) people say Half-Orcs are good Barbarians. they get a brutal critical at level 1. if you were playing a fighter or paladin, I would take either the maul or Greatsword. I would lean towards the maul because there are more monsters resistant and immune to slashing than with bludgeoning.
I personally like the idea of Polearms.
I blame Hellion from Darkest Dungeon.