I see lots of people keep talking about Krupp and Sigma values for ships and various guns.
There are lots of misconceptions about what these values actually are and how they are used by the game engine.
There was a recent discussion on the US forum and Little White Mouse gave excellent explanations of these values. I thought it might be a good idea to share them here.
Krupp is a constant used to tweak shell penetration values. Normally, shell penetration calculations look something like this:
VL = (K)(C)TtDd/[Ww COSa(Ob)]
This is the formula used in the real world to calculate the (T) thickness of armour penetrated, where V is the striking velocity, at a given angle of impact (Ob), naval gun rounds of various (D) diameters and (W) weights. (C) describes the quality of armour -- Japanese steel, historically, was more brittle and of lower quality than American steel, for example. Finally, we have coefficient (K) which was a constant.
World of Warships probably uses something similar to this formula but simplified. All armour quality is the same in our game, so we don't need that. But we do concern ourselves with shell impact angles, shell velocity, shell mass, etc. In order to make some shells better at penetrating than others, the Krupp value was put in place to modify the penetrative power of all shells. Simply put, it's a direct modifier which tweaks all of the other values as a multiplier to reign in or empower a given ship's AP performance to be better or worse as needed.
The classic example of this is Murmansk compared to Omaha & Marblehead. These ships all fire effectively the same shell. Same muzzle velocity, same damage. The only thing that's different between them is 100g of weight and their Krupp value.
OMAHA: 1,772 Krupp
MURMANSK: 2,692 Krupp
If you math this out (Murmansk / Omaha) you get a ratio of approximately 1.5x. Effectively what this means is that Murmansk can penetrate 1.5x the amount of armour that Omaha can. For example, at 10km Omaha can penetrate approximately 100mm of steel while Murmansk can do nearly 150mm.
Wargaming has used Krupp as a hidden stat, to tweak the penetration behaviour of a lot of ships. This allows them to do things like have guns with ridiculous high muzzle velocity but still have very little penetration power. Or guns that are lower velocity to have a lot of muscle when it comes to punching through heavy plate. On its own, Krupp isn't terribly important -- it's just a coefficient -- a small piece of the penetration puzzle. It's a difference between the what and the why. The what players should be concerned with is Shell Penetration; how good is it? Once they understand that a ship has good or poor penetration, Krupp can help explain why that is.
Sigma Gross Oversimplifications
Presently, sigma is a buzzword for accuracy -- so much so that some players erroneously attribute it for an exact measure of accuracy; with the higher the better. Unfortunately, in reality its more complicated.
In World of Warships terms, sigma describes the scatter of shell distribution within a ship's shot dispersion field. The higher the sigma value, the more likely shells will land towards the center of the field, and inversely, the less likely they are to fly to the extremes of the dispersion area.
Now if you were paying attention, you'll have noticed a key factor here: dispersion area. Without knowing that, sigma is useless. Ships in this game have a dispersion area, defined by its horizontal dispersion and vertical dispersion. In port, we're given one value -- the horizontal dispersion at the ship's maximum range. Through the use of modifications and upgrades it was discovered that horizontal dispersion scales in a linear fashion. For example, all German battleships will have the same horizontal dispersion values at 10km, 15km, 20km, etc. Upgrades, like Aiming Systems Modification 1 will reduce this number by 7%, while disruption camo on a target will increase this by 4%. This makes it rather easy to plot and you may have seen graphs describing it before.
This, again, got erroneously simplified to describe ship accuracy, with Japan being more accurate than Warspite & Hood, which were more accurate than other British, Soviet and American battleships, which were more accurate than German and French battleships. Sigma was used as a "tiebreaker" to describe which ships within these respective lines was more accurate than the other (Yamato with 2.1 sigma > Nagato with 2.0 > Kii with 1.7 > Fuso with 1.5, etc). This again was a gross oversimplification. We have only two pieces of the puzzle and we were missing the third: vertical dispersion.
People have been trying to calculate vertical dispersion for a while without much success. The values data mined don't describe it as it's really a function of shell velocity and shell fall angle. The faster the shell and more shallow the dive, the wider the dispersion field. Thus higher velocity guns tend to have a larger vertical dispersion than those that are lower velocity. I discovered this when I started mapping dispersion, spending a couple of hours per ship to plot shell fall. Vertical dispersion changes with range in a non-linear fashion, with it being especially exaggerated relative to horizontal dispersion at close range, and becoming less disproportionately large at longer ranges.
Three battleship guns, each with 1.8 sigma, firing 180 shells at 15km. Accuracy modes applied where applicable. Red is Japanese (high velocity), Teal is German (high velocity), Purple is American (low velocity). Note how much more 'squat' the American dispersion is while the German and Japanese shells tend to overshoot and undershoot the target.
Putting it All Together
So if you've got all of that, now we can focus on sigma and what it looks like and what it's effects are in terms of game play. Sigma WILL NOT prevent you from having wonky dispersion. Sigma DOES NOT describe how often you will hit your target. Sigma WILL NOT make you aim better. All it does is say how likely it will be that your shells will land towards the center of where you're aiming. Individual volleys can still scatter all over the place -- dispersion is calculated at the whims of RNG. All sigma does is guarantee that OVER TIME a ship with higher sigma will more likely land shots in the middle of its dispersion area. Consider these two images:
These are two volleys from the same gun (Japanese battleship at 15km). Red looks more accurate, doesn't it? Once you plot twenty volleys total, I got this:
1.8 sigma (red) vs 2.1 sigma (orange). Same gun, same range, same vertical and horizontal dispersion.
Notice how oranges shells tend to cluster more towards the middle? These guns have the same maximum dispersion field -- you can almost get a hint of what it might be by so many of Red's shells landing towards the periphery (even then, they're not many). However, it's undisputed that orange's shells are more accurate between the two. But you would be unable to see this in a single volley. It's only over time that sigma makes itself felt with many, many, many samples.
In game play terms, good sigma makes it more likely that you'll have that one off shot where all of your shells cluster together and deliver the perfect, monstrous alpha strike with multiple citadels. I stress, that it makes it more likely. A ship with lower sigma can still pull off the same feat -- it will happen less frequently, mathematically speaking. Once you toss in the variable of human error, it's unlikely that most players can discern the difference except after a lot of experience. If dispersion is perfect but your aim is off, you won't land that massive hit, so trying to weigh in on sigma after even a handful of games is ludicrous. It takes time to notice the trend -- a lot of time in some cases and some people never really appreciate the difference.
Source - https://forum.worldofwarships.com/topic/148963-why-won’t-wg-list-shell-penetration-krupp-values-sigma-etc-for-ships/