ADR's and Surv's

  For some time, the ADR and Surv's program has been available for download; however, I am aware that many of you no longer have any version of BASIC on your computers, and many of those of you who still have QBASIC left over from DOS 6.22 don't know it is there or how to use it.  I use to calculate ADR's and Surv's long-hand for years before creating the program to do it for me.  I

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expect that those with the CGI skills of Mr. Wei-Hwa Huang could probably create an interactive HTML form which would do the calculations, but I haven't studied that aspect of programming as yet.  I more recently used my meager skill in spreadsheets to create a functional version in MicroSoft Excel, but it contains no explanations and no examples.  So I am taking this space to explain what the ADR's and Surv's are, how they work, why they are useful, and how to calculate them.  If you have the program, you probably don't need this--most of it is explained in the program, except the formulae themselves, which you don't need to understand to run the program.  However, for those of you who look at the program itself for information, you might find it confusing, since it's set up to gather a lot of information early that it doesn't need until later--based on the assumption that you'll start with the character paper in hand, and copy the needed information from that, and then draw the other necessary information, such as weapon damage, from various books.  So this might help clarify what the program does more than actually looking at the program.

  ADR stands for Average Damage per Round.  The concept here is based on probability theory.  The to hit roll gives the chance that the attacker will hit a particular opponent.  So if his hit roll is 11, there are 10 chances that he will hit (11 through 20), and 10 chances he will miss (1 through 10)--and given a million attack rolls he will hit as many times as he misses, 50%.  If he does hit the opponent, he rolls another die or set of dice for damage.  If it's a d8 for damage, then the character will roll 1 point as often as 8 points, or any other roll, and (barring extraordinary dice luck) has a calculable average damage per hit, in this case (1+2+3+4+5+6+7+8)/8=4.5 points average damage per hit.  Given that our character hits only 50% of the time, the average damage per attack is 4.5x50%=2.25 points average damage per attack.  But if the character is a tenth level fighter swinging at zero level goblins, he gets ten attacks in the round, each of which has the same chance to hit and the same probable damage, thus 2.25x10=22.5 points average damage per round.

  Of course, there are a lot of variables here.  Once when tutoring a young gamer in algebra, I posed the problem of a paladin fighting a dragon, and, given the armor class, chance to hit, damage range, and number of attacks of each of the adversaries, asked who would survive and who would die.  It's perfectly possible to run the program with a specific adversary in mind; but I decided that it was more useful to have general information which could be compared, character by character, so the DM--and the party leader--would know which characters really had the most combat attack force.  Thus I chose a standard armor class to be the basis of the ADR's.

  There is a strong preference in AD&D to take note of the chance to hit armor class zero.  This is in part due to the history of the development of the game--originally, a fighter's chance to hit was determined by subtracting the sum of the opponent armor class and the attacker level from twenty.  All of the other tables were based on penalties against this number.  However, it was decided in creating the tables that a twenty should be adequate to hit armor classes as good as negative five--six places in a row.  That means that low-level characters who are bonused or penalized one or two points will not gain or lose much benefit in relation to others at armor class zero.  However, at armor class five there is adequate variation between classes, even with bonuses and penalties, until very near the top of the fighter tables.  Thus the ADR's have always used the character's chance to hit armor class five as the basis for a standard comparative result.

  It is possible for a high-level fighter, or one with significant bonnuses to hit, to have a chance to miss lower than two (remember that a one always misses).  Although the resultant average damage per round may be an amount of damage greater than the character can inflict on the given number of attacks, it is still a valuable number for comparative purposes.  Similarly, a low-level magic user with penalties for such as low strength or low dexterity or non-proficiency could conceivably have a chance to miss greater than twenty, which would also create an unreal number (since a twenty always hits) valid only for comparison.  Note that if the armor class of the opponent changes, the chance to hit will move and so become valid; thus the damage values which do not represent real average damage per round are better comparisons.

  For each character, in addition to the standard chance to hit, there are numerous standard adjustments--bonuses or penalties for strength and dexterity.  This has to be included to get a useful value.  Then for each weapon, bonuses for specialization, magical properties, and other adjustments will be included.  Missile attacks will usually require adjustments for range.  Some attacks, such as those in martial arts or the attacks of some monsters, will be "additional attacks"--such as the damage done by a bite attack which is rolled in the same round as two claw attacks, or an iron fist attack which can be done with the left hand after a weapon-based attack with the right.  The chance to hit of any character can be a complex calculation involving many factors.

  The average damage is not so much trouble.  In Dungeons & Dragons, all damage rolls are either a linear roll or a balanced curve--that is, the chance of maximum damage is equal to the chance of minimum damage, and lower rolls are always perfectly balanced by higher rolls.  Given such curves, the average damage on any attack is equal to the maximum damage plus the minimum damage, divided by two.  It doesn't matter whether the bonus or penalty damage is added to the maximum and minimum before calculating the average, or whether it is added to the average of the die roll after it is calculated.

  Surv is short for Survival Rating.  In brief, the character's armor class is a measure of the probability that he will be hit, and his hit point total is a measure of the number of hits he can withstand.  At some level, the character with the better armor class but lower hit points and the character with the poor armor class but many hit points are equal; but arriving at who is able to survive combat longer is not easy to thumbnail.  Party leaders and DM's alike may wish to know which member of the party is toughest, and which is most vulnerable, and the Surv calculation derives a reasonable measure of that.

  The character's armor class must be used.  However, a shield is a common part of this value--and a magic shield may be a significant part.  But shields are limited under the game rules in the number of attacks against which they may be counted.  The value of a small shield only counts one attack in the round, and the largest shield only counts against three attacks.  Thus in calculating Surv, the assumption is made that there will be four attacks in a round, and the shield will be negated eventually.

  A hypothetical adversary is created.  This monster attacks four times per round, so as to negate the shield.  It's chance to hit armor class zero is eleven (as a twenty-first level magic user).  This number is used because it means that the character with an armor class of ten will be hit every round (ignoring the rule that a one always misses in order to create a statistical accuracy), and a character with an armor class of negative nine will only be hit once in twenty attacks.  The surv is not as useful for characters with armor classes better than negative nine, who are treated as having a neg nine AC for this calculation (or, viewed the other way, the same roll of 20 which is required to hit a negative nine AC will also hit any better AC).

  To make the number as useful as possible, each attack which connects does one point of damage.

  Calculate an average damage per attack, as described in discussing ADR's.  This is now the statistical probable damage which is done to the character each time the enemy swings.  If the character has no shield, it is a simple matter to divide the number of hit points by the damage per attack and round up.  If the character's AC changes during the round due to the loss of a shield bonus, it is preferred that the damage be subtracted individually for each attack until the character's remaining hit points are not greater than zero.  (In the alternative, the four values representing the average damage for each attack may be averaged, and the total hit points divided by this arithmetic mean.  This will occasionally yield a different number, but will not be far from the other result.)  The Surv is equal to the number of attacks which this hypothetical opponent statistically will have to launch before the character is dead.

  For comparison several monsters have been selected from the Monster Manual (TM TSR) and processed.  Here are the melee ADR's of some significant monsters:

60.8	Tiamat (6 attacks)
38.95	Bahamut (3 attacks)
31.2	Loxodont (Elephant)
28	Elephant (Asiatic)
25.5	Huge Ancient Gold Dragon
22.6625	Storm Giant
17.85	Cloud Giant
15.6	Ettin
14.4375	Fire Giant
11.2	Frost Giant
7.7	Small Very Young White Dragon
7.6	Owlbear
7.35	Stone Giant
6.325	Centaur
6.3	Hill Giant
6.175	Minotaur
5.525	Manticore
5.225	Carnivorous Ape
3.85	Bull
3.85	Yeti
3.575	Doppelganger
3.025	Ogre
2.75	Werewolf
2.5	Bugbear
2.5	War Dog
2.5	Gnoll
1.8	Dwarf
1.8	Hobgoblin
1.575	Orc
1.375	Elf
1.225	Gnome
1	Goblin
0.875	Halfling
0.5	Giant Rat
0.5	Brownie (with Dagger)

  Melee ADR's are based on weaponless attacks unless specified.  All ADR's shown are against armor class 5 man-sized opponents and are the creature's best ADR for such opponents.  Note that ADR's set hit probability against average damage (multiplied by attacks per round).  Where hit probability is higher (e.g., due to worse defender armor class) greater average damage is more valuable.  Conversely where hit probability is low small differences in it (plusses to hit) have greater effect on ADR.  ADR's are valuable for character comparisons and weapon selection.  DM's can use them for estimating the attack strength of an encounter.

  Missile ADR's are best compared at select ranges.  We will consider:

1 game inch:	Point Blank
3 game inches:	Short
6 game inches:	Midrange
12 game inches:	Long
21 game inches:	Extreme

  Missile types are identified.  Standard fire rates are applied.

  Missile ADR at Point Blank (1 game inch):
 

13.65	Manticore Tail Spikes
11.05	Cloud Giant Rock Toss
9.075	Fire Giant Rock Toss
8.8	Frost Giant Rock Toss
7.35	Stone Giant Rock Toss
6.3	Hill Giant Rock Toss
3.85	Centaur Composite Long Bow
3.5	Gnoll Long Bow
2.8	Hobgoblin Composite Long Bow
2.45	Gnome Short Bow
2.45	Orc Long Bow
2.1	Elf Long Bow
2.1	Halfling Short Bow
1.75	Bugbear Spear, Hammer, or Hand Ax
1.4	Dwarf Spear or Hammer
1.05	Goblin Spear
0.875	Kobold Spear or Javelin

  Missile ADR at short range (3 game inches):
 

13.65	Manticore Tail Spikes
11.05	Cloud Giant Rock Toss
9.075	Fire Giant Rock Toss
8.8	Frost Giant Rock Toss
7.35	Stone Giant Rock Toss
3.85	Centaur Composite Long Bow
3.5	Gnoll Long Bow
2.8	Hobgoblin Composite Long Bow
2.45	Gnome Short Bow
2.45	Orc Long Bow
2.1	Elf Long Bow
2.1	Halfling Short Bow
1	Dwarf Light Crossbow
0.875	Bugbear Spear, Hammer, or Hand Ax
0.75	Goblin Sling Stone
0.525	Kobold Javelin

  Missile ADR for midrange (6 game inches):
 

13.65	Manticore Tail Spikes
11.05	Cloud Giant Rock Toss
9.075	Fire Giant Rock Toss
8.8	Frost Giant Rock Toss
7.35	Stone Giant Rock Toss
6.3	Hill Giant Rock Toss
3.85	Centaur Composite Long Bow
3.5	Gnoll Long Bow
2.8	Hobgoblin Composite Long Bow
2.45	Orc Long Bow
2.1	Elf Long Bow
1.75	Gnome Short Bow
1.4	Halfling Short Bow
1	Dwarf Light Crossbow
0.5	Goblin Sling Stone
0	Kobold Spear or Javelin

  Missile ADR at long range (12 game inches):

11.55	Manticore Tail Spikes
11.05	Cloud Giant Rock Toss
9.075	Fire Giant Rock Toss
8.8	Frost Giant Rock Toss
7.35	Stone Giant Rock Toss
6.3	Hill Giant Rock Toss
3.15	Centaur Composite Long Bow
2.8	Gnoll Long Bow
2.1	Hobgoblin Composite Long Bow
1.75	Orc Long Bow
1.4	Elf Long Bow
0.75	Dwarf Light Crossbow
0.7	Gnome Short Bow
0.35	Halfling Short Bow
0.125	Goblin Sling Stone

  Missile ADR at extreme range (21 game inches):
 

11.05	Cloud Giant Rock Toss
7.35	Stone Giant Rock Toss
2.1	Centaur Composite Long Bow
1.75	Gnoll Long Bow
1.05	Hobgoblin Composite Long Bow
0.7	Orc Long Bow
0.35	Elf Long Bow
0.2625	Dwarf Heavy Crossbow

  Note that Manticores throw 6 spikes per round.  Giants take no range penalties.  Breath weapons and magic attacks rely on defender save instead of armor class and hit probability.

  Outside the stated armor class range the SURV becomes artificial for comparison.  Against opponents with higher hit probability survival probability is leveled for worse armor class defenders (against whom only a 1 misses).  Conversely, against those with lower hit probability leveling occurs for better armor class defenders (hit only on a 20).  Leveling means armor class has no real effect on survival beyond a point.

  SURV of select monsters:
 

479	Bahamut
359	Huge Ancient Gold Dragon
255	Tiamat
130	Storm Giant
97	Cloud Giant
84	Stone Giant
81	Fire Giant
69	Ettin
67	Frost Giant
61	Loxodont (African Elephant)
56	Asiatic Elephant
53	Hill Giant
42	Manticore
37	Minotaur
32	Owlbear
28	Carnivorous Ape
27	Werewolf
27	Yeti
25	Ogre
23	Centaur
23	Doppelganger
21	Bull
19	Bugbear
13	War Dog
11	Gnoll
7	Small Very Young White Dragon
7	Elf
7	Hobgoblin
6	Dwarf
5	Gnome
5	Goblin
5	Orc
4	Halfling
3	Brownie
2	Kobold
2	Giant Rat

  SURV's of dragons are calculated on exact number of hit points for the type and size and age dragon specified.  All other monsters are calculated based on average hit points for the stated hit dice.

  I hope these statistic analyses will improve your gaming.  Questions and comments may be sent to the author at Dragon@MJYoung.Net.

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