Distribution of secondary wand boosts

[Manxome]

You know how each student gets +1 in a couple random skills, aside from their main wand skill? I seem to get those secondary boosts in lightning & air about twice as often as in other elements. Anyone else notice something like this?

[Pwnyboy-Curtis]

I'm doing a mostly all-water run, and I noticed fire for sure seems rare. I don't know if I got a wild RNG distribution, or if it's bias against opposite elements. Literally a couple dozen initiates without ever seeing a +1 fire.

I did a single wizard start though, and that starting wizard had +1 fire on a water wand, so it's certainly possible. If I had to guess I'd say the distribution is bias toward closer elements. I'd love to hear something more definitive though. I mean it wouldn't be that hard to test, but I don't want to.

[perfectevil]

Originally posted by Manxome:

You know how each student gets +1 in a couple random skills, aside from their main wand skill? I seem to get those secondary boosts in lightning & air about twice as often as in other elements. Anyone else notice something like this?

Secondary wand? You cant get more than one wand, though if you mean leveling the wand to level 3, this will open for that students two trials and skill points

[Pwnyboy-Curtis]

10 water wand initiates. This was the distribution of random skills. This pretty much tracks with what I'd noticed already. Zero fire.

Nature / Earth 4
Air / Lightning 3
Earth / Lightning 1
Earth / Air 1
Dark / Earth 1

[Manxome]

Originally posted by Pwnyboy-Curtis:

10 water wand initiates. This was the distribution of random skills. This pretty much tracks with what I'd noticed already. Zero fire.

Nature / Earth 4
Air / Lightning 3
Earth / Lightning 1
Earth / Air 1
Dark / Earth 1

You're saying you got the exact combination of Nature+Earth four times? 70% of your wands used only 2 out of the 15 possible 2-element pairings they could have had? Suspicious. Were these 10 successive students, or did you reload a save to try to get more data?

In the 7-element circular system they use, each element has 2 opposites, and the opposites of water are fire & earth. You got earth 7/10 times so this doesn't seem to support an "opposite are less likely" hypothesis.


I've been recording data for 19 students so far. I only recorded individual frequencies and not pair frequencies, and I didn't record the main wand elements for each, but the largest group had water wands, there was a substantial number of earth wands, a few fire wands, and one air wand. My secondary elements were:

4 Fire
1 Water
4 Dark
4 Nature
6 Earth
10 Lightning
9 Air

Water being low is possibly because water was the most common wand element in this test set, but that doesn't explain Lightning/Air being way higher than everything else.

I also recorded 10 elements from tier 2 wand upgrades:

1 Dark
3 Nature
3 Lightning
3 Air

This is just data since I started formally recording; I was already suspicious about air & lightning being more common, but didn't record the mages that made me suspicious because that could introduce selection bias. But, for instance, out of my first 6 students at the start of the game, the one with an air wand had lightning as a secondary element, and 4 of the remaining 5 students had the exact combo of air+lightning. (These aren't included in my totals above.)

I suppose if I'm going to continue maybe I should start recording exact element combinations of each student, but mostly I wanted to check my gut impression that air + lightning were significantly more common, and my data so far seems to support that.

[Manxome]

Originally posted by perfectevil:

Secondary wand? You cant get more than one wand, though if you mean leveling the wand to level 3, this will open for that students two trials and skill points

Every new student starts with a skill max of 3 in their wand element, 2 in two other elements, and 1 in the remaining elements. I'm talking about the two elements that have a skill max of 2.

then, upgrading wand to tier 2 gives you +1 to the wand element and +1 to another element, and then upgrading to tier 3 gives you +1 to the wand element and +1 to two other elements again.

There is a breakdown of this in the top-left corner of the mage's character screen.

[ulzgoroth]

My faculty wands, obviously significantly biased. Except for the last one, all were hired before level 3 wands became available though a few have been upgraded:
Primary element, | element adds grouped by wand level

N | DA D EL
A | FE F FE
D | EL L
F | LA E
L | DA N
E | FW N DF
W | EA N
F | DL A
N | EL L AE

My current cohort of students, who haven't been selectively thinned though their primary elements certainly aren't random and they're just a small sample:

N | WA W DA
E | WN D FN
W | EL N DN
A | EL L DN

My tallies of extra elements across all mages:
A: 8
D: 9
F: 6
E: 10
L: 10
N: 8
W: 4

8 is the median and very nearly also the mean. Lightning was high, but not outrageously, alongside Earth. Water being rare is the big outlier in my wands, though given the small sample probably not a meaningful one.


The lightning is rather overrepresented in tier 1, though, for what that's worth, with 6 of the bumps being there, which is a pretty big overrepresentation out of 26.

[Pwnyboy-Curtis]

My sample was with level 1 water wands only, on fresh initiates, all summoned in a row. The combinations were the two elements they rolled together for the level 1 wand in addition to the +2 water you get automatically.

[D'Angelo]

I can confirm the rarity of Fire as a secondary skill. I got suspicious because I was specifically fishing for them and never seemed to get any. I started tracking each new cohort of 5 new students, rerolling them while rotating each student through each of the wand types. I didn't think Fire would ending up being as rare as it was (I thought perhaps I'd just gotten the short end of the RNG stick), else I would've kept a comprehensive record and tally of all of these rolls, but suffice to say I've gone through seven or eight of these groupings now, and within the previous 200-230ish students I have gotten exactly one student with Fire as a secondary wand skill.

edit: While my results suggested a frequency for Fire of below 1%, it looks like this was far outside the norm and that other variables likely play a role.

[Pwnyboy-Curtis]

Do you know if the type of student matters? Was it a fairly equal distribution, or was it all one type, like how my sample was all water?

[D'Angelo]

Originally posted by Pwnyboy-Curtis:

Do you know if the type of student matters? Was it a fairly equal distribution, or was it all one type, like how my sample was all water?

Each time I cycled through every wand type five at a time. Sometimes at the end if there were no desirable race/primary/secondary combos I would attune the first summon to a specific race which changes the subsequent RNG.

I should also note that Water as a secondary trait seems to be at a reduced frequency too, probably half the rate of the other elements, but again still far far more common than Fire.

I hope it's not too indecipherable, but this is the way I have been keeping track: https://i.imgur.com/Rifj8YF.png

The frequency of Nature as a secondary stat in this batch is atypical.

[ulzgoroth]

Originally posted by D'Angelo:

I can confirm the rarity of Fire as a secondary skill. I got suspicious because I was specifically fishing for them and never seemed to get any. I started tracking each new cohort of 5 new students, rerolling them while rotating each student through each of the wand types. I didn't think Fire would ending up being as rare as it was (I thought perhaps I'd just gotten the short end of the RNG stick), else I would've kept a comprehensive record and tally of all of these rolls, but suffice to say I've gone through seven or eight of these groupings now, and within the previous 200-230ish students I have gotten exactly one student with Fire as a secondary wand skill.

Since I don't have the exact numbers, I can't calculate my p-values or deviations or what have you, but just as a back-of-the-envelope guesstimate I don't think it's unreasonable to suggest that Fire shows up *at most* at a frequency of 1 in 100 students and is more likely at a frequency of somewhere around 1 in 250, if that.

I am almost sure my dataset disagrees with that...

Achievements say I've summoned 66 students. Most of them are gone, of course, but in my remaining set I've got:

An air mage with +fire at all 3 ranks, for a real outlier.
An earth mage who got +fire at 1 and 3.
A water mage who got +fire only at 3 (might count as a negative in your dataset)

That's all out of the 13 mages currently in my school and I have no way to check if there are examples in my retired pool, but 2 out of 66 is already not agreeing well with 'at most 1 in 100'.

[Manxome]

Since my previous post, I've summoned 36 new students, and their secondary boosts are:

2 fire (6%)
2 water (6%)
11 dark (31%)
10 nature (28%)
15 earth (42%)
17 lightning (47%)
15 air (42%)

I've started keeping exact track of the wand element as well, so I can tell you that this data includes only 1 fire wand (with EA secondary) and 2 water wands (NA and LA), so in a fair distribution fire and water should appear slightly above average as secondary elements (since they had more chances).

If you combine that with the 19 students from before I started recording wand elements, my totals are now:

6 fire
3 water
15 dark
14 nature
21 earth
27 lightning
24 air

This isn't consistent with fire being as low as a 1% chance, but it is looking like fire and water are less common, and earth/lightning/air are more common.

(These are all different students; I'm not using reloading tricks to retry.)

So far, I'm NOT seeing similar patterns at higher tiers. Out of 30 new tier two wands, I have:

3 fire (10%)
5 water (17%)
7 dark (23%)
7 nature (23%)
4 earth (13%)
0 lightning (0%)
4 air (13%)

And from 9 tier three wands, I have a basically flat distribution, but that's not a lot of data yet.

I'm starting to wonder if there's some hidden variable, like the distribution depends on the skills of the staff member who is doing the summoning or something.

[D'Angelo]

It is entirely possible that I have simply gotten the short end of the stick then. I've been looking for them specifically, so I definitely didn't miss any. That said, I'm interested in looking into the skill of the summoning staff member. Most of mine were summoned by staff with < 2 Fire.

Fire as low as 1% or lower didn't seem like purposeful design, so I'm glad to see that my results were outside of the norm. I'll edit my previous post to reflect that.

I also agree that Fire does not seem to be as rare in T2 and T3 wands.

[Manxome]

Note that I almost never summon with my lightning specialists, but lightning is my most common secondary boost, so it's probably not just "higher chance of the summoner's best elements."

I'm just grasping for explanations for why people don't seem to be getting quite the same results. Do the odds change depending on time of day? Does your entire school get an intentional skew that's randomized when you first start the game? Are they just using a terrible RNG with long sequences of similar outputs?

But maybe I'm looking too hard. Pwny-boy only posted 10 wands' worth of data (none of which had fire), ulzgoroth is mostly posting data of mages they happen to have on hand (which is filtered by who they chose to keep), and you, D'Angelo, are doing a save-scum thing that could be biasing the results.

In fact, looking again at that table you posted, I see a lot of vertical lines in your color matrices, which suggests that your results are highly correlated between retries. For instance, the first mage in your table got lightning + air literally every single time except when that would have been impossible, and earth in both of the remaining times.

Maybe you actually have a lot less data than you think you have. If you used each wand at least 7 times, and some of them more than that, then maybe your ~200 data rows are more like 20 distinct wands, in which case getting fire once is roughly in line with my data. This analogy doesn't quite work if you retried that wand 7 times and it only got fire in 1 out of those 7 tries, though.

[ulzgoroth]

Originally posted by Manxome:

But maybe I'm looking too hard. Pwny-boy only posted 10 wands' worth of data (none of which had fire), ulzgoroth is mostly posting data of mages they happen to have on hand (which is filtered by who they chose to keep), and you, D'Angelo, are doing a save-scum thing that could be biasing the results.

Yeah, I'm not running intentional experiments, and only a modest fraction of my samples are unfiltered students (and, as it happens, zero of the ones with tier 1 fire).

I would contest that I did not select any of those mages on the basis of their fire skill...but that doesn't change that they're definitely a non-random sample. (Sadly, that mage with +3 secondary fire was not apprenticed in fire.)


I'm terribly curious why the distribution would be as uneven as your data rather strongly suggests.