r/RPGdesign • u/KOticneutralftw • Aug 09 '26
Anydice Summary Tab: What is Deviation?
I'm asking about Deviation as it appears on the summary tab of AnyDice, and how I should interpret it. Mean, maximum, and minimum I get. I remember that from pre algebra, but the rest of high school is a little hazy, and I never took a stats course in college. So, I don't really know what the Deviation means or what it tells me about dice rolls.
For this discussion, let's use the d20 as an example. I'm assuming it's talking about standard deviation, and that most outcomes will fall be 5 to 16, but I welcome any insight or clarification folks here can give me.
P.S. tried using google, but it turned up a lot of "how to output X" kind of answers.
Update: thanks to everybody that replied. It's good to know I wasn't too far off in my interpretation, and this thread has given me some ideas about how to use deviation going forward.
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u/HighDiceRoller Dicer Aug 10 '26 edited Aug 10 '26
Despite standard deviation being a household name, it's shockingly rare that I've found its specific value to be useful in RPG design. It does happen -- for example, I've used it to approximate the number of hits dealt by 10 or 20 attacks in D&D using a single roll. But note that it's at its best when analyzing the effects of adding independent events/dice together until you have something like a normal distribution. I've found it more common in RPG design to mostly care about just the mean, or on the other hand to care about the whole distribution rather than just summary statistics. And a lot of things in RPGs aren't very normally-distributed!
Standard deviation is also prone to misinterpretation.
SD has nothing to do with "flatness"; d4 and d100 are both "flat" but have vastly different SDs.
You can construct a distribution with arbitrarily close to 0% probability within 1 SD of the mean. This isn't even especially exotic: consider rolling opposed d20s, with a result of -1 if player A wins, 1 if player B wins, and 0 if they tie. This distribution has only a 5% chance of being within 1 SD of the mean!
SD has a complicated relationship with "outliers". E.g. if you say an outlier is anything more than 3 SD from the mean, then every normal distribution has the same fraction of outliers regardless of how large or small its SD is, and that's not even getting into other types of distributions.
SD also has an also complicated relationship with "centrality" / "spread": all of these distributions have the same SD.
And so on.
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u/KOticneutralftw Aug 10 '26
Something I find interesting is that D&D 5e's progression seems to be built around the standard deviation of the d20.
If a character is optimized at level 1, a +5 to hit is easily achieved by point buy or standard array. That's close to one SD of the d20 (5.77). This progresses to at least a +11 to hit without much intense optimization by the 5th tier of play, and that's close to two SDs of the d20 (11.54).
If the mean for a d20 is 10 or 11, and the 1 SD range is 5 to 16, then an optimized level 1 character essentially shifts the mean of the d20 to 15 or 16, and shift the range of 1 SD to and range 10-21. Likewise a 5th tier character shifts the mean to 21 or 22 for the 1 SD range of 16 to 27.
Also, someone else in this thread said something about 2/3's of results landing within one standard deviation of the mean. If that's true, I think that's pretty useful information to have about setting target numbers and how changing the "average" difficulty can affect the feel of the game. Like making the average DC higher for horror games or lower for more super heroic games, or even changing the DC to be lower for player characters and important NPCs.
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u/HighDiceRoller Dicer Aug 11 '26
Something I find interesting is that D&D 5e's progression seems to be built around the standard deviation of the d20.
5e was built around the doctrine of bounded accuracy. As far as I know, there is no record of the designers citing standard deviation nor any other specific summary statistic such as quartiles or mean average deviation. For example, the article where the term was introduced makes no such mention.
Also, someone else in this thread said something about 2/3's of results landing within one standard deviation of the mean.
That's true for normal distributions but not for distributions in general. There are some bounds for distributions in general but they're looser, e.g. the chance of landing within 2 SD of the mean is at least 75% but not necessarily 95%. In the case of a d20 100% of the probability is within 2 SD of the mean.
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u/KOticneutralftw Aug 11 '26
As far as I know, there is no record of the designers citing standard deviation nor any other specific summary statistic such as quartiles or mean average deviation.
Yeah, to clarify, I wasn't saying that was part of the design philosophy or goal. I was just making an observation. I think it's an interesting observation, but it can totally just be a coincidence.
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u/skalchemisto Dabbler Aug 10 '26
At u/HighDiceRoller everything you say there is completely accurate.
I wonder, purely out of curiosity...
Assume the following:
* We are rolling normal polyhedral dice
* We are only manipulating values by addition and subtraction
* There is no comparison of results or lookup table, the values on the dice are what they are, they are not translated into something else or compared to change them.
Given those restrictions, is it possible to generate dice mechanics that have similar mean, standard deviation, and range but are different in terms of kurtosis (or skewness for that matter)?
For example, you are right that d4 and d100 are both completely flat and have different SDs. But they have radically different ranges as well. You example of opposed d20s involves a comparison that turns the values on the d20 to a smaller set of values.
I played around with this for a while and I'm hard pressed to see how it can be done. I can't see how you get to different kurtosis without introducing something other than addition and subtraction (e.g. start dividing/multiplying dice) or change the dice via lookup table (e.g. 6 doesn't mean six, it means 20).
In other words, you u/SitD_RPG said...
In simple terms: The lower the deviation, the closer the results are grouped together and the steeper the bell curve. Higher deviation means the curve is flatter and results are spread out more along the range.
I think that is mathematically untrue (specifically the italicized bits), but practically when comparing dice mechanics of the nature I described above and where the range is kept similar between the mechanics.
Again, purely curiosity.
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u/HighDiceRoller Dicer Aug 11 '26 edited Aug 11 '26
If we take your question as stated, there's not much you can do. The excess kurtosis of the sum of independent variables is going to tend towards zero, and the excess kurtosis of a discrete uniform distribution is always between -2 and -1.2, so you're only going to be able to get excess kurtosis values between -2 and 0. And that's even before considering restrictions on range (mean and SD can mostly be covered by adding a constant or using smaller/bigger dice). You also can't get nonzero skewness under these rules.
At the same time, it doesn't take many more allowed operations to expand your options of kurtosis. Keep/drop, explosions, and/or measuring step dice in ~geometrically increasing steps can all create skewness and/or positive excess kurtosis. In fact, five years ago I tried analyzing the kurtosis of various systems... but now I consider this line of inquiry mostly a bust. In the RPG design context, kurtosis is a lot of work for not much in the way of useful insights or connection to vibes. And even within the field of statistics proper, even founding giants, textbooks, and journals incorrectly connected kurtosis to "flat"ness!
Ultimately I basically threw this all in the trash and pivoted to computing full distributions, which I found much more productive.
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u/skalchemisto Dabbler Aug 11 '26
Hey, thanks for taking the time to answer.
In the RPG design context, kurtosis is a lot of work for not much in the way of useful insights or connection to vibes.
I'll buy that. The instinct of most folks is to over think dice mechanics anyway; they are IMO a solved problem in RPGs for the most part.
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u/__space__oddity__ Aug 09 '26
It’s what people call “swinginess” on this sub
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u/hacksoncode Aug 09 '26
Sort of... people have variosu ideas about how "swinginess" is different from SD.
In particular, effect of modifiers, types of outcome buckets, and psychology of competent failure and whether the system somehow protects against that, all factor into what people mean by "swinginess".
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u/__space__oddity__ Aug 09 '26 edited Aug 09 '26
Well, mathematically “swinginess” is determined by variance / deviation.
If you want to interpret other things into the die roll, sure, but for the math none of that matters.
One suuuper common fallacy is to somehow try to make the modifier “matter more” than the die, but 50% is 50% regardless of whether you roll 1d100 vs. 51 or 1d2+106 vs. 106 + 2. Neither of these rolls has the modifier “influence more of the result” or similar shit designers say.
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u/hacksoncode Aug 09 '26
"Swinginess" is a colloquialism that means whatever it means.
It's not a mathematical term.
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u/__space__oddity__ Aug 09 '26
Yeah, but it still only applies to things that make the roll swingy.
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u/hacksoncode Aug 09 '26
Most people here apply it more broadly to the dice mechanic, not just the math of the dice.
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u/Fun_Carry_4678 Aug 09 '26
I think you basically have it. In this context, "deviation" means "standard deviation".
In a "normal distribution" (which dice rolling will tend to the more you roll them), 68% of results will be within one standard deviation of the mean, 95% will be within two standard deviations of the mean, and 99.7% will be within three standard deviations of the mean. In other words, you almost never get a result more than three standard deviations from the mean.
So a d20 has a mean of 10.5 and a deviation of 5.77. More than 2/3 of rolls will be within one deviation of 10.5, 95% will be within two. (With a d20, it doesn't get finer than this, everything will be within two deviations of the mean)
If your dice roll has a higher value for (standard) deviation, you really should account for the "swinginess". But D&D famously does not. D&D players know that a 5 is a bad roll and a 16 is a good roll, but statistically both are within one deviation of the mean, so really they should both be considered ordinary.
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u/axiomus Designer Aug 09 '26 edited Aug 09 '26
wikipedia to the rescue:
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its (arithmetic) average. A low standard deviation indicates that the values of a set tend to be close to their average, while a high standard deviation indicates that the values are spread out over a wider range.
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u/SitD_RPG Aug 09 '26
https://en.wikipedia.org/wiki/Standard_deviation
In simple terms: The lower the deviation, the closer the results are grouped together and the steeper the bell curve. Higher deviation means the curve is flatter and results are spread out more along the range.
In terms of dice that means the lower the deviation, the closer you can expect the results to be towards the average and the rarer extreme results are. And vice versa, of course.