Keep in mind I'm going to ignore BSB rerolls - at least for this discussion. If you want to skip the math and go right to the analysis, look for the "So what does all this mean" section.
So, the first part is relatively easy - calculating the 2d6 percentages. Basically this boils down to what is the probability that a certain leadership score will pass based on the unit's leadership. Basically given 2 dice, there are 36 possible combinations, and they add together for the following:
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Total Possible Rolls Number Fraction Percentage
===== ==================== ====== ======== ==========
2 1,1 1 (1/36) 2.78%
3 2,1:1,2 2 (3/36) 8.33%
4 1,3:2,2:3,1 3 (6/36) 16.67%
5 1,4:3,2:2,3:4,1 4 (10/36) 27.78%
6 1,5:2,4:3,3:4,2:1,5 5 (15/36) 41.67%
7 1,6:2,5:3,4:4,3:5,2:6,1 6 (21/36) 58.33%
8 2,6:3,5:4,4:5,3:6,2 5 (26/36) 72.22%
9 3,6:4,5:5,4:6,3 4 (30/36) 83.33%
10 4,6:5,5:6,4 3 (33/36) 91.67%
11 5,6:6,5 2 (35/36) 97.22%
12 6,6 1 (36/36) 100%
This is straight forward - the possible rolls that total to a given number are listed, along with number of rolls equal or less than the number. (i.e. the probabilities are additive, so the probability of rolling an 8 or less, for example, is the probability of rolling a 2 + the probability of rolling a 3 + .. etc.)
Asleep yet? No? Ok, moving on.
So, the 3d6 chart is more complicated since you discard the highest die rolled, but the probabilities are calculated in the same way. There are 216 combinations (6*6*6). Here's the start of the chart - personally, I cheated and generated the numbers with a short perl script.
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2 - 1,1,(1-6):1,(2-6),1:(2-6),1,1
3 - 1,2,(2-6):2,1,(2-6):1,(3-6),2:2,(2-6),1:(3-6),1,2:(3-6),2,1
4 - 1,3,(3-6):1,(4-6),3:(3-6),1,3:2,2,(2-6):2,(3-6),2:(3-6),2,2:3,1,(3-6):3,(3-6),1:(4-6),3,1
.
.
.
12 - 6,6,6
Numbers in parentheses denote it not mattering what the roll is as long as it's in the right range of numbers (like 4-6). So, for counting purposes, there are 16 possible combinations out of 216 that will result in exactly a 2 being rolled. The three dice being 1/1/1 , 1/1/2 , 1/1/3, etc. This results in the following numbers when you count them up:
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Total Number Fraction Percentage
====== ======= ========== ==========
2 16 (16/216) 7.41%
3 27 (43/216) 19.91%
4 34 (77/216) 35.65%
5 36 (113/216) 52.31%
6 34 (147/216) 68.06%
7 27 (174/216) 80.56%
8 19 (193/216) 89.35%
9 12 (205/216) 94.91%
10 7 (212/216) 98.15%
11 3 (215/216) 99.54%
12 1 (216/216) 100%
Ok, so what does all this mean?
Basically this gives us a measure of how much better an equal leadership score really is for the lizardmen (i.e. Ld8 for Lizardmen means what if it were one of our units?) and also lets us estimate how easy or difficult it is to make unit types flee ...
So, lets compare - I found the numbers to be quite interesting. Lizardmen skinks with leadership 5 are going to run away, right? Well, compared to Harpies for example - not as often. Lizardmen leadership 5 is 52.31% chance to succeed, while Druchii Harpies at leadership 6 is a 41.67% chance to succeed. (Keep in mind I'm ignoring modifiers for right now). Skinks are closer to leadership 7 from a probability standpoint. Not quite as good a leadership as warriors though, but it's close. From a simplicity standpoint, you can more or less assume lizardman leadership is +1 to +2 what's listed as a rule of thumb estimate and be fairly close.
The middle to high end Lizardmen leadership is more of a difference in my mind since that's where more of the probability curve bunches up. Take a block of Saurus for example. They will succeed at a leadership check 89.35% of the time based on lizardman leadership of 8, whether supported by a general or not. We get around the same odds for a unit within 12" distance of a highborn general (91.67%), while our leadership 8 troops succeed checks 72% of the time. So keep this in mind - an unsupported block of lizardman infantry isn't necessarily going to be easier to break - and one in range of the general/BSB is going to be worse.
For a unit of Temple Guard (stubborn) with a Slann, they succeed leadership checks 95% of the time - regardless of casualties! The point is, no matter how many leadership checks you force on a TG unit w/ Slann, don't count on them breaking. (You can auto-break them with fear/terror and outnumber if they lose combat, but that doesn't require a test.) Either plan on killing this unit to a man (errrr...lizard), autobreak them through fear, or avoid them. I think that's important to know if you are fighting a Toad.
The other point is - Lizardmen units can be broken through combat, but it's harder to do than with the units of most armies. This is already an obvious point, but you can use the table to determine what level of success you are going to have - which means you can intelligently pick your fights since you have superior mobility.
example: 20 warriors vs 16 Saurus, warriors charge. Ok, from the chart, leadership 8 has a 89% chance to save - to get the odds in your favor you'll want < 50% chance of passing the test ... which means winning the combat by 4. Same number of ranks, lets assume musician and banner ... so you need to do 4 more wounds than your opponent. With 5 attacks, str 3 vs toughness 4 and 5+ save, it's not likely to happen. In fact, the warriors are likely going to lose the combat anyway unsupported.
Ok, lets use the same magic number (win combat by 4) with 20 warriors and a unit of 10 witch elves flanking - roughly the same points. The flank attack cancels ranks, so we're at +3 to start. Assuming the warriors take 3 wounds while dealing out 1 wound, the witch elves need to do at least 3 wounds more than what they suffer to win the combat by enough to have a good chance at forcing the Saurus to flee. (With 10-15 poisoned attacks, this is about even odds roughly that the Saurus will break and run - but still not definite).
Keep in mind this discussion is that of probabilities - not certainties. But with my dice rolls, I'd rather depend on stacking the odds in my favor rather than luck.
For summary, essentially the main points are:
1) The lizardman leadership score is deceptive due to the 3d6 rolls to test. Use the above charts to get a feel for how hard a unit will be to break. Or, as a rule of thumb, you can assume they have a leadership +1 or +2 points of what's listed.
2) Temple Guard with a Slann aren't likely to miss a leadership test - period. You can autobreak them however with fear/terror causing units being involved in the combat. Plan appropriately.
3) Choose your fights wisely - determine whether or not you'll think you have the edge in combat before committing your troops to a fight whenever possible. Better mobility means it's possible to choose your fights, but it adds a lot of complexity to your movement phase - making it that much more important.
4) "Shooty" armies, assuming 3 turns of shooting, need to do a lot of damage - don't depend on Lizardmen running due to panic tests. Even an unscreened block of saurus (for example) have a 70.5% chance of passing all panic tests from shooting - even assuming you do enough casualties every turn to force a check. (If anyone is interested, I can estimate the number of shots it will take to have over a 50% of forcing a check every turn per Saurus block ... or it would be another good topic for discussion.)
Additional points from follow up comments:
From Reika:
Although it is implied by the numbers and whatnot, I just thought I'd mention something not specifically stated: expect any broken/panicked units to rally the next turn. As the numbers Zader kindly calculated for us show, even without a musician, pretty much the whole of the lizardman army has a better than average chance of rallying. Be sure movements/pursuits on your turn reflect this assumption.
Comments or constructive criticism are appreciated.