Misconceptions about growth rates

[Fore]

Context: There are a number of posts on this forum (not just this board) about growth rates and assertions about their related probabilities. They are typically in the form of 

Quote

Unit X has a growth rate of n, and unit Y has a growth rate of (n - 0.1). After ten level-ups, on average, unit X will have grown one additional point in speed relative to unit Y.

Equivalently stated, the assertion that is being made is that a single point of growth, over the course of 100 level-ups, is worth one stat point. While intuitive, this is not a true statement and is not a good way to think about growth related probabilities. 

---

Skimming over a few details, a Bernoulli random variable is a function which maps to either 0 or 1, according to some probability density function. In simpler terms, we we can say that P(X=1) = 0.25 if there is a 25% chance that random variable X maps to 1. (You can equivalently define X by stating that P(X=0) = 0.75.)

Suppose we are tossing a fair coin. We define random variable X to be the act of tossing the coin, and a heads is represented by a 0 and a tails is represented by a 1. Then, P(X=0) = P(X=1) = 0.5, and thinking about a single toss is straightforward. Now we have a way of formally asking, "when unit X levels up, how frequently do they a point of speed?".

Consider a game where we toss a coin 10 times and a success is defined as the coin landing on heads. The number of possible successes is [0, 10] inclusive. Let G be a random variable which describes the outcome of playing this game. We can define the probability of G as the summation of X over 10 trials, where P(X=1) = 0.50. Now we can formally ask, "when unit X levels up 10 times, how frequently will they gain exactly m points of speed?" by computing P(G=m). We can also ask more useful questions such as, "when unit X levels up L times, how frequently will they gain at least m points of speed?" by computing P(G_L>=m), where G_i is the random variable of the coin tossing game over i throws.

That was a mouthful, but we are almost there! A binomial distribution is defined as the sum of Bernoulli random variables -- in other words, the sum of evaluating a Bernoulli random variable over some number of trials. (Formally there are two requirements about those variables, namely that they be independent and identically distributed, but this is not important in the context of this discussion!) 

Okay, so what does that mean for us?

A unit, U_1, has a speed growth of 0.5. Over 10 levels, the probability that they will gain at least 5 points of speed is roughly 62%. In notation, P(G_U1>=5) = 0.623, the probability that over 10 Bernoulli trials, there will be at least five successes.

A unit, U_2, has a speed growth of 0.4 (i.e., the same unit is a Paladin). The probability that they will gain at least 4 points of speed is roughly 61%. Additionally, P(G_U2>=5) = 0.36, whereas P(G_U1>=6) = 0.38. 

The further you deviate from the center of the distribution, the more extreme these differences will become. The intuitive case is when you go from a growth of 0.1 to a growth of 0.0 over any number of level-ups. I encourage you to experiment with these numbers on your own, as probability can be extremely unintuitive.

https://www.wolframalpha.com/examples/mathematics/probability/bernoulli-trials/

Thanks for reading, this is a pet peeve of mine.

 

tl;dr: 1% of growth != 1 stat gain over 100 level-ups, and stat growths can be perfectly modeled using Bernoulli trials or a binomial distribution. 

Edited December 10, 2019 by Fore

[Eltosian Kadath]

 

2 hours ago, Fore said:

Quote

Unit X has a growth rate of n, and unit Y has a growth rate of (n - 0.1). After ten level-ups, on average, unit X will have grown one additional point in speed relative to unit Y.

Equivalently stated, the assertion that is being made is that a single point of growth, over the course of 100 level-ups, is worth one stat point. While intuitive, this is not a true statement and is not a good way to think about growth related probabilities. 

You may notice the use of the phrase "on average" (I have even made it bold for emphasis) in the quote being use, that is a key word that makes it clear that people are talking about an average, also known as the mean, the first central moment, and far more importantly the expected value. What you claim is an equivalent statement is not equivalent because it describes the situation as if it were a certainty, instead of the expected value. While some people might be using the phrase inaccurately (and it is nice of you to inform others about Bernoulli numbers and binomial distributions), I think you have misunderstood what people were trying to say. For those experimenting with the Wolfram Alpha examples, they might notice that the "expected number of successes" number to be the average that people are referencing in the quote.

[Jotari]

Base stats are more important anyway tbf.

[Shanty Pete's 1st Mate]

I think it's worth keeping in mind that there are multiple, valid ways of looking at probabilities. "Expected value" is one such way (ex. 50% growth means, on average, 5 points over 10 level-ups).

6 hours ago, Fore said:

The further you deviate from the center of the distribution, the more extreme these differences will become. The intuitive case is when you go from a growth of 0.1 to a growth of 0.0 over any number of level-ups. I encourage you to experiment with these numbers on your own, as probability can be extremely unintuitive.

Have to disagree here. The average stat difference between 0% growth and 10% growth, over X levels, is equal to the average stat difference between 10% and 20% growths over X levels. In an arithmetic game like Fire Emblem, this means these margins are equivalent.

I'm not opposed to looking at the binomial distribution, and I think there's definitely value in considering potential stat variability (ex. A stat with a 50% growth has a higher chance of departing from the norm, positively or negatively, than a 10% or 90% growth). But I think it's a mistake to shun expected values, when it's the best tool we have to take an (admittedly simplified) look at a unit's average stat performance. 

[Whisky]

You are right in what are you talking about, and you did a pretty good job of explaining it. Thank you for explaining this to people that may not know. Hopefully they find it helpful.

 

However,

As others have said, I think you misunderstand what others are talking about when they talk about average stats. I think most people completely understand that the ‘average’ is not at all a certainty, and that it can deviate from there.

 

I agree with Shanty Pete about this. I my self do use Binomial Distribution when analyzing and comparing units in Fire Emblem.
 

There is absolutely no certainty to getting the expected value but that value is the one in the middle, with the most likely probability out of any value. And it deviates from there in either direction, with the probability decreasing, the further away you get from the center.

 

Units are very likely to have within 2 points in either direction of their average. This gives us a good idea of what to expect, knowing full well that this still isn’t a guarantee (I’m sure we have all died to 1% Crits. Murphy’s law.). Just listing the averages of units is much easier to write and read. I’m sure most people understand that units are not guaranteed to have that average. And because units are pretty much equally likely to be above or below that average, it’s a pretty good estimate.

 

If someone treats the average or expected value as a guarantee in unit analysis, then I agree that that would be a mistake. Variance in units stats should be considered in unit analyses and comparisons. This is why people say that bases are more important than growths. Because they are a guarantee. However, I would not call It incorrect to use averages.

Edited December 10, 2019 by Whisky

[Jotari]

12 minutes ago, Whisky said:

You are right in what are you talking about, and you did a pretty good job of explaining it. Thank you for explaining this to people that may not know. Hopefully they find it helpful.

 

However,

As others have said, I think you misunderstand what others are talking about when they talk about average stats. I think most people completely understand that the ‘average’ is not at all a certainty, and that it can deviate from there.

 

I agree with Shanty Pete about this. I my self do use Binomial Distribution when analyzing and comparing units in Fire Emblem.
 

There is absolutely no certainty to getting the expected value but that value is the one in the middle, with the most likely probability out of any value. And it deviates from there in either direction, with the probability decreasing, the further away you get from the center.

 

Units are very likely to have within 2 points in either direction of their average. This gives us a good idea of what to expect, knowing full well that this still isn’t a guarantee (I’m sure we have all died to 1% Crits. Murphy’s law.). Just listing the averages of units is much easier to write and read. I’m sure most people understand that units are not guaranteed to have that average. And because units are pretty much equally likely to be above or below that average, it’s a pretty good estimate.

 

If someone treats the average or expected value as a guarantee in unit analysis, then I agree that that would be a mistake. Variance in units stats should be considered in unit analyses and comparisons. This is why people say that bases are more important than growths. Because they are a guarantee. However, I would not call It incorrect to use averages.

I'd also add that I think people do take this into consideration. I can at least pull out the example of Eliwood, who has very middling growths across the board, but generally isn't considered a good unit because of how well he performs is subject to a lot of variance (and his legendary weapon is way too heavy for him). People generally do not consider 40-50% a good growth rate even though it's a stat every other level (depending on the stat and relative growth rate in the game of course). It's the units with 60-70% growths in one or two of their key stats and 20-30% in their other stats that are usually considered good growth units. Hector only has a 10% higher total growth rate than Eliwood (and that extra 10% is in HP, which is the least valuable stat point for point) and Eliwood even beats him by 1 in bases, yet Hector is considered a far better unit because his stats are more focused (and also he can use a hand axe).

Edited December 10, 2019 by Jotari

[Whisky]

13 minutes ago, Jotari said:

I'd also add that I think people do take this into consideration. I can at least pull out the example of Eliwood, who has very middling growths across the board, but generally isn't considered a good unit because of how well he performs is subject to a lot of variance (and his legendary weapon is way too heavy for him). People generally do not consider 40-50% a good growth rate even though it's a stat every other level (depending on the stat and relative growth rate in the game of course). It's the units with 60-70% growths in one or two of their key stats and 20-30% in their other stats that are usually considered good growth units. Hector only has a 10% higher total growth rate than Eliwood (and that extra 10% is in HP, which is the least valuable stat point for point) and Eliwood even beats him by 1 in bases, yet Hector is considered a far better unit because his stats are more focused.

This is what I've seen in my anecdotal experience as well. I have had very different versions of Eliwood in different playthroughs. With some characters, you have a good idea of what to expect. With Lyn, you pretty much know that she will be fast. Hector will be strong. But with 'well rounded' characters like Eliwood, you could end up with pretty much anything. He's kind of a wild card in that sense.

 

Higher growth rates are generally more reliable than lower growth rates. Especially if a character is likely to reach  their cap in a stat, or if they are likely to reach a higher value than they need for a specific milestone. In those situations, even if they get unlucky level ups, and end up below their average, they will likely still have good enough of that stat to do what they need to do. Hopefully, getting RNG screwed in their lower growth stats doesn't hurt them too much though, like Lyn with her low Str and nonexistent durability.

Edited December 10, 2019 by Whisky

[Shanty Pete's 1st Mate]

If anyone is looking for a Bernoulli trial calculator that works on mobile devices, here it is.

2 hours ago, Whisky said:

Higher growth rates are generally more reliable than lower growth rates.

Have to disagree a smidge - the "least reliable" growth rate, in the sense of most variable, is 50%. Very low growth rates tend to produce reliably little gains, whereas high growth rates produce reliably high gains.

2 hours ago, Whisky said:

 

Units are very likely to have within 2 points in either direction of their average. This gives us a good idea of what to expect, knowing full well that this still isn’t a guarantee (I’m sure we have all died to 1% Crits. Murphy’s law.). Just listing the averages of units is much easier to write and read. I’m sure most people understand that units are not guaranteed to have that average. And because units are pretty much equally likely to be above or below that average, it’s a pretty good estimate.

Building off of this, perhaps there would be value in IDing "stat thresholds" for certain units. Like, at level 11, Eliwood's (borrowing him from @Jotari's example) strength is above X, and below Y, 95 percent of the time. Knowing that, or some equivalent to "standard deviation", could show how often to expect variability from averages in a given unit.

[Whisky]

46 minutes ago, Shanty Pete's 1st Mate said:

If anyone is looking for a Bernoulli trial calculator that works on mobile devices, here it is.

Oh nice! That’s cool. Thank you.

Quote

Have to disagree a smidge - the "least reliable" growth rate, in the sense of most variable, is 50%. Very low growth rates tend to produce reliably little gains, whereas high growth rates produce reliably high gains.

Yes that’s true. What I meant by more reliable is that you can rely on Lyn having good Spd more than you can with Eliwood.  Even if Lyn gets unlucky she will still have pretty good Spd. (I’m not arguing that Lyn is better than Eliwood. This is just an example.)

Quote

Building off of this, perhaps there would be value in IDing "stat thresholds" for certain units. Like, at level 11, Eliwood's (borrowing him from @Jotari's example) strength is above X, and below Y, 95 percent of the time. Knowing that, or some equivalent to "standard deviation", could show how often to expect variability from averages in a given unit.

Yeah, we could do that. I think there is definitely merit to that. Averages are just a lot easier to use.

[eclipse]

1. This topic is as old as growth rates.
2. Unless you're doing a run that needs exact stats to do certain things, growth rates are a benchmark.
3. Yes, we know that growths can deviate.
4. However, "freaks of nature" (like my 9 Res Deke in a FE6 run) don't really contribute to a general discussion of which character will do better.
5. Since there's some auto-leveling involved, growth rates do matter, BUT. . .
6. Over the course of 39 levels, a 10% growth rate difference is a maximum difference of roughly four stats.  Most of the time, we're not hitting level 40.  And in those places where that growth is crucial (usually early-on), it isn't going to make a huge impact unless number 2 on this list is relevant.  The first time a 10% difference growth rate shows its worth will be ten levels from base.
7. Therefore, why are you telling the greater community this?  Are you sure you're on the right forum?  Because if this is a "general PSA", then direct it to the actual site where you saw this problem.  If it's here, then respond to the posts directly.  Not this.  I don't care for passive-aggressive nonsense, and this is exactly it.
8. And while you're at it, post it to the correct subform next time.

EDIT: Words are hard.

Edited December 10, 2019 by eclipse

[AnonymousSpeed]

2 hours ago, eclipse said:

Over the course of 39 levels, a 10% growth rate difference is a maximum difference of roughly four stats.

Actually, a unit with a 10% higher growth could have as much as 39 more in a stat- it's extremely unlikely, but 4 is far from the maximum. An increase of 5 or 6 is far from unimaginable in this instance.

***

@Fore Having recently taken a statistics class, I too was wondering how what I learned could be applied to Fire Emblem. I think there was one point I considered making an excel sheet to calculate how likely a unit was to reach a certain threshold by a certain level, which would be an even more reliably metric than averages.

The debate scene is unfortunately a bit dead, but this would be very interesting to compare between units. 

[X-Naut]

2 hours ago, AnonymousSpeed said:

Actually, a unit with a 10% higher growth could have as much as 39 more in a stat- it's extremely unlikely, but 4 is far from the maximum. An increase of 5 or 6 is far from unimaginable in this instance.

The only way that would be true is if you leveled both units exclusively in FE12 Drill Grounds, where gains are bracketed by growth total. But in most cases how many stats you're getting per level is up to RNG- for what it's worth, the 10% lower growth unit could end up ten points ahead of his peer, possibly even before they promote.

[eclipse]

19 hours ago, AnonymousSpeed said:

Actually, a unit with a 10% higher growth could have as much as 39 more in a stat- it's extremely unlikely, but 4 is far from the maximum. An increase of 5 or 6 is far from unimaginable in this instance.

In debates, averages are used.  Otherwise, I'm going to argue that Wil is the best archer due to that one I got in a draft that maxed Str/Spd.

I don't think it was advanced enough for deviations, but:

1. Units didn't get their full 39 levels (usually due to an early promotion)
2. Unless it's a unit with a bunch of 50% growths in places that matter, it wasn't that big of a deal

IIRC it was more about bases and join time anyway.  Growths had some bearing, but that didn't invalidate the token prepromote's contributions early-on (or in Seth's case, the entire game).

[AnonymousSpeed]

3 hours ago, eclipse said:

In debates, averages are used.  Otherwise, I'm going to argue that Wil is the best archer due to that one I got in a draft that maxed Str/Spd.

What you said was this:

On 12/10/2019 at 3:59 PM, eclipse said:

Over the course of 39 levels, a 10% growth rate difference is a maximum difference of roughly four stats.

A "maximum difference", not an "average increase"

***

3 hours ago, eclipse said:

IIRC it was more about bases and join time anyway. 

I won't dispute that bases are better than growths. Bases are king.

***

20 hours ago, X-Naut said:

The only way that would be true is if you leveled both units exclusively in FE12 Drill Grounds, where gains are bracketed by growth total.

Wait, you mean something like stat caps? I'm afraid I don't follow this part- I haven't played FE12, which may leave me a bit out of the loop.

[X-Naut]

3 hours ago, AnonymousSpeed said:

Wait, you mean something like stat caps? I'm afraid I don't follow this part- I haven't played FE12, which may leave me a bit out of the loop.

Drill Grounds basically takes you GRT/100 and gives you that many points when you get a level. A unit with 330% total growths and nothing capped will get three points with a 30% chance of a fourth.

And you could do more to explain your "a unit with a 10% higher growth could have as much as 39 more in a stat" because I don't know how you're getting an exact figure when growth procs are RNG.

[AnonymousSpeed]

24 minutes ago, X-Naut said:

And you could do more to explain your "a unit with a 10% higher growth could have as much as 39 more in a stat" because I don't know how you're getting an exact figure when growth procs are RNG.

What I was trying to say is that, because growth rates are random, you could have a scenario like this:

Let there be two units, one of whom has a 10% higher growth rate in some stat than another. Over the course of 39 levels, it is possible (though improbable) for the unit with the higher growth rate to increase that stat every level and for the unit with the lower growth rate to never increase that stat.

Sorry if that wasn't clear.

[eclipse]

3 hours ago, AnonymousSpeed said:

What you said was this:

A "maximum difference", not an "average increase"

 

Using the lens of proper debate, which is strictly averages.  So using that context, it is a maximum four point difference.

Reality will most likely be different, but for the purposes of this topic (the root of which is character evaluation), it's irrelevant.  If you want to introduce standard deviation, do the math for that specific debate.

Edited December 12, 2019 by eclipse

[Whisky]

9 minutes ago, eclipse said:

Using the lens of proper debate, which is strictly averages.  So using that context, it is a maximum four point difference.

Reality will most likely be different, but for the purposes of this topic (the root of which is character evaluation), it's irrelevant.  If you want to introduce standard deviation, do the math for that specific debate.

I’m going to have to agree with Anonymous here. You specifically said “maximum” not ‘average’. You didn’t say the difference would be ‘around’ 4 points. Or anything of the sort. You specified “maximum”, which is completely inaccurate. It’s actually rather unlikely for the difference to be exactly the average value. It is very likely to be within 2 points form the average in either direction. The average is a good estimate, but it is inaccurate to assume that the difference will exactly equal the average value.

[eclipse]

3 minutes ago, Whisky said:

I’m going to have to agree with Anonymous here. You specifically said “maximum” not ‘average’. You didn’t say the difference would be ‘around’ 4 points. Or anything of the sort. You specified “maximum”, which is completely inaccurate. It’s actually rather unlikely for the difference to be exactly the average value. It is very likely to be within 2 points form the average in either direction. The average is a good estimate, but it is inaccurate to assume that the difference will exactly equal the average value.

Did you miss "the lens of debate, where standard deviation has yet to be meaningfully introduced, and growths are strictly averages"?  Because if you're going to ignore this, then you're not arguing in good faith.

[Whisky]

1 minute ago, eclipse said:

Did you miss "the lens of debate, where standard deviation has yet to be meaningfully introduced, and growths are strictly averages"?  Because if you're going to ignore this, then you're not arguing in good faith.

I’m confused. We all understand that units are unlikely to have their exact average. The average is just that, the average. It’s a good estimate for unit analysis, and unless as you said, we are trying to find when a unit will reach a very specific benchmark, then the average is probably a good enough estimate. But if “lens of debate” means assuming that growths are in fact strictly exact averages, then that’s a completely flawed format. Also you said “lens of proper debate”. So you’re enforcing a “proper” way to have a debate? That seems pretty close minded.

[eclipse]

2 minutes ago, Whisky said:

I’m confused. We all understand that units are unlikely to have their exact average. The average is just that, the average. It’s a good estimate for unit analysis, and unless as you said, we are trying to find when a unit will reach a very specific benchmark, then the average is probably a good enough estimate. But if “lens of debate” means assuming that growths are in fact strictly exact averages, then that’s a completely flawed format. Also you said “lens of proper debate”. So you’re enforcing a “proper” way to have a debate? That seems pretty close minded.

Yes, it is closed-minded.  But lemme explain how debates worked way back when:

Every character had a bunch of things about them, such as join time, bases (both regular stats and weapon ranks, among others), class, and growths.  Costs included "how much does this unit benefit from X as opposed to Y?" (makes more sense if you put this in terms of something like "the Energy Ring from Lyn Mode" or "the first Master Seal in FE12"), "how much does this unit contribute?", "is it worth putting experience in this unit?", etc.  The last question was a little weird for FE7 ranked runs, since there was an experience rank, so even low-leveled late-joiners had a purpose.

Generally, bases, join time, and class put units roughly in order on a tier list.  Growths were more important for the early-joining units who could contribute during that critical phase, and were generally used in comparison with a later-joining unit of the same class.

I can attempt an argument for two units that fall under all the criteria, namely Kent vs. Isadora (assuming Lyn Hard Mode, because I'm insane).  Kent will most likely be leveled to 10 and promoted before the end of Lyn Mode.  Thus, he'll most likely be 10/1 on joining (not sure if killing the boss on Chapter 10 would be enough to get him to 10/2).

Averages kick in if you compare Kent's average stats (no he's not getting either stat booster) when he rejoins versus Isadora's starting stats.

Name     HP  Str  Skl  Spd  Lck  Def  Res
Kent     29   10   11   12   3    9    4
Isadora  28   13   12   16   10   8    6

All this shows IMO is that Isadora's facing less crit than Kent forever.

This is what "strict averages" mean, so the debate would be "can Kent gain enough experience such that he can close the gap between him and Isadora during the five or so chapters he's there and she's not?" for these kinds of stats, using his growth rates.

The idea of standard deviation, if implemented correctly, would probably do some good.  For example, Kent's HP growth is 85%, so he's fairly likely to hit (if not exceed) his HP average.  Meanwhile, his Luck is a mere 20%, so he's more likely to undershoot that average instead of overshoot it.

If you could implement average stats with standard deviation, and argue it properly, it would make an interesting read (especially given the growth rates in the more recent releases).  IMO it's a lot of work.

[Jotari]

1 hour ago, eclipse said:

Yes, it is closed-minded.  But lemme explain how debates worked way back when:

Every character had a bunch of things about them, such as join time, bases (both regular stats and weapon ranks, among others), class, and growths.  Costs included "how much does this unit benefit from X as opposed to Y?" (makes more sense if you put this in terms of something like "the Energy Ring from Lyn Mode" or "the first Master Seal in FE12"), "how much does this unit contribute?", "is it worth putting experience in this unit?", etc.  The last question was a little weird for FE7 ranked runs, since there was an experience rank, so even low-leveled late-joiners had a purpose.

Generally, bases, join time, and class put units roughly in order on a tier list.  Growths were more important for the early-joining units who could contribute during that critical phase, and were generally used in comparison with a later-joining unit of the same class.

I can attempt an argument for two units that fall under all the criteria, namely Kent vs. Isadora (assuming Lyn Hard Mode, because I'm insane).  Kent will most likely be leveled to 10 and promoted before the end of Lyn Mode.  Thus, he'll most likely be 10/1 on joining (not sure if killing the boss on Chapter 10 would be enough to get him to 10/2).

Averages kick in if you compare Kent's average stats (no he's not getting either stat booster) when he rejoins versus Isadora's starting stats.

Name     HP  Str  Skl  Spd  Lck  Def  Res
Kent     29   10   11   12   3    9    4
Isadora  28   13   12   16   10   8    6

All this shows IMO is that Isadora's facing less crit than Kent forever.

This is what "strict averages" mean, so the debate would be "can Kent gain enough experience such that he can close the gap between him and Isadora during the five or so chapters he's there and she's not?" for these kinds of stats, using his growth rates.

The idea of standard deviation, if implemented correctly, would probably do some good.  For example, Kent's HP growth is 85%, so he's fairly likely to hit (if not exceed) his HP average.  Meanwhile, his Luck is a mere 20%, so he's more likely to undershoot that average instead of overshoot it.

If you could implement average stats with standard deviation, and argue it properly, it would make an interesting read (especially given the growth rates in the more recent releases).  IMO it's a lot of work.

What's insane about Lyn hard mode?

[Whisky]

12 hours ago, eclipse said:

Yes, it is closed-minded.  But lemme explain how debates worked way back when:

Every character had a bunch of things about them, such as join time, bases (both regular stats and weapon ranks, among others), class, and growths.  Costs included "how much does this unit benefit from X as opposed to Y?" (makes more sense if you put this in terms of something like "the Energy Ring from Lyn Mode" or "the first Master Seal in FE12"), "how much does this unit contribute?", "is it worth putting experience in this unit?", etc.  The last question was a little weird for FE7 ranked runs, since there was an experience rank, so even low-leveled late-joiners had a purpose.

Generally, bases, join time, and class put units roughly in order on a tier list.  Growths were more important for the early-joining units who could contribute during that critical phase, and were generally used in comparison with a later-joining unit of the same class.

I can attempt an argument for two units that fall under all the criteria, namely Kent vs. Isadora (assuming Lyn Hard Mode, because I'm insane).  Kent will most likely be leveled to 10 and promoted before the end of Lyn Mode.  Thus, he'll most likely be 10/1 on joining (not sure if killing the boss on Chapter 10 would be enough to get him to 10/2).

Averages kick in if you compare Kent's average stats (no he's not getting either stat booster) when he rejoins versus Isadora's starting stats.

Name     HP  Str  Skl  Spd  Lck  Def  Res
Kent     29   10   11   12   3    9    4
Isadora  28   13   12   16   10   8    6

All this shows IMO is that Isadora's facing less crit than Kent forever.

This is what "strict averages" mean, so the debate would be "can Kent gain enough experience such that he can close the gap between him and Isadora during the five or so chapters he's there and she's not?" for these kinds of stats, using his growth rates.

The idea of standard deviation, if implemented correctly, would probably do some good.  For example, Kent's HP growth is 85%, so he's fairly likely to hit (if not exceed) his HP average.  Meanwhile, his Luck is a mere 20%, so he's more likely to undershoot that average instead of overshoot it.

If you could implement average stats with standard deviation, and argue it properly, it would make an interesting read (especially given the growth rates in the more recent releases).  IMO it's a lot of work.

When you explain it like that, I don't have anything to disagree with. That all makes sense to me. Of course that is only one situation. We could also compare them without promoting Kent in LHM or even skipping LHM completely. 

For the most part, Averages give us a good enough estimate. In one playthrough, someone might get an unlucky Kent and Isadora will be better. Someone else might get a lucky Kent that is definitely better than Isadora. But there is a reason people don't use personal experience in unit comparisons. Averages are literally the average of all possible outcomes. You may get lucky or unlucky in any given playthrough. Maybe if Kent gets unlucky in your playthrough, you will decide to bench him for Isadora, but in unit comparison, we have to assume that Kent will close to his average. The only thing we would be able to say, is that it is possible for Kent to get unlucky, and not reach his average. Where as Isadora's base stats are guaranteed. I don't know how much that matters though.

I think for the most part, the only way implementing Standard Deviation would actually matter is if there are specific stat benchmarks to reach by a specific level. We can calculate the probability of Kent being above or below specific stat benchmarks, but I don't know what those benchmarks would be, or why they need to be reached by a specific level. If a unit's average is a certain amount, such as Kent having 12 Spd in the above example, he may or may not reach 12 Spd by that level. But even if he doesn't, he probably will in a few more levels. We could estimate the level that Kent will be, at the time that Isadora joins, and then calculate the probability of Kent having better or worse stats than Isadora at that time. But I'm still not sure how much this would matter. Besides, we already know that if his average his above her base stats, then he will most likely be have higher stats than her. That's probably a good enough estimate.

[eclipse]

13 hours ago, Jotari said:

What's insane about Lyn hard mode?

Figuring out where those stat boosters go (definitely to Wallace).

There's also that issue of "which cavalier do I promote" and whatnot.

2 hours ago, Whisky said:

When you explain it like that, I don't have anything to disagree with. That all makes sense to me. Of course that is only one situation. We could also compare them without promoting Kent in LHM or even skipping LHM completely.

Both of these will knock Kent down significantly.  But you have the right idea!

2 hours ago, Whisky said:

For the most part, Averages give us a good enough estimate. In one playthrough, someone might get an unlucky Kent and Isadora will be better. Someone else might get a lucky Kent that is definitely better than Isadora. But there is a reason people don't use personal experience in unit comparisons. Averages are literally the average of all possible outcomes. You may get lucky or unlucky in any given playthrough. Maybe if Kent gets unlucky in your playthrough, you will decide to bench him for Isadora, but in unit comparison, we have to assume that Kent will close to his average. The only thing we would be able to say, is that it is possible for Kent to get unlucky, and not reach his average. Where as Isadora's base stats are guaranteed. I don't know how much that matters though.

For debate, not at all.  For an Actual Run, you'd throw a couple of levels into both Sain and Kent, and see if one pulls ahead.  Then, go with that one.  Or, you put everything into Wil.  😛

It's why bases are far more important!

2 hours ago, Whisky said:

I think for the most part, the only way implementing Standard Deviation would actually matter is if there are specific stat benchmarks to reach by a specific level. We can calculate the probability of Kent being above or below specific stat benchmarks, but I don't know what those benchmarks would be, or why they need to be reached by a specific level. If a unit's average is a certain amount, such as Kent having 12 Spd in the above example, he may or may not reach 12 Spd by that level. But even if he doesn't, he probably will in a few more levels. We could estimate the level that Kent will be, at the time that Isadora joins, and then calculate the probability of Kent having better or worse stats than Isadora at that time. But I'm still not sure how much this would matter. Besides, we already know that if his average his above her base stats, then he will most likely be have higher stats than her. That's probably a good enough estimate.

Kent's speed growth is 50%, which IIRC is where the biggest deviations occur.  Which means Kent needs to be on his averages, or he'll fall behind in raw stats. . .

. . .except GBA FEs are a little deceptive when it comes to speed.  Kent has 3 more CON than Isadora, meaning that he gains speed if he and Isadora wield a weapon with more than 6 WT - like Javelins.

The last bit of methodology is correct.  Don't know if it'll translate to 3H due to auto-leveling and variable recruitment time.  But if you wan to give it a shot, have fun!