Wow! Lots of new critters! Time for a stocktake and review...
Interlude: A Bunch Of Monsters!
Well in the last week or so a whole bunch of monsters have made their way to us! There was an update to Horde Of The Dragon Queen, the Player's Handbook was released, a new version of Basic D&D was released with over 150 monsters and there have been a number of Monster Manual creature spoilers released.
As a result my sample size jumped from 33 monsters to 209!
On top of that there have been several tweaks to PC data via the PC Stats thread on the Wizards of the Coast forums.
All this means one thing... It's time to stop and review what we have done so far. This will delay the next scheduled post a few days, but it's important to do.
I think it's best to cover this all in one big post, so forgive me if this one gets a bit long. To make it up to you here's a handy-dandy index...
Most of the mechanical items we talked about in the CR section of Part 3 have either been ratified or remain unchanged. The DM Basic Rules v0.1 released at Basic D&D does provide some insight and a couple of useful tables that replace some of what we discussed.
The Proficiency Bonus By Challenge Rating table on page 4 shows that we were correct about using CR to select a monster's proficiency. It also handily extends monster Proficieny below CR1 and above CR20 for us.
The Experience Points By Challenge Rating table on page 5 shows that we were very close with our XP values, at least in the CR0 through CR10 range. If we take our formula and extend out past CR10 to CR30 we find some inaccuracies creep in, though. As I previously said, our small set of sample data made it unwise to project out past CR10.
Rather than smply work off a formula for the curve Wizards of the Coast have, in effect, spliced three curves togethor. What I can see by CR is this...
- CR <1: These creatures look like they are hand-fashioned off of CR1 creatures. One could probably model the first and second curves togethor by going to a fairly complex polynomial.
- CR 1 - 20: Matches a polynomial 3 equation similar to this:
XP = 1.1553x^3 + 31.119x^2 + 166.47x - 43.52
- CR 21+: Another curve greatly increases XP gain for defeating monsters, which is appropriate since PCs will supposedly be fighting creatures far more powerful than themselves. This appears to be a polynomial equation similar to the following:
XP = 564.39x^2 - 15148x + 101214
This is pretty much of academic interest, but it does serve to illustrate the dangers of computing data values based on too small a sample size.
Reflection: I predicted that we would see creatures of CR30 or even beyond. The Tarrasque spoiler from the Monster Manual is CR30, confirming my prediction.
The Hit Dice By Size on page 3 of the DM Basic Rules v0.1 shows that our belief regarding the link between Size and Hit Dice was correct and that we estimated the Hit Dice size for Huge and Gargantuan creatures correctly.
Naturally there were no references to the linkage between Size and Speed.
Reflection: It's a bit hard to draw any conclusions about my prediction about monster sizes at higher levels, since we still have very few higher-CR monsters to look at. But at this stage I see no reason to doubt my prediction.
On reviewing Ability Score values and referring back to the Ability Score section of Part 4 I find that the greater sample data has upheld the findings of that post. In fact the trendlines we use have firmed up very considerably and now naturally align quite closely to the formulae and table we constructed in there! I can only see this as statistical ratification.
I'd be quite comfortable extending this out to CR15 based on the data we now have.
Reflection: While there is no definative proof of the prediction I made that peak attributes would reach 30 past CR20 I feel very comfortable clarifying that this will become the common peak Ability Score closer to CR30 than CR25.
Before considering the impact the new mass of monsters has on the Hit Points section of Part 4, let's stop a moment and consider the community input on my assessment of PC statistics. It's important to consider this first because, as we said in part 4, these aspects of monster math are built on PC math.
Over the last couple of weeks the Wizards of the Coast community has discussed this on my PC Stats thread over on the WotC forums. A number of changes have been suggested to the way I have calculated this data and most of it has been incorporated into my math. But the funny things is, at the end of the day it hasn't significantly changed the curves I described in part 4. Yes, some of the variations in the progression changed. But in the end the curve formula remained unchanged. Which makes things a bit easier in this review.
So PC Damage remains as previously defined...
PC Damage=4.55 x Level ^ 0.72
|Note: Typical boundaries shown.|
Now the new monsters... Gaining such a significant injection new data showed up something interesting things about the relationship between PC Damage and monster Hit Points! I found the new data pretty interesting and spent a fair bit of time re-analyzing HP as a result.
It turns out the PC Damage:Monster HP ratio isn't a static 5, as previously posted. It's very clear that the multiplier actually starts somewhere down near 3.5 or 4 and trends up very quickly to an average of 7 then scales out to around 12 at CR30.
I did some pretty extensive analysis of this, attacking the problem from several different angles. By the end I had several potential formulae that could potentially be used to build monster HP. But profiling these I found most of my methods of correlation pointed towards the use of a Logarithmic expression to determine the multipiler. I believe it's close to the following...
Multiplier=0.31 x LN(CR) + 3.7
Which yields a monster HP formula something like this...
HP=(4.55 x CR ^ 0.72) * (0.31 x LN(CR) + 3.7)
It's a little convoluted, but testing showed some positive flags...
- A close relationship to observable HP data on both scatter and summary graphs.
- A match HP summary data.
- General tightening up of data clusters, aligning with sampled data.
- Very close matches when use in my provisional CR assessment formula.
- Data not matching my provisional CR assessment formula is fairly evenly split above and below targets.
It's unlikely any single formula can adequately match all sampled data - which is a significant factor behind my belief that CR assessment is a measure of composite data. However this formula achieves a high proportion of match, with its trendline being mainly centred within sample data.
And the result of this relates very well to a simple linear equation which can be used in its place seamlessly...
HP=20 x CR + 8
Now that I had reset the monster HP progression values I had to re-baseline the monster HD table. This is a job that takes a couple of hours or so by hand and I had already done it three times. So I invested a couple of hours writing up a VBA macro to compute it all. Now with the press of a button it's all recalculated in less than a second.
Being an ex-programmer pays off from time to time!
Reflection: I indicated that I believed monster Hit Points would be more variable than monster Damage, once we were able to analyse the Monster Manual. I've seen nothing to suggest that I should recant this.
The significant increase in sample size has served to clarify AC significantly. In fact sample size is now big enough for us to realistically estimate AC independently of PC Attack Bonus, if we choose to do so. This opens up several avenues of correlation that we did not have in the Armor Class section of Part 5.
Examining trendlines on scatter and summary graphs reveals a very linear progression of monster AC, with values still fairly tightly grouped at +/-5 of trendlines. Note that when looking past CR20 we do need to be careful of skew as both the CR24 Ancient Red Dragon and the CR30 Tarrasque likely have an AC near the top of their range.
Reassessing our data from a PC Attack Bonus perspective also shows that the relationship between it and AC has clarified. Between CR1 and CR20 we can see that same-level PCs consistently need to roll an 8 or better to hit the monster. Considering that PC Attack Bonus is a curve how do we resolve this to a linear monster AC progression?
Well, it's not as difficult as some readers might expect.
Simply plotting the two on the same graph provides some understanding of this. The red line on the graph shows PC Attack Bonus plus 8 (in other words, it's a pure derivation). The blue line represents a linear equation closely matching the summary data I have crunched in my spreadsheet. You'll se that the results of the two will be quite close, particularly after rounding.
Now some readers might feel concerned about the small disparities between the two. But as we previously noted Bounded Accuracy means that AC varies more within a given CR than it does across all CRs. And it's very normal to see AC variance up to +/-5 of normal within a given CR's monsters. We even see variation beyond this occasionally. So it really is something we simply do not need to be concerned with.
The linear equation itself is quite simple, of course...
AC=0.32 x CR + 12.5
Readers will note that several resulting AC values are different on the new table, mainly at the lowest levels. But the differences are pretty minimal and this again highlights the reasons behind our decision to restrict ourselves to CR10 and below until the Monster Manual is released.
Reflection: It looks like average monster AC may peak at 22 at CR29 or CR30, rather than at CR25.
The Misc Stats area is another place where the increase in sample data has served only to suggest that our findings in the Miscellaneous Statistics section of Part 5 are correct.
While the scatter graph contained many more data points its trendline remained completely unchanged. The summary graph also gained more data points (because we have sample monsters at levels where we previously did not) and this changed the high-value end of the trendline... Only a small amount.
Reflection: I've not seen anything to suggest that I should change the predictions I made in this area.
Example: Human Pyromancer
Let's see how our findings impact our sample monster. We'll also make a couple of other minor tweaks/corrections to the Pyromancer...
|Medium humanoid (human), any alignment|
|Armor Class 12|
|Hit Points 88 (16d8+16)|
|Saving Throws Dex +5, Int +7|
|Skills Arcana +7, Perception +4|
|Damage Resistances Fire|
|Lanuguages Common, Ignan|
|Challenge 5 (1,800 XP)|
While our chosen Challenge Rating of 5 remains unchanged we do need to update the Pyromancer's XP reward from 1,700 to 1,800.
Size and Ability Score values remain unchanged.
Based on the updated data the creature's Hit Points are probably a bit too low, even for a glass cannon. We could either increase number of HD or we could increase the Pyromancer's Constitution Score. Since changes in HD generally result in smaller incremental changes I tweaked this up by two to 16, resulting in an average HP of 88.
Although we have made some updates to Armor Class in this post it's important we remember that AC tends to vary more within a given CR than it does when across all CRs, thanks to Bounded Accuracy. As such I believe that AC does not need to be updated for the Pyromancer.
While we don't need to make any changes to Miscellaneous Stats base don this post, I did notice an error in the Pyromancer as listed last installment. I had intended to give it proficiency in Perception, which is much more useful for monsters than Arcana. And I accidentally left off the actual Arcana bonus. I've corrected these in this version and will go back and edit my previous post shortly.