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In the last article Surf unearthed the math behind hitpoints. Today we look at the last of these major monster stats - Damage....
Meh to crud with the intro... let's dive right in this time!
High-Level Data
During this particular analysis we consider a creature's “At-Will” damage and it's “Damage Per Round” (aka DPR).
| At-Will | DPR | |
|---|---|---|
| Average: | 15.47 | 17.00 |
| Stdev: | 15.83 | 16.80 |
| Var: | 251.08 | 282.32 |
As we saw with character classes, back in Part 2, damage is lower than hitpoints and thus it's variability is lower than hitpoints. Because of Bounded Accuracy it's still considerably higher than AC or Attack bonus though. We expect that monster damage should scale relative to PC hitpoints so this is a good sign.
| StdDevp of At-Will Damage by XP | |||||
|---|---|---|---|---|---|
| Level | Easy | Average | Tough | Solo | Total |
| 1 | 1.56 | 1.31 | 0.00 | 1.62 | |
| 2 | 2.12 | 1.85 | 2.56 | 2.41 | |
| 3 | 2.68 | 3.01 | 0.00 | 2.94 | |
| 4 | 0.00 | 3.59 | 3.77 | 3.70 | |
| 5 | 0.00 | 5.46 | 4.88 | 5.22 | |
| 6 | 6.89 | 4.00 | 6.87 | ||
| 7 | 9.29 | 7.55 | 8.36 | ||
| 8 | 9.80 | 9.80 | |||
| 9 | 2.24 | 2.24 | |||
| 10 | 14.33 | 14.33 | |||
| 11 | 7.59 | 7.59 | |||
| 12 | 9.17 | 9.40 | 11.85 | ||
| 13 | 9.09 | 11.11 | 10.61 | ||
| 14 | 31.60 | 31.60 | |||
| 15 | 0.00 | 0.00 | |||
| 16 | |||||
| 17 | |||||
| 18 | 2.00 | 2.00 | |||
| 19 | |||||
| 20 | 6.00 | 6.00 | |||
| Total | 2.27 | 4.88 | 10.20 | 21.84 | 15.83 |
When looking at the stdev table we need to remember to ignore the pink data points. The low amount of data makes these cells very "swingy". That's why we have some cells with a value of 0.00 (there's only one sample, or the small number of samples have the same value). In other cases it results in quite large values (like 31.60).
Again, there is the linear increase in variation each level.
We'll face similar issues to our hitpoint analysis here. That's mostly associated with sparseness of data at higher levels. But, as before, the blue datapoints at levels 10, 12 and 13 can be used to guide us and the pink data points may be handy as very rough indicators that we are in the neighbourhood of what we need.
| At-Will Damage Average by XP | |||||
|---|---|---|---|---|---|
| Level | Easy | Average | Tough | Solo | Total |
| 1 | 3.61 | 4.77 | 8.00 | 4.08 | |
| 2 | 6.25 | 6.50 | 8.70 | 7.10 | |
| 3 | 8.04 | 9.32 | 12.00 | 8.77 | |
| 4 | 9.00 | 10.96 | 12.09 | 11.43 | |
| 5 | 11.00 | 14.44 | 14.17 | 14.28 | |
| 6 | 18.48 | 13.00 | 18.17 | ||
| 7 | 23.00 | 27.96 | 26.61 | ||
| 8 | 27.70 | 27.70 | |||
| 9 | 36.40 | 36.40 | |||
| 10 | 36.61 | 36.61 | |||
| 11 | 58.33 | 58.33 | |||
| 12 | 40.07 | 56.25 | 44.93 | ||
| 13 | 54.00 | 56.57 | 55.80 | ||
| 14 | 66.50 | 66.50 | |||
| 15 | 86.00 | 86.00 | |||
| 16 | |||||
| 17 | |||||
| 18 | 94.00 | 94.00 | |||
| 19 | |||||
| 20 | 82.00 | 82.00 | |||
| Total | 4.50 | 9.28 | 15.29 | 40.90 | 15.47 |
Average Tables
What we find when we start digging through the averages is a lot of similarity with hitpoints. The values may be lower, but the patterns are very similar. Some of this is due to our categorisation methods, but many of these similarities persist between the "by HP" and "by XP" views. This not unexpected and another good sign that our theories on the relationships between the different PC and monster stats are on the right track.
Again, it's very obvious that there's a relationship between each type of monster at a given level. If you are having trouble seeing this try looking at the bottom total. That's an obviously skewed summary, one that inflates Solo creatures somewhat. But it does give you a feel of the relationships.
The same apparent miscategorisation we noticed in the hitpoints analysis is also present. Closer examination shows us there really don't seem to be any Tough creatures at levels 12 and 13. For those we'll again just use the Total in the Solo column's place.
| DPR Average by XP | |||||
|---|---|---|---|---|---|
| Level | Easy | Average | Tough | Solo | Total |
| 1 | 3.75 | 4.98 | 8.00 | 4.25 | |
| 2 | 6.52 | 6.81 | 8.98 | 7.39 | |
| 3 | 8.27 | 9.97 | 12.00 | 9.21 | |
| 4 | 9.00 | 11.67 | 12.67 | 12.07 | |
| 5 | 11.00 | 15.17 | 15.18 | 15.11 | |
| 6 | 20.05 | 13.00 | 19.64 | ||
| 7 | 27.71 | 30.17 | 29.52 | ||
| 8 | 35.93 | 35.93 | |||
| 9 | 38.57 | 38.57 | |||
| 10 | 39.57 | 39.57 | |||
| 11 | 59.42 | 59.42 | |||
| 12 | 41.32 | 59.69 | 46.83 | ||
| 13 | 57.00 | 56.92 | 56.94 | ||
| 14 | 69.88 | 69.88 | |||
| 15 | 86.00 | 86.00 | |||
| 16 | |||||
| 17 | |||||
| 18 | 94.00 | 94.00 | |||
| 19 | |||||
| 20 | 106.00 | 106.00 | |||
| Total | 4.67 | 9.75 | 16.40 | 43.78 | 16.54 |
As with the hitpoints analysis all four tables look like promising ways of reconstructing monster damage data.
And, again, I'll opt to build all four and then compare them. I won't bore my dear readers with an explanation of why.
Time to check out the graphs.
Graphs
Right now many of you are probably sitting there saying to yourselves "Er. If the Damage data is so similar to the Hitpoints data, why do the graphs look so different?!?" And that's a fair question.
There are several reasons for the apparent differences. The scale is, of course, different and this accounts for some of the variation. The way the actual averages fall out within the scale is a little different to, which is to be expected but nonetheless contributes to the apparent differences.
The most obvious difference is in the trendlines, but we do expect some of our trendlines to wander because of the scant data above level 7. But if we remove the trendlines we see that the general placement of all the datapoints is very similar. And, as mentioned in the Average Tables section, the distances between the different types at each level are relatively close.
The patterns are close. And although the trendlines are really only a useful guide or indicator, they look more useful for damage analysis than for hitpoint analysis.
These graphs also give us another important clue. If you go back and look at the graphs for hitpoints and think about how the datapoints are clustered around the trendlines you'll see that it's about the same for the two "by HP" graphs and the same again for the two "by XP" graphs - the clustering is only very slightly "tighter" for the "HP Calc" graph in each pair... But these graphs are a different story. In these graphs the data for the DPR version of each pair is a good deal more tightly clustered around it's trendline, which is predictably most noticable with the Solo datapoints and trendline.
This is a pretty solid indicator that the math behind monster damage is built around projected DPR, rather than "At-Will" DPR. I'll still create all four tables, but that's my tip for this one.
Choosing Between Damage Tables
After building the four tables I like to go back and examine each. How close is the table to the original? Is there a section of the table that doesn't match the table we are trying to duplicate? Or are there small variations scattered throughout? What about the base numbers? Are they unusual numbers like 4.23791? Or are they close to more "natural" numbers for humans to use? And the derived numbers?
What's quickly obvious is that the "by HP" tables aren't a great match. We can set up forumals so that most of the table matches fairly well - but ther's always a section of the resultant table that "drifts" away from the source table. That's a pretty good indication that we are on the wrong track.
The "by XP" tables, on the other hand, tend to line up easily and naturally. We do have small variations scattered throughout the table, but that's something we'd expect - it's highly unlikely our sample data is will just spell the table out for us, we expect variation. The "DPR by XP" table, in particular, is a very close match that uses natural values and increments. It's the best match, so my tip panned out on this occasion. Sometimes we get lucky, other times we have to keep slogging through until we find what we need.
Damage Table
| Level | Easy | Average | Hard | Solo |
|---|---|---|---|---|
| 1 | 4.00 | 5.00 | 6.00 | 10.00 |
| 2 | 6.00 | 7.50 | 9.00 | 15.00 |
| 3 | 8.00 | 10.00 | 12.00 | 20.00 |
| 4 | 10.00 | 12.50 | 15.00 | 25.00 |
| 5 | 12.00 | 15.00 | 18.00 | 30.00 |
| 6 | 14.00 | 17.50 | 21.00 | 35.00 |
| 7 | 16.00 | 20.00 | 24.00 | 40.00 |
| 8 | 18.00 | 22.50 | 27.00 | 45.00 |
| 9 | 20.00 | 25.00 | 30.00 | 50.00 |
| 10 | 22.00 | 27.50 | 33.00 | 55.00 |
| 11 | 24.00 | 30.00 | 36.00 | 60.00 |
| 12 | 26.00 | 32.50 | 39.00 | 65.00 |
| 13 | 28.00 | 35.00 | 42.00 | 70.00 |
| 14 | 30.00 | 37.50 | 45.00 | 75.00 |
| 15 | 32.00 | 40.00 | 48.00 | 80.00 |
| 16 | 34.00 | 42.50 | 51.00 | 85.00 |
| 17 | 36.00 | 45.00 | 54.00 | 90.00 |
| 18 | 38.00 | 47.50 | 57.00 | 95.00 |
| 19 | 40.00 | 50.00 | 60.00 | 100.00 |
| 20 | 42.00 | 52.50 | 63.00 | 105.00 |
The damage table is even simpler than the hitpoints table! Average damage for a level is obtained by taking level, adding one and multiplying the result by 2.5. An easy creature only inflicts 80% of an Average creature's damage, while a Hard creature inflicts %120 of that damage and a Solo inflicts 200% of that damage.
This can be expressed as...
- Average_Dmg = (level + 1) x 2.5
- Easy_Dmg = Average_HP x 0.8
- Tough_Dmg = Average_HP x 1.2
- Solo_Dmg = Average_HP x 2.0
This particular table aligns quite nicely with our source table, with most of the green/blue values being very close. A few do, of course, deviate from our source data. But this is easily accounted for by variations in the source data and certainly within expectations.
So how much can we adjustment a creature's damage without causing problems? We do see variation of 25%-30% of DPR within any given level and type, but that will normally be accompanied by adjustment of other attributes - as it should be. I'll stick with what I said about hitpoints - I wouldn't recommend adjusting by more than +/-2% of damage without compensating elsewhere.
My biggest concern with damage is, again, Solo creatures. As with hitpoints, there's a disparity between their XP award and their damage output. I do understand that having a creature that does an average of almost an entire PC's hitpoints in damage is a problem. But I'm not sure that the answer to that is docking their damage output. We'll look at how we can improve Solo creatures in a later installment of this series.
Validation
Our primary validation of monster damage is against PC hitpoints. If we divide the Damage for an Average creature at a given level by the hitpoints of an average PC of the same level we get a fraction. That fraction is the percent of PC hitpoints that Average monster does. If there is a direct relatonship between the two then this number should stay fairly constant through the levels. And what we see is that this number averages 0.23 (23%) with a variance of 0.000 and a stdev of 0.008. Which is pretty compelling!
Our secondary validation was against PC damage. Monster damage starts out at about 41% of PC damage and slowly progress to 131% of PC damage. This sounds weird, but if we look more closely we see that most of this disparity is in the first couple of levels. At level three it's 75% of PC damage and progresses evenly.
So, while Solo creatures ahve some shortcomings that need to be addressed this table should closely reflect the current damage progression and be suitable for most DMs to use.
Check back tomorrow for the next installment Part 8: Putting It All Together...
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