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D&D 5E On Die Averages and Hit Points in 5e

OptionalRule

Explorer
I posted up a bit of a nerdy and complex breakdown of why die average numbers are what they are, since this seems to be a point of confusion that comes up now and again.

The confusion seems to get more mixed when people apply it to rules or in generating player hit points vs monster hit points.

On Die Averages and Hit Points in 5e
 

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el-remmen

Moderator Emeritus
This just reminds that while I often simply choose the monsters' hit points from within the range depending on how tough I want them to be to kill, when I choose "average" I always give them max hps for the first die (assuming they have more than one) and average for the rest because what is good for the PC is good for the monster. :unsure: ;)
 


NotAYakk

Legend
For the "roll vs fixed value", if you aren't worried about cheating, simply let PCs reroll 1s on their HD.

1d8 reroll1s has an average of 5, identical to the "take the average" choice. And it lets people who want to have fun rolling HD not be worse off, on average.

At level 10, 9d8 drop 1 has an average of 45 and a Variance of 48/12*9=36, thus a SD of 6 and a 95% CI of 33 to 58. So a constitution 14 cleric using "pick take average" has 73 HP, while the cleric who rolls has 61 to 86 95% of the time (and has an identical average).
 

Charlaquin

Goblin Queen
For the "roll vs fixed value", if you aren't worried about cheating, simply let PCs reroll 1s on their HD.

1d8 reroll1s has an average of 5, identical to the "take the average" choice. And it lets people who want to have fun rolling HD not be worse off, on average.

At level 10, 9d8 drop 1 has an average of 45 and a Variance of 48/12*9=36, thus a SD of 6 and a 95% CI of 33 to 58. So a constitution 14 cleric using "pick take average" has 73 HP, while the cleric who rolls has 61 to 86 95% of the time (and has an identical average).
Oh my god, I can’t believe I never thought of this before! It’s always bugged the crap out of me that the fixed value rounds up from the average for rolling, making it the clear better choice. But this totally fixes that problem! And it removes the feel-bad moment of deciding to roll HP and getting a 1. I’m totally adopting this.
 
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Cadence

Legend
Supporter
For the "roll vs fixed value", if you aren't worried about cheating, simply let PCs reroll 1s on their HD.

1d8 reroll1s has an average of 5, identical to the "take the average" choice. And it lets people who want to have fun rolling HD not be worse off, on average.

At level 10, 9d8 drop 1 has an average of 45 and a Variance of 48/12*9=36, thus a SD of 6 and a 95% CI of 33 to 58. So a constitution 14 cleric using "pick take average" has 73 HP, while the cleric who rolls has 61 to 86 95% of the time (and has an identical average).

I like that reroll 1 a lot. Thanks for the idea!

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Would it be better to call it a prediction interval and not confidence interval (you aren't making an interval for estimating a parameter from a sample, you're giving the 95% range where an individual observation from doing the experiment is likely to be).

The upper interval is off by 1 (9*5+12=57).
 

NotAYakk

Legend
Oh my god, I can’t believe I never thought of this before! It’s always bugged the crap out of me that the fixed value rounds up from the average for rolling, making it the clear better choice. Bur this totally fixes that problem! And it removes the feel-bad moment of deciding to roll HP and getting a 1. I’m totally adopting this.
I played with a DM who did this, and I was like "wait a second, that fixes the average". I blame 4e's Brutal1 weapon mechanics for it being obvious.

...

An interesting variation is to remove con-to-HP, and instead have it give rerolls.
 

Charlaquin

Goblin Queen
I posted up a bit of a nerdy and complex breakdown of why die average numbers are what they are, since this seems to be a point of confusion that comes up now and again.

The confusion seems to get more mixed when people apply it to rules or in generating player hit points vs monster hit points.

On Die Averages and Hit Points in 5e
Umm… Ironically, you’ve made an error in your attempt to clear up confusion. Rolling player HP is the only case where 5e rounds up on the average for each die. In both mister HP and the average damage value given for mister attacks, they take the true average, multiply that by the number of dice, and round the final result down to the nearest whole number. A bugbear, for example, has 5d8 hit dice and +1 con, for an average of 27 hp (5.5 * 5 = 27.5, which rounds down to 27). If they rounded up the average on each die, they would have 35 hp (7 * 5 = 35).
 


OptionalRule

Explorer
Umm… Ironically, you’ve made an error in your attempt to clear up confusion. Rolling player HP is the only case where 5e rounds up on the average for each die. In both mister HP and the average damage value given for mister attacks, they take the true average, multiply that by the number of dice, and round the final result down to the nearest whole number. A bugbear, for example, has 5d8 hit dice and +1 con, for an average of 27 hp (5.5 * 5 = 27.5, which rounds down to 27). If they rounded up the average on each die, they would have 35 hp (7 * 5 = 35).
Hmm, making a mistake isn't new to me but I don't see it from what you described here. I mention in the post that there aren't any 0.5 increments on a die and they roll up for players. The to examples I give have those values (bugbear at 27 hp).
 
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aco175

Legend
I allow players to roll HP each level and they can choose average rounded down after. So, if you roll a d8 the lowest you can get is 4. I also just give the monsters roughly 3/4 HP instead of half. This mostly started with 4e in that a typical monster started to get better versions instead of just saying that there is a leader with max HP like in 1e/2e. So instead of having 10 orcs and one has max HP so he is the leader, we now have basic orcs and a orc shooter and a orc sergeant, and an orc...
 

Charlaquin

Goblin Queen
Hmm, making a mistake isn't new to me but I don't see it from what you described here. I mention in the post that there aren't any 0.5 increments on a die and they roll up for players. The to examples I give have those values (bugbear at 27 hp).
I see. I misread you, I thought you were saying they round up in general, not that they only round up for the players, and your explanation of how monster HP are calculated didn’t specifically say they are t rounded up there. I also misunderstood the way you wrote the formula - it looked like you were saying the d8 average was 5.0 rather than that you were multiplying the average by 5.0

My mistake, sorry.
 


Umbran

Mod Squad
Staff member
1d8 reroll1s has an average of 5, identical to the "take the average" choice. And it lets people who want to have fun rolling HD not be worse off, on average.

Except, for any particular character, you're probably going to roll maybe a dozen of them, and a maximum of 20. The "on average" holds when you consider statistically relevant numbers of characters. Any one individual character, with only a handful of hit dice, has a pretty high likelihood of being far from average.
 

NotAYakk

Legend
Except, for any particular character, you're probably going to roll maybe a dozen of them, and a maximum of 20. The "on average" holds when you consider statistically relevant numbers of characters. Any one individual character, with only a handful of hit dice, has a pretty high likelihood of being far from average.
Yes, I covered that. Did you see the bit where I talked about variance? That is a measure of how likely extreme rolls are.

The 95% confidence interval (ie, 95% of characters) will be within about 80% to 120% of the "take the average" PC's HP at level 10.

By the time you roll even 3 dice and add them up, the central limit theorem's implied cheat works pretty damn well.

At level 5, with 14 con, a d8 HD PC who takes the average has 43 HP. The variance of the d8 drop 1s is 20, so +/-8.76 HP. So about 80% to 120% of HP even by level 5, 19 times out of 20.

Now, the extremes exist. As the option is level by level, a PC who rolls minimum 2 times in a row could just start taking average if they wanted to.
 


Charlaquin

Goblin Queen
Except, for any particular character, you're probably going to roll maybe a dozen of them, and a maximum of 20. The "on average" holds when you consider statistically relevant numbers of characters. Any one individual character, with only a handful of hit dice, has a pretty high likelihood of being far from average.
True, but you’re still more likely to get a lower result than the fixed value when you roll than you are to get a higher result.
 



Charlaquin

Goblin Queen
I mention this later in the article when it comes to a house rule to reroll HP. Basically, replacing a 1 or 2 makes almost no difference with Great Weapon Fighter. There have been a number of articles about this, I wrote a thing about it awhile back too.
It makes a pretty small difference on each attack, sure. The reason you’d take it is because you’re planning to use two-handed melee weapons (so you won’t benefit from archery, dueling, two-weapon fighting, or protection) and you want an offensive fighting style (so defense won’t be satisfying.) The benefit, while small, also compounds the more attacks you make, so it works best with Polearm Master and Sentinel.

It’s definitely the weakest fighting style (well… maybe second weakest after protection), but since it’s the only one that boosts the damage of two-handed weapon users, which is an extremely strong build, it still has its niche.
 
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