A Few Questions/Observations From an Old Player

[Timo Pietilä]

Standard deviation approximately is average value over square root of number of dice in case of NdM type distributions. It is more accurate for larger N. E.g. 100d100 have mean of 5050 and deviation of 505, so chance that actual value will be greater then 6060 is about 2.5%.

How did you get 2.5% out of that and values of 505, 5050 and 6060? My question is how do I use that info in practical way. Do I have to look it from a table Patashu posted, or is there a easy way to use it in calculations? I hope answer doesn't include integrals or laplaces.

[LostTemplar]

Standard deviation approximately is average value over square root of number of dice in case of NdM type distributions. It is more accurate for larger N. E.g. 100d100 have mean of 5050 and deviation of 505, so chance that actual value will be greater then 6060 is about 2.5%.

[Timo Pietilä]

If you know the mean and standard variation of your normal distribution, then

...you know already a lot more than just deviation. Which makes my point, standard deviation without additional info is just useless number.

[Timo Pietilä]

As I said, the odds of a monster being two standard deviations above the mean are about 2.5%. The actual math is something I don't personally remember off the top of my head.

And this "in our example we use an SD of 1/8th of the monster's average HP" is what determines the "standard deviation" in reality and not just as example (in case of angband)?

[Patashu]

OK, and how likely that is? How do I use that info about "standard deviation" to calculate that? To me that is entirely pointless number without any real use.

I don't feel like going back several decades to learn something like this again.

If you know the mean and standard variation of your normal distribution, then
1) calculate how far from the mean the target value is
2) divide by the distribution's standard deviation, to get the number of SDs away from the mean the target value is
3) Use this image: http://www.cs.bilkent.edu.tr/~korpe/...2/z-table.jpeg to determine the probability of being between the mean and that number of SDs (which you can multiply by 2, invert, etc. to get the other probabilities, like the probability of being higher than that number of SDs, or the probabily of being higher than that number of SDs in both directions)

This works for any distribution as long as the distribution is similar to a normal distribution (= has a bulge in the middle). If it does not (for example a completely flat distribution, or an exponential/logarithmic distribution), then you need different math.

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OK, but how do you calculate standard deviation?

First, calculate variance:

Var = E[X^2] - E[X]^2
where E[something] is the expected (mean) value of something. so E[X^2] is the mean of the squares, while E[X]^2 is the mean, squared.
Standard deviation is then Sqrt(Var).

In addition, variance has a useful property: Var(X+Y) = Var(X) + Var(Y). So if you need to figure out the standard deviation of 2d4, you can figure out the variance of 1d4, double it then sqrt it to get the standard deviation.

[half]

For some reason I thought there were 10 stars of health, not 5.

There are 10 stars, but you can sometimes knock off one with 19% of the monster's hp. It is because you can't knock off 1.9 stars -- it is a rounding issue.

[Derakon]

As I said, the odds of a monster being two standard deviations above the mean are about 2.5%. The actual math is something I don't personally remember off the top of my head.

[Timo Pietilä]

Once we have the number of standard deviations away from the average we are, we can use that value plus knowledge of the distribution to determine how unlikely it is that an AMHD would have that many hitpoints.

OK, and how likely that is? How do I use that info about "standard deviation" to calculate that? To me that is entirely pointless number without any real use.

I don't feel like going back several decades to learn something like this again.

[Oramin]

I was just using Til-i-arc (Fire Brand and Cold Brand) on a 5-headed Hydra. The damage message said I was burning the Hydra.

Since Hydras have a vulnerability to Cold, shouldn't I be freezing the Hydra instead?

[Derakon]

Wikipedia link

In layman's terms, the standard deviation measures how "scattered" a distribution is. For example, a 1d12 is more scattered (has a higher standard deviation) than a 2d6, which has a higher standard deviation than a 3d4. A low standard deviation means that you consistently get values close to the average / rarely get values far from the average. The standard deviation is also a value, though, so you can gauge how rare a given result is by measuring how many standard deviations away from the average you are. For example, in our example we use an SD of 1/8th of the monster's average HP.

1848 / 8 = 231 (the standard deviation is 231)
2300 - 1848 = 452 (finding how far we are from the average)
452 / 231 ~= 1.95 (that value in standard deviations)

Once we have the number of standard deviations away from the average we are, we can use that value plus knowledge of the distribution to determine how unlikely it is that an AMHD would have that many hitpoints.

[Oramin]

http://en.wikipedia.org/wiki/Standard_deviation

[Timo Pietilä]

That's 1.95 standard deviations away; I don't remember how to do the relevant stats myself but a friend informs me it should be about 2.5%, i.e. 1 in 40 will have at least that much more health. (Odds of being at least two standard deviations away from the mean are 4.55%, but half of those are two SDs below the mean)

OK, I'm going to have to ask this: What an earth is "standard deviation"?

[Derakon]

That's 1.95 standard deviations away; I don't remember how to do the relevant stats myself but a friend informs me it should be about 2.5%, i.e. 1 in 40 will have at least that much more health. (Odds of being at least two standard deviations away from the mean are 4.55%, but half of those are two SDs below the mean)

[Timo Pietilä]

Timo: uniques don't have varying HP. And I think if you're running the stats sufficiently heavily to rely on, say, an Ancient White Dragon doing no more than 211 damage with its breath attack, then you're trying too hard.

Sometimes I do take on monsters that I know are very dangerous, but who start losing their breath power immediately, like sneaking on sleeping AMHD without rPoison trusting that after I hit it with my arrows it will not be able to breathe more than around 600 points of damage.

It is the first move that counts. If it doesn't use it's poison breath immediately I might be able to kill it. If it does, I bail out immediately. However, if that first breath is 700 points even after that attack I might be dead.

So how unlikely it is that AMHD could have, say 2300 HP instead of monster.txt indicated 1848? That's only 24% increase, and if that sorcerer had 610 instead of 457 it is 33% increase.

[Derakon]

Presumably it just has to have more than 610 HP, then it might take off (e.g.) 19% of its health which might remove just one star.

For some reason I thought there were 10 stars of health, not 5.

Timo: uniques don't have varying HP. And I think if you're running the stats sufficiently heavily to rely on, say, an Ancient White Dragon doing no more than 211 damage with its breath attack, then you're trying too hard.