Update Notice: Overlord has said that 2.5 Standard Deviations is the cut off. Edits in Bold have been added to reflect this.

As you may know, it ALWAYS feels like you're getting min-damage rolls. And the funny thing is, you're not entirely wrong. A few people have been making posts on the forums and the subreddit talking about the cause of these issues. Since the intent of this blog is to delve into the basic stats the biases of the game, I figured that this would make a good post.

If you understand the way that the Damage and Penetration (and even the aim-deviation) RNG works, it makes a lot of sense that you would see a lot more minimum damage rolls than you would expect. If you're looking for conspiracy theories, you might even think that this is done on purpose.

If you've been paying attention, you know that Damage and Penetration values vary +/-25%. If this was a uniform distribution of values, the chance of getting any damage value would entirely dependent on your mean damage. The probability of each integer damage value X would be, given a mean damage D, simply:

For a 100 damage gun? P(X) = 1.9% for any damage value between 75 and 125. A 200 penetration gun? P(X) = 0.9% for any damage value between 150 and 250.

Now, that would seem like an easy, logical choice of a random value scheme. In fact, it's actually rather computationally "cheap", which leaves me confused by this next part:

That's not what Wargaming did. Wargaming decided that they wanted to concentrate the damage values around the average, but still provide a degree of randomization. To do this, they decided that the damage (and penetration) values should be pulled from a normal distribution, but limited to a range of
values +/- 25% from the mean. Now, in case you don't know what a normal distribution looks like, Wikipedia provides a nice example of 3:

I certainly didn't make this, but I could have. I think.

The so-called 'Standard Normal' (doesn't that sound redundant?) distribution is shown in red. It has a Mean of 0 and a Variance of 1. The important thing about this guy is that you can take any other normal distribution and convert it into a Standard Normal distribution with some simple calculations. Further, because of difficulty of integrating an arbitrary normal distribution, there exist standard tables of values, called Z-tables, for calculating what percentage lies under parts of the curve.

Okay, that's all well and good, but what does that tell us about are damage values? I swear I'm getting to it. One additional property about the normal distribution is important: It extends from -∞ to +∞. That means that in order to limit the variation to +/- 25% of the mean, you have to set some hard cut offs. The way Wargaming did this was to say "If we get a value below X, we'll just call it X. If we get a value about Y, we'll call it Y." What this did was to take all of the probability that was below X and give it to X and similarly take all of the probability above Y and give it to Y.

The white areas would be the probabilities rolled into Min and Max values
Now I come to a bit of a conundrum, there is a hidden parameter here that we don't know. Where we know the mean damage values for each gun, we don't know what values Wargaming has chosen for Variance (Edit: Overlord has mentioned that it is a 2.5 Standard Deviation Variance). Since Variance determines how flat or pointed the distribution is, we can't say exactly what proportion of hits should be minimum damage. However, by choosing some logical limits, we can give a good idea of what it should look like.

The obvious choices for the +/-25% limits are at the 1, 2 and 3 standard deviation lines.

Cut Off (#SDs) % Min Value Hits

Obviously, one can pull those values off the standard Z-tables. For each of these cases, we can determine the standard deviations of the distributions of based on each gun's damage. It's a pretty simple equation:

(Mean Damage)/(4*#SDs)= Damage Standard Deviation

There is one final question to be answered, how much more often do these extreme values occur than any other value? To answer this, I'm going to find the ratio of Minimum damage hits to Mean value hits. I'm going to assume that WG rounds fractions to the nearest integer, so the 'Mean' damage hits actually occur in when a value +/-0.5 from the mean is selected.

Gun Type Mean Damage 1 SD 2 SD 2.5 SD 3 SD
76mm M1A1
88mm L/56
105mm T5E1
122mm BL-9
130mm S-70A
155mm SA 58 AC
170mm PaK46

EDIT: It has come to my attention that Overlord has clarified that the +/- 25% is a 2.5 Standard Deviation limit.
Added Damage Distribution Charts: All Y-Axis Scales are the same for easy comparison.

88mm L/56 Damage Distribution
76mm M1A1 Damage Distribution
Since the mean value is supposed to be the most likely roll, it will always be more likely than any other roll (by varying amounts). The interesting part though is the trend. As your average damage goes up, these extreme values (Min and Max rolls) increase in proportion.

If we go with the 2 Standard Deviation case , the one I find to be most likely (Particually since I remember WG saying that the aim circle was the 98% line), you'll see that a Jagdpanzer E-100 will see a min or a max damage roll 7.2 (1.58 2.5 SDs) times more often than they will see any other number! More typically armed high tier tanks will see min or max rolls 2-4 times more often (Half to 80% as often) as any other number, and all tanks with average damage greater than ~140 will see minimum and max rolls more often than average rolls.
170mm PaK46 Damage Distribution

120mm M58 Damage Distribution

Keep in mind though, for the 2 SD case, that Minimum and Maximum rolls will still only make up 4.4% of the total rolls we see. For the 2.5 SD case, they would be 1.2% of all rolls. But they are more likely than any OTHER most other specific values, especially those a bit removed from the mean, so they will seem to show up a lot (plus, how many people remember how often they hit for 159 damage? There is a bit of a confirmation bias effect going on here too).

As for Max damage rolls, there is one additional reason we see less of them than Min rolls: Low Health Tanks. When you hit a tank that has hitpoints less than your Max damage, you have a chance, dependent on their health, of wasting a roll higher than their health to kill them. If you compile all of your shots over a number of battles and don't drop kill shots, you'll find that your average damage per shot is lower than the gun's mean damage due to this effect reason.

Now, if we had a data set to help us determine the rule WG uses to truncate their random numbers, we could figure out just how badly off my assumptions are! I'll see about getting Xylenes to get some data on that...

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