Crit calculations

[Jackinthegreen]

After a bit of fiddling, I've made a probably very basic observation: A weapon with 20/x2 crit range will have its total average damage be exactly its weapon damage due to the inherent 5% miss from 1's automatically failing.  Because of that, the calculations done at http://www.seankreynolds.com/rpgfiles/rants/keen_medium.html are off, but thankfully it's all 5% off so the comparisons are still accurate.

These tables show the total average damage for the weapons assuming they hit on a 2 and likewise confirm crits on a 2.

Longsword versus scimitar, base crit:

            [/row]
                                                                                                                                                                                                                                                                                                                           

Str	longsword 1x crit	scimitar 1x crit	longsword 1.5x crit	scimitar 1.5x crit
10	4.70250	3.82375	4.70250	3.82375
12	5.74750	4.91625	5.74750	4.91625
14	6.79250	6.00875	7.83750	7.10125
16	7.83750	7.10125	8.88250	8.19375
18	8.88250	8.19375	10.97250	10.37875
20	9.92750	9.28625	12.01750	11.47125
22	10.97250	10.37875	14.10750	13.65625
24	12.01750	11.47125	15.15250	14.74875
26	13.06250	12.56375	17.24250	16.93375
28	14.10750	13.65625	18.28750	18.02625
30	15.15250	14.74875	20.37750	20.21125
32	16.19750	15.84125	21.42250	21.30375
34	17.24250	16.93375	23.51250	23.48875
36	18.28750	18.02625	24.55750	24.58125
38	19.33250	19.11875	26.64750	26.76625
40	20.37750	20.21125	27.69250	27.85875
42	21.42250	21.30375	29.78250	30.04375
44	22.46750	22.39625	30.82750	31.13625
46	23.51250	23.48875	32.91750	33.32125
48	24.55750	24.58125	33.96250	34.41375
50	25.60250	25.67375	36.05250	36.59875

It takes 19 extra damage for the scimitar to pull ahead of the longsword with base crit for both.

[Jackinthegreen]

After a bit of fiddling, I've made a probably very basic observation: A weapon with 20/x2 crit range will have its total average damage be exactly its weapon damage due to the inherent 5% miss from 1's automatically failing.  Because of that, the calculations done at http://www.seankreynolds.com/rpgfiles/rants/keen_medium.html are off.

Note: Since posting this information I've refined and corrected the calculations with the help of Garryl, Halinn, and Maat Mons.  The information below is inaccurate, but I am keeping it around for others to see how the calculations were refined.


A longsword with 4.5 average damage and 19-20/ x2 crit will have an average damage of 4.725, a difference of .225 instead of the .45 the chart claims on 100% crit confirms.  Likewise, if Keen and Imp Crit stack a 4.5 longsword (with 30% crit) will deal an average of 5.625 damage and not the 5.9 (actually 5.85) claimed in his second chart.

Here is a chart of the average damage taking into account the base miss chance.  The two points where a Scimitar outdoes a Longsword are in red.  This also assumes 100% critical confirm, so this is the best case scenario.

            

Weapon	Str	Str 1x, with IC + Keen	1.5x Str, with IC + Keen
Longsword	10	5.625	5.625
Scimitar	10	4.9	4.9
Longsword	12	6.875	6.875
Scimitar	12	6.3	6.3
Longsword	14	8.125	9.375
Scimitar	14	7.7	9.1
Longsword	16	9.375	10.625
Scimitar	16	9.1	10.5
Longsword	18	10.625	13.125
Scimitar	18	10.5	13.3
Longsword	20	11.875	14.375
Scimitar	20	11.9	14.7
Longsword	22	13.125	16.875
Scimitar	22	13.3	17.5
Longsword	24	14.375	18.125
Scimitar	24	14.7	18.9
Longsword	26	15.625	20.625
Scimitar	26	16.1	21.7
Longsword	28	16.875	21.875
Scimitar	28	17.5	23.1
Longsword	30	18.125	24.375
Scimitar	30	18.9	25.9

Changing the calculations to account for inherent miss chance shows us that one-handing a scimitar with IC + Keen only becomes better than a likewise IC + Keen longsword (barely) at 20 strength and higher, or +5 damage.  Two-handing the scimitar becomes better than a longsword at 18 strength, which is +6 damage.

The formula to calculate these numbers was (((weapon damage)*(non-crit hit chance percent/5)) + ((wpn)*crit multiplier * (crit chance/5)) all over 20, as Halinn did.  Note that non-crit hit chance/5 and crit chance/5 totaled up to 19 to account for the 5% miss chance.

I'll add the numbers for IC and Keen not stacking later.  I might do similar for the greatsword and falchion.

[Jackinthegreen]

Reserved for future use.

This should be the last reserved post I'll need.

[Maat Mons]

Accounting for the automatic miss on a 1 doesn't really change anything as long as you account for it on both the initial attack roll and the roll to confirm the critical.  I'll do the math without assuming success. 

(click to show/hide)

We can safely say the average damage is equal to the following. 

[chance of non-critical hit] * [non-critical damage] + [chance of critical hit] * [critical damage]



We can also safely say these two things. 

[chance of critical hit] = [chance of critical threat] * [chance of hit]

[critical damage] = [non-critical damage] + [extra critical damage]



If we assume all the numbers in your threat range are enough to hit, we can say the following. 

[chance of non-critical hit] = [chance of hit] - [chance of critical hit]



With some substitution, we get this. 

([chance of hit] - [chance of critical threat] * [chance of hit]) * [non-critical damage] + [chance of critical threat] * [chance of hit] * ([non-critical damage] + [extra critical damage])



Expand that out. 

[chance of hit] * [non-critical damage] - [chance of critical threat] * [chance of hit] * [non-critical damage] + [chance of critical threat] * [chance of hit] * [non-critical damage] + [chance of critical threat] * [chance of hit] * [extra critical damage]



And cancel. 

[chance of hit] * [non-critical damage] + [chance of critical threat] * [chance of hit] * [extra critical damage]



Now factor. 

[chance of hit] * ([non-critical damage] + [chance of critical threat] * [extra critical damage])

Since chance to hit will be the same for both weapons, it won't have any effect on which is larger. 

[Jackinthegreen]

Halinn mentioned his equation didn't factor in the 5% automatic fail on confirm rolls, but my understanding of it is it'd be easy to add in.  I'll tweak my excel stuff and edit it in the OP.

Come to think of it, the automatic fail for confirms lowers a 20/x2 weapon's total observed average damage to slightly below its sheet base damage.

[Vicerious]

You can't use a rapier in two hands to get 1.5x Str.  A scimitar has the same stats (1d6, 18-20/x2) and can be used two-handed, so your numbers are still valid.

Edit:  Question: how do the numbers compare to the high-multiplier weapons? Battleaxe (1d8, 20/x3) and heavy pick (1d6, 20/x4)?

[Jackinthegreen]

You can't use a rapier in two hands to get 1.5x Str.  A scimitar has the same stats (1d6, 18-20/x2) and can be used two-handed, so your numbers are still valid.

Seems like yet another screwup for the Sean K Reynolds guy, as well as myself of course.  I'll change the weapon.

[Halinn]

I'd be interested in seeing the math done for an x4 weapon as well. The performance of the scythe surprised me when I did the math on two-handers.

[Jackinthegreen]

I'd be interested in seeing the math done for an x4 weapon as well. The performance of the scythe surprised me when I did the math on two-handers.

Any specifics on this?  I can compare a higher base damage 20/x3 weapon to a lower base 20/x4 to see how it goes if you'd like.  I think I can even make it a dynamic calculation thanks to your math.

[Halinn]

I suggest the two mentioned by Vicerious, the Battleaxe and the Heavy Pick. Then you'll have covered the four basic types of one-hander - 1d8, 19-20x2; 1d6, 18-20x2; 1d8, x3; and 1d6, x4

[brujon]

Will two-handers be next?
I'd love to see the weapons of choice for chargers statted out...

[Halinn]

Will two-handers be next?
I'd love to see the weapons of choice for chargers statted out...

Halberd of Vaulting, upgraded with the valorous enhancement. Get enough damage that criticals won't really matter

[Jackinthegreen]

I suggest the two mentioned by Vicerious, the Battleaxe and the Heavy Pick. Then you'll have covered the four basic types of one-hander - 1d8, 19-20x2; 1d6, 18-20x2; 1d8, x3; and 1d6, x4

What I'll eventually be doing is statting them all out with base crit, then IC or Keen, then IC + Keen.  With base crits for the battleaxe and heavy pick though, the pick doesn't pull ahead until 46 strength, or +18 damage.  For 2h it's 34 strength, or +12 x 1.5.  Note that I've reformatted the table from the top to make it easier to read.  The heavy pick is noted as red where it's better than the battleaxe, assuming 95% chance to hit and all crits are confirmed.

Base crit:
         [/row]
                                                                                                                                                                                                                                                            

STR	btlaxe 1x	hvy pick 1x	btlaxe 1.5x	hvy pick 1.5x
10	4.7250	3.8500	4.7250	3.8500
12	5.7750	4.9500	5.7750	4.9500
14	6.8250	6.0500	7.8750	7.1500
16	7.8750	7.1500	8.9250	8.2500
18	8.9250	8.2500	11.0250	10.4500
20	9.9750	9.3500	12.0750	11.5500
22	11.0250	10.4500	14.1750	13.7500
24	12.0750	11.5500	15.2250	14.8500
26	13.1250	12.6500	17.3250	17.0500
28	14.1750	13.7500	18.3750	18.1500
30	15.2250	14.8500	20.4750	20.3500
32	16.2750	15.9500	21.5250	21.4500
34	17.3250	17.0500	23.6250	23.6500
36	18.3750	18.1500	24.6750	24.7500
38	19.4250	19.2500	26.7750	26.9500
40	20.4750	20.3500	27.8250	28.0500
42	21.5250	21.4500	29.9250	30.2500
44	22.5750	22.5500	30.9750	31.3500
46	23.6250	23.6500	33.0750	33.5500
48	24.6750	24.7500	34.1250	34.6500
50	25.7250	25.8500	36.2250	36.8500

[Jackinthegreen]

If we go with Improved Crit OR Keen, we find that the heavy pick does exactly the same damage as a battleaxe at +8 damage, which happens 1-handed at 26 strength.  Going 2-handed shows the heavy pick ahead at 22 strength, which is +9 damage.

Improved Crit or Keen:
         [/row]

STR	btlaxe 1x	hvy pick 1x	btlaxe 1.5x	hvy pick 1.5x
10	5.1750	4.3750	5.1750	4.3750
12	6.3250	5.6250	6.3250	5.6250
14	7.4750	6.8750	8.6250	8.1250
16	8.6250	8.1250	9.7750	9.3750
18	9.7750	9.3750	12.0750	11.8750
20	10.9250	10.6250	13.2250	13.1250
22	12.0750	11.8750	15.5250	15.6250
24	13.2250	13.1250	16.6750	16.8750
26	14.3750	14.3750	18.9750	19.3750
28	15.5250	15.6250	20.1250	20.6250
30	16.6750	16.8750	22.4250	23.1250
32	17.8250	18.1250	23.5750	24.3750
34	18.9750	19.3750	25.8750	26.8750
36	20.1250	20.6250	27.0250	28.1250
38	21.2750	21.8750	29.3250	30.6250
40	22.4250	23.1250	30.4750	31.8750
42	23.5750	24.3750	32.7750	34.3750
44	24.7250	25.6250	33.9250	35.6250
46	25.8750	26.8750	36.2250	38.1250
48	27.0250	28.1250	37.3750	39.3750
50	28.1750	29.3750	39.6750	41.8750

[Jackinthegreen]

Woohoo triple post!

I've done the spreadsheet for all the variations of 1d6 x4, 1d8 x3, 1d6 18-20, and 1d8 19-20 weapons for normal crit, IC or Keen, and then IC and Keen stacking, and up to +30 damage.

However, the potential problem is it accounts for the 5% automatic miss chance on normal rolls, but does not account for the 5% miss on crit confirms, if the rule that a 1 is an automatic miss applies to confirms.  I'm not entirely sure how to go about it though because putting the 5% fail in the crit section gives me a different output than distributing the 5% fail throughout the equation, assuming I did it right.  Since I'm doing this spreadsheet, I might as well get it right.

For example, a +0 longsword without the autofail crit does 4.725 average damage.

If I put the 5% fail in only the crit area, the output is 4.68.  The equation for that was =(((wpndmg)*17)+(0.95*(wpndmg)*2*2))/20.
Putting the 5% fail throughout results in the longsword doing 4.7025 damage.  That equation is =0.95*((((wpndmg)*18)+((wpndmg)*2*2)))/20

Maat, Halinn, are either of those two correct for this situation?

[Garryl]

The percentage increase over the base damage due to criticals (after accounting for critical confirm rolls and the chance of missing, which happen to be one and the same) is 5% * [effective threat range] * [critical multiplier - 1]. So a longsword deals an actual average of 10% above normal, like a battleaxe and any other 19-20/x2 and 20/x3 weapons. 20/x2 weapons deal 5% extra, and 18-20/x2 and 20/x4 weapons deal 15% extra damage. Keen and similar effects that double your crit range also effectively double this average bonus damage, regardless of the balance of critical threat and critical multiplier.

Note that if your attack bonus is really low, your effective threat range may be lower than normal since you only threaten a crit on a roll good enough for a hit. Also, due to D&D math and stacking multipliers, critical hits are less of a bonus if you already have some sort of multiplication going on.

After a bit of fiddling, I've made a probably very basic observation: A weapon with 20/x2 crit range will have its total average damage be exactly its weapon damage due to the inherent 5% miss from 1's automatically failing.  Because of that, the calculations done at http://www.seankreynolds.com/rpgfiles/rants/keen_medium.html are off.

If you're looking at the average damage per attack where you hit on a 2, the actual average damage done by a 20/x2 weapon is 0.9975 of the average non-critical damage on a hit. The average damage on a hit is 1.05 of average non-critical damage on a hit, but the chance of hitting is only 0.95. However, since the chance of a hit is solely determined by the extremely variable comparison of your attack bonus vs. the target's AC (damage and critical threat and multipliers having nothing to do with it), it becomes easier to run the math comparing two weapon choices by canceling it out on both sides. Sean K. Reynold's chart is accurate for inter-weapon comparison along the 100% confirm column, even if his reasoning for how he got there is way off.

[Maat Mons]

Maat, Halinn, are either of those two correct for this situation?

There are two ways to score a non-critical hit.  You can have an attack roll that doesn't threaten a critical, or you can threaten a critical, but not confirm it.  Now that you're not assuming critical confirmation, you need to account for the latter. 

The modified equation is =(((wpndmg)*17)+(0.95*(wpndmg)*2*2)+(0.05*(wpndmg)*2))/20

This should give you the same results as =0.95*((((wpndmg)*18)+((wpndmg)*2*2)))/20. 

[Jackinthegreen]

Maat, Halinn, are either of those two correct for this situation?

There are two ways to score a non-critical hit.  You can have an attack roll that doesn't threaten a critical, or you can threaten a critical, but not confirm it.  Now that you're not assuming critical confirmation, you need to account for the latter. 

The modified equation is =(((wpndmg)*17)+(0.95*(wpndmg)*2*2)+(0.05*(wpndmg)*2))/20

This should give you the same results as =0.95*((((wpndmg)*18)+((wpndmg)*2*2)))/20.

Plugging both of those in shows they are indeed equal.  I'll update my spreadsheet and tables with that to reflect actual average damage.

@Garryl: I'm doing this mostly because I'm OCD and have the goal of eventually getting a handbook of sorts together to give guidance on weapon hit chance and damage.  Now that I've verified the exact process it'll be fairly easy.  You're right, his chart is still accurate for comparison purposes since everything is 5% off pretty much.

Much thanks to all of you!

[Garryl]

@Garryl: I'm doing this mostly because I'm OCD and have the goal of eventually getting a handbook of sorts together to give guidance on weapon hit chance and damage.  Now that I've verified the exact process it'll be fairly easy.  You're right, his chart is still accurate for comparison purposes since everything is 5% off pretty much.

His chart is 100% accurate for your average damage on a hit, but I don't think that's what you're talking about. I'm not actually sure what we're talking about exactly right now? I'm just going to lay down some formulas and hope this sorts itself out.

(click to show/hide)

Note: These calculations don't include additional sources of damage on criticals (such as Flaming Burst weapons), or that aren't multiplied on criticals (such as Sneak Attack), nor do they consider special modifiers to critical confirmation rolls (such as the Power Critical feat).

[average damage per attack] = [average damage per hit] * [chance to hit]
[average damage per hit] = [average non-critical hit damage] + [bonus damage on critical hit] * [chance on hit to critically hit]
[chance on hit to critically hit] = [chance to confirm crit] * [effective critical threat chance] / [chance to hit]
[chance to confirm crit] = [chance to hit]
[bonus damage on critical hit] = [average critical hit damage] - [average non-critical hit damage]
[average critical hit damage] = [average non-critical hit damage] * [critical multiplier]

[chance on hit to critically hit] = [chance to confirm crit] * [effective critical threat chance] / [chance to hit]
   = [effective critical threat chance]

[average damage per attack] = [average damage per hit] * [chance to hit]
   = ( [average non-critical hit damage] + [bonus damage on critical hit] * [chance on hit to critically hit] ) * [chance to hit]
   = ( [average non-critical hit damage] + ( [average critical hit damage] - [average non-critical hit damage] ) * [effective critical threat chance] ) * [chance to hit]
   = ( [average non-critical hit damage] + ( ( [average non-critical hit damage] * [critical multiplier] ) - [average non-critical hit damage] ) * [effective critical threat chance] ) * [chance to hit]
   = ( [average non-critical hit damage] + ( ( [average non-critical hit damage] * ( [critical multiplier] - 1 ) ) ) * [effective critical threat chance] ) * [chance to hit]
   = ( 1 + ( [critical multiplier] - 1 ) * [effective critical threat chance] ) * [chance to hit] * [average non-critical hit damage]

[chance to hit] is generally a value between 0.5 and 0.95, although concealment and other miss chances can lower it further. However, between weapons, it will always be the same, and so can be ignored when comparing average damage per attack.
[average non-critical hit damage] is based on a wide variety of factors. Strength usually contributes to it, and base weapon damage always does, but to simplify it to those two parameters is, well, to oversimplify. Assuming equal conditions, the miscellaneous modifiers other than base weapon damage will generally be the same between weapons, and so can be considered as a whole rather than being broken down into their components.
[critical multiplier] is the weapon's critical multiplier normally. Due to D&D math, if you already have a source of damage multipliers, it's actually a bit less. Specifically...
   [critical multiplier] = ( [weapon critical multiplier] + [total effective other multipliers] - 1 ) / [total effective other multipliers]

[Jackinthegreen]

His chart is correct for average damage on a hit.  However, I'm adding base accuracy into the equation, which lowers his numbers 5%.  Thinking about it more I suppose it's inappropriate to include accuracy results in for those who just want to see the pure average damage without the accuracy factor, but getting accuracy in there will eventually help calculate how much damage on average a character can expect to do with various attacks.  In short, I want to get the statistics of how much accuracy and damage play together to eventually put a handbook together that will give accurate information on what kind of gear to use to maximize damage.

And it's an exercise in getting the table coding to cooperate.