Looking for some advice on balancing factions

These win probabilities look similar, though it depends on exactly how many skirmishes were involved. The Gaussian fitting is obviously affecting something because the wins don't average out to 50%. Why not use the raw win percentages? Or do about 0.7% of skirmishes end in a draw?

(Quick note: The +/- figure often quoted with poll results is a 95% confidence interval, which is about 1.96 times the standard deviation for large samples.)

You can perform a statistical test to see if the win probabilities are unequal, though one of the assumptions is violated because these trials are not strictly independent of one another. You could get independence by throwing out all head-to-head skirmishes when comparing two factions, though we are testing play balance here rather that taking an econometrics exam. Personally I would leave in the head-to-head skirmishes.

This still leaves 24 or 30 comparisons, and with a 95% confidence interval we expect a false positive in about 1 out of every 20 tests. The method of handling expected screw-ups is called Šidák correction. This raises the bar for detecting a bias on the theory that we only have one set of evidence and we don't want to "falsely convict" the game. Alternatively, you could use raw t-tests and just run the entire simulator a lot of times to see if any persistent differences remain.

Now, suppose there is a persistent difference. For example, maybe Faction 3 is clearly better than Faction 6. If you can do so without eviscerating your theme, try swapping soldiers between those two while maintaining the point balance. Then try the simulator again and see if that helps.

For example suppose it's a hand of five cards, but the player can discard/redraw up to two. I'd have the simulator just deal seven cards and pick the five "best" by some rubric. Not a perfect emulation of play, but it gets closer to testing the skirmishes more likely to occur in a game.

I have run further diagnostics in Excel. By subtracting one % of the other %. To see how much surplus each faction has over another. In a weird explanation: Every + is a rock over scissors, every - is a scissors over rock.

Then I added these all up for each faction. And I suggest, removing faction 6 AND 4.

And here is the tldr train of thought:

I know that 2 vs 5 is very extreme. But both are the top 2 of all 6.
Faction 5 has a surplus of 9.6 over faction 2. But faction 2 has for example a surplus of 8.2 over faction 6. If faction 6 had a surplus over another faction, the circle could go round. Unfortunately, it doesn't. Faction 6 is always the weakest.

Here is the list that I got as total strength of each faction:

Faction 1: -0,6
Faction 2: 4,1
Faction 3: 9,9
Faction 4: -1
Faction 5: 11,4
Faction 6: -23,8

Clearly, faction 6 is useless. In fact, I count no surplus for faction 6. It has -5.
When I remove faction 6. We know things will shift a lot:

Faction 1: -1,9
Faction 2: -4,1
Faction 3: 4,5
Faction 4: -5,2
Faction 5: 6,7

We see that faction 5 has lost a lot of leverage. And so has faction 3. In fact, when removing 1 out of 6 factions, you expect the numbers to reduce with 1/6th. But as you can see, the numbers plummet more. Which is a good sign.

But this list only shows the "bucket imbalance". We need to look back at the differences. That is, if we want to look at the choices that players make. In this case, a yes/no question is going to be answered. That is why we need those + and -

5 will still remain strong compared to 2. But has lost 6, which was a +. In fact, if it comes to choosing a faction. Faction 5 has 2+ and 2-. This faction is a properly balanced faction, regarding choices. To get a true balance in choices of factions. You need an equal amount of + and - for each faction. However, a 3/1 or 1/3 is also allowed, since you have an unique choice in that regard. And it still counts as choice.
Having a 0 however means a choice that will never happen.

The list of + and -
Faction 1: +2 -2
Faction 2: +3 -1
Faction 3: +3 -1
Faction 4: +0 -4
Faction 5: +2 -2

Faction 4 clearly shows, it has no value. So, I tried to see, what happens if I remove that faction as well.
And now, the magic begins.

Faction 1: +1 -2
Faction 2: +2 -1
Faction 3: +2 -1
Faction 5: +1 -2

As you can see, there are no "none" choices left.
Of course, faction 2 and 3 look like the better choices. I have the percentages in my table. But it is better to talk in RPS.

Faction 1 beats 5.
Faction 5 beats 2.
Faction 2 beats 1 and 3.
Faction 3 beats 1 and 5.

The last table:
Faction 1:-3,5
Faction 2:-4,7
Faction 3:2,3
Faction 5:5,9

Faction 5 looks like the strongest and faction 2 like the weakest. But faction 2 beats 2 other factions. The weakest faction sir! Beats 2 out of 3 factions. And lets not forget, faction 1 is rather weak too, and this one beats faction 5.

So, marvel at your system while faction 4 and 6 are not to be heard of ever again.

Of course, this math doesn't include other things that you might change. The whole system could be turned up side down, since you are already that close to the 50.

Pm me your email. And I'll send it.
I saved it for that question.
Please keep in mind, I didn't include the SD. That is a secondary list of + and -. Which might soften the discarding of a faction.

It was fast indeed, but I am used to doing those comparrisons. It doesn't come with the explanation in it.
Sometimes looking back in one. I don't even know what I was doing in the first place. So I guess I need to clean up just a tad and punt in some colors and decision making.

I have a 16 hour work day today. So expect the email Saturday (dutch time)

I only went from imbalanced. To practical balance.

Perfect balance doesn't exist.

If you want to change the factions by picking the remains of faction 6. I suggest to work on faction 4 only. And with every switch, take note on the changes.