"2D6 distribution is NOT a bell curve ..." Topic

Regularly, people claim that a 2D6 roll produces a 'nice bell curve'.

These wildly exaggerated claims have to stop!

A 2D6 distribution produces a triangular ditribution, which is a far way off from a 'nice bell curve''. If you just plot the probabilities of each number showing up, you get a nice pointed triangle shape, not the famous Gaussian bell curve.

When increasing the number of D6 (3D6, 4D6, …), one approaches the Gaussian distribution (the real bell curve) closer and closer, but with a 3D6, one does still have 4 fairly obvious linear segments in the distribution joined together.
Perhaps a 10D6 distributions starts to approximate a bell curve reasonably well, but no sooner than that! In theory, one needs an infinite number of dice to get to the real bell curve.

Ok, I'm back off my soapbox now.

:-)

Nevertheless, 2D6 is closer to a Gaussian distribution than 1D6.
I'm prepared to use the term as a shorthand – not being a statistician. But I know there are plenty of things that annoy me about the way 'most people' use language. I am metaphorically incandescent with rage whn people misuse the term 'literally', for example.

It's not "shorthand", except in the way that "Tiger" is shorthand for "Panzer IV".

As Phil has pointed out, the Gaussian distribution is unbounded, and therefore extends to + and – infinity. It's also a continuous distribution, whereas anything with dice is a discrete distribution.

I think I shall start referring to the 2d6 distribution as the "dunce hat" distribution.

FWIW I understand that a discrete approximation to the Gaussian distribution that is adequate for most purposes can be achieved with 7 or more d6s.

All the best,

John.

Interesting, was not aware.

Regards,

J. P. Kelly

Agreed that it is not a Normal distribution but it is a 'bell curve' in the more general meaning of the term in that it is symetrical and peaks in the centre reducing continually to minimums at the extremes.

I wouldn't get too hung up on using a layman's term such as Bell Curve to have such a precise correspondence to terms such as Normal or Gaussian – recognising the rough shape is hard enough for some !!

No discrete event probability distribution can be Normal anyway though, as John says, multiple d6's can give a rough approximation (and it is pretty rough, no matter how many dice are used).

However, what 2D6 does is produce a series of results where the upper and lower quartile results are extreme and the two central quartile results are more probable.

So, for example, rolling 2, 3 or 4 should happen just 16.7% percent of the time, as should rolling 10, 11 or 12. Whereas a roll or 5, 6, 7, 8 or 9 should happen 66.7 % of the time. In other words results around the median are likely to occur two thirds of the time, whereas results at the extremes only happen a third of the time.

Personally I find this a better reflection of reality – where the norm is the norm and extreme results are extreme, rather than the D10 where one is just as likely to roll a 1 as a 5 as a 10, because probability of any result is equal.

The use of multiple D6 does allow a player to estimate what he feel the possible range of results could be. For example, if I have a tank which can use 2D6 for movement, with each pip being 1", I can estimate that it should move 70" over a ten turn period. In a turn I can predict that it should move around 5 to 9", more likely 6 to 8". And two thirds of the time I'd be right.

This is a feature which is shared by a bell curve, hence people adopting the term.

toofatlardies wrote:

However, what 2D6 does is produce a series of results where the upper and lower quartile results are extreme and the two central quartile results are more probable.

Interesting dice you must have. In my universe, each of the quartiles holds exactly 25% of the results, for any distribution whatsoever.

I wish I could remember which of our education ministers it was who said it was unacceptable that 50% of students continued to score less than the median exam mark.

All the best,

John.

All the wold loves a smartarse.

You know precisely what I mean.

You know precisely what I mean.

Yes, he probably does, but there are probably quite a few people who read TMP who don't …

Looks like a bell curve to me.

Is it a reasonable distribution (or any kind of curve) if I still roll 1s no matter what? Curse of the dice guides blows it all out of the water…

:)

I prefer Lake Bell's curves :)

2d6 may not be a NICE bell curve

but is certainly more closely related to a bell curve
than it is to a linear single die 'curve'

also FWIW I prefer Kristen Bell curves

But a true gaussian distribution isn't a true bell curve as, by definition, it can never be symetrical around the peak point. Below it starts\stops at zero, but trails above to infinity. Meanwhile 2D6 clearly is a Bell as it's possesses the appropriate symetry and gradient requirements.

Bell Curves =/= Gaussian Distribution

I'm an ex-Brit in the US now, it's "lovely" to see the word "smartarse" every now and again….but that's another topic…. ;-)..as for dice, no matter what the curve, I always seem to be at the bottom of it ;-)

Seriously, who cares. Get a life.

I would have expected a user named 6sided to have been on accuracy's side!!

Serious, calling it "bell curve" is good enough at the table. Most people understand the implications of that. Explaining how it's really not Gaussian and debating the definitions of quartiles should really be left for when you're finishing your beers after the game.

Perhaps we can just call it a "Dinner Bell Curve" and you can get a good nights sleep.

Canister is not grapeshot, though many military men of the time referred to it as grapeshot…

Presumably they weren't gunners.

Yeah, Lake Bell curves…

Bellish curve? :)

2d6 is a good-enough approximation of normal for gaming purposes. Most things that are assumed to be "normal" for the purposes of statistical inference really aren't, but are close enough for government work. I should know, I'm a professional statistician working for government.

If you feel that strongly about it, keep ranting.
I keep ranting about using an apostrophe to form a plural, and nothing has changed. The ignorant continue to shockingly disregard my attempts to edumicate them.

It's not "shorthand", except in the way that "Tiger" is shorthand for "Panzer IV".

I would say something, but I suspect that is exactly what you meant to say.

As the OP, I would like to say I don't feel very strongly about this issue. I thought I made that clear by the tone of my post. I know this a geeky issue even amongst geeks, but still …

Nevertheless, I have encountered gamers who assume all sort of things when something is called a 'bell curve'. Granted, these are the mathematically-challenged, so maybe one should not care.

But calling any non-constant distribution with a symmetrical peak in the middle a bell curve is simply incorrect. One can as well say an average die produces a bell curve.
Moreover, any die with any selection of numbers on them, when thrown in large enough quantities, will produce a bell curve. So, is it then fair to say that any two such dice are a good enough approximation of a bell curve?

Phil Dutre did you ever see any of the old Z&M publishing games that converted 2d6 to percentiles?

No, never saw them …

I wonder if there are any ladies out there called "Belle Curve"?

OK Phil, you seem to want to be both rigorous and informal both at the same time – not a viable position I'm afraid.

A bell curve is an informal name for a normal distibution curve and is used for other distributions that have similar (but not identical) properties.

You then insist that the 2d6 distribution doesn't fit the description of a bell curve (which you have modified from my original – mine would not fit an average die).

The description of a bell curve is clearly wide enough to cover a range of distributions and that includes multiple d6's (thanks to the central limit theorem).

All that is at issue here is how few D6 do you need to still use the term 'bell curve' as a reasonable description. I'd say 2 was reasonable for laymen, if I was talking to others who were not laymen the number would be significantly greater.

Phil Dutre if you want to see them send me an email this screen name at yahoo. I can send the stats.

toofatlardies wrote:

All the wold loves a smartarse.

I am pleased to have won the affection of an area of forested woodland, but it's obviously pretty easy to be labelled a smartarse if one can earn the distinction just by recalling a little O-level maths.

You know precisely what I mean.

I suspect that you might have meant a quarter of the range of results, and am intrigued to know how you slice up 11 discrete results into quarters.

Dayglo Sword wrote:

But a true gaussian distribution isn't a true bell curve as, by definition, it can never be symetrical around the peak point. Below it starts\stops at zero, but trails above to infinity.

No, wrong. The normal distribution is by definition symmetrical, and, as has already been mentioned, is unbounded in both directions. The standard normal distribution is centred on zero. What you are talking about is a truncated normal distribution. Of course, this is one of the reasons why, as MSerafin says,

Most things that are assumed to be "normal" for the purposes of statistical inference really aren't,

When statisticians assume that people's height is normally distributed, they don't usually worry too much about the lack of people with heights of 18.3 metres or -0.8 metres.

GildasFacit wrote:

A bell curve is an informal name for a normal distibution curve and is used for other distributions that have similar (but not identical) properties.

Since when? I have never heard the term "bell curve" used to refer to anything but the normal distribution. I would be most interested to see any citations of the alternative usage you claim. Also, be sure to edit the Wikipedia article saying that "bell curve" is synonymous with a Gaussian curve.

If you want to describe something that is thin at one end, much thicker in the middle, and thin at the far end, then I suggest it should be called an Anne Elk distribution:

YouTube link

…and, yes, I was annoyed at people saying "brontosaurus" when they meant "apatosaurus" when I was eleven.

All the best,

John.

Yawn.

I have seen triangular bells.

I have seen triangular bells.

..and I've heard tubular ones!

If we can continue to use "25mm" and "15mm" as figure "scales," then we can call the distribution of results of a roll of 2D6 a bell curve.

You To-may-to and I say to-Mah-to…

Let's call the whole thing off.

I say let's just cancel it.

Call it what you want, but it is annoying when people missuse 2d6 and make silly statements about probabilities, or apply die roll modifiers without a clue about how the distribution is affected. If you know nothing about probabilities, please design only rules using linear distributions (1d6,1d10, 1d20, 1d100, 1dwhater). Many games use sums of dice without any benefit from it being non-linear anyway and could just as well switch to 1d100 (or 1d66…).

anydice.com is a nice tool that can visualize many different dice distributions.

Of course, not all bells are curvy, so where does that leave us?