[OT] Averages

[randomling]

Cool.

But presumably the "normal" way of rolling stats (ie, 4d6 and drop the lowest die) is different. Are mean, median and mode then different? I mean, presumably they are different from straight 4d6, but are they different from each other?

And is that mean, median or mode Joe?

 

[bolen]

just remember . . .

there are lies, damn lies and statistics

 

[Agback]

OK so here's where we hit on-topic.

When you talk about the "average" roll of, say, 4d6, are we talking about a mean, median, or mode?

None of them. We are talking about the expected value, which is like a mean but subtlely different.

The expectation of a discrete distribution (like 4d6) is calculated by multiplying each possible outcome by its probability of occurring, and then adding together the products.

If you roll 4d6 n times, sum the n rolls, and divide by n, you will have the mean of those particular rolls. For given n, that will be sometimes more and sometimes less: it will be a random variable (it is called the sample mean, and its symbol is an X with a bar across the top). But if you make n really big, the chances are very small that X-bar will be very far from mu, where mu is the expected value. (For discrete distributions, the expected value corresponds to a thing called the population mean, but the point is not extensible to continuous distributions, and tends not to get a lot of attention in real statistics.)

Example:

2d6 has eleven possible outcomes (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). Their probabilities are (1/36, 1/ 18, 1/12, 1/9, 5/36, 1/6, 5/36, 1/9, 1/12, 1/18, 1/36). Multiply each possible outcome by its own probability and you get (2/36, 6/36, 12/36, 20/36, 30/36, 42/36, 40/36, 36/36, 30/36, 22/36, 12/36). Add those up and you get 252/36 = 7. The expectation of 2d6 is 7.

Now I roll 2d6 eleven times and get (5, 4, 7, 11, 3, 8, 3, 9, 7, 8, 7). That sample has a mean of 6 6/11 (and it has a median of 7 and a mode of 7), but it doesn't have an expectation because it is a sample, not a distribution. Meanwhile, 2d6 has an expectation of 7, doesn't have a mean, and doesn't have a median, because it is a distribution, not a sample. (A distribution does have a mode, but in a slightly different sense from that in which a sample does. The mode of 2d6 is 7.)

Regards,


Agback

 

[Agback]

The mean, median, and mode of a multi-die roll are the same -- at least if you ignore rounding issues. It's a "normal" distribution.

Sorry, that's a very misleading statement. The normal distribution is a particular distribution (also called the Gaussian distribution). That is not a term for any distribution that has mean = median = mode.

The normal distribution is a particular distribution of a *continuous* variable with range from minus infinity to infinity, bell-shaped etc. But not every bell-shaped distribution of a *continuous* variable with range from minus infinity to infinity is normal. And no ddistribtuion of a discrete variable is normal.

Regards,


Agback

 

[Sir Whiskers]

Randomling, to answer your second question: Yes, when you roll 4d6, drop lowest, the results will not be the same as rolling 4d6 and adding them all together. I'm not an expert with statistics (in fact, I keep hoping someone will post a *simple* formula for determining averages when rolling extra dice), but I'll take a stab at it.

To understand the reason you can't just use the mean of all the dice you roll, try this: Roll 6d6 for each stat, dropping the lowest three. The average of 6d6 is 21 (avg 3.5 per die, times 6 dice). But you can only have a maximum of 18 on 3d6 (the three you keep), so the straight average ("mean") of 6d6 doesn't help you much.

But, as a general rule of thumb (by no means scientific), I've found that for every extra die you add, you tend to bump the average by about 1 point. So 3d6 averages 10.5, 4d6 averages 11.5, etc. (The amount added to the average actually decreases per die with the number of dice added, since 18 is the max value, but now things start getting complicated...)

Any math experts out there - feel free to correct me if I'm wrong.

 

[frisbeet]

Applause to Agback!

Here's the # of ways of rolling an outcome 3 through 18 for the 3d6, 4d6 drop lowest, and 5d6 drop two lowest methods. These numbers can be generated by a lot of tedious counting or by using a complicated counting formula.

oc____3d6_____4d6_____5d6
18_____1______21______276
17_____3______54______610
16_____6______94______935
15_____10_____131_____1111
14_____15_____160_____1155
13_____21_____172_____1055
12_____25_____167_____881
11_____27_____148_____665
10_____27_____122_____470
9______25_____91______296
8______21_____62______170
7______15_____38______90
6______10_____21______41
5______6______10______15
4______3______4_______5
3______1______1_______1

By dividing by the total # of ways (216 for 3d6, etc.), you can get a probability for each outcome. By multiplying this P by the outcome and summing for every outcome product, you'll get the expected "average" value. Here they are:

3d6
10.50

4d6
12.24

5d6
13.43

 

[Darkness]

just remember . . .

there are lies, damn lies and statistics

Yeah, and also remember 73.2% of all statistics are made up on the spot!

 

[Bagpuss]

By multiplying this P by the outcome and summing for every outcome product, you'll get the expected "average" value. Here they are:

3d6
10.50

4d6
12.24

5d6
13.43

This is one case where the mode or median is probably a better average to use as I have yet to see a single player with 12.24 in any attribute.

Mode gives
3d6
10 or 11

4d6
13

5d6
14

Median gives
3d6
10 or 11

4d6
12

5d6
13

 

[randomling]

This is one case where the mode or median is probably a better average to use as I have yet to see a single player with 12.24 in any attribute.

LMAO!

Wow, thanks everyone! Some of this is a little beyond my ability to understand, which just goes to show I need to review mathematics. But that was all very interesting, which just goes to show an artsy type can get into maths if she really tries.

I can't add up my d20 rolls on sight, either. You'd think D&D would improve a girl's mental arithmetic, wouldn't you?

Oh, well...

(edit: Heh. Always forget I'm allowed my sig outside the Hive! )

 

[JayOmega]

(in fact, I keep hoping someone will post a *simple* formula for determining averages when rolling extra dice)

Unfortunately, the "formula" winds up being a lot more complicated than the output of a spreadsheet. There's a post on rec.games.frp.dnd from way-back-when that has tables for 3d6 drop 0 to 9d6 drop 6. It's right here.

HTH.