Establishing how Critical Rate works

DrYoshiyahuDrYoshiyahu
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edited 11:46PM May 15, 2018 in Archer
If you're wondering whether it's better to put your ability points into Critical Rate or Dexterity, use the attack score in the character page to determine which is better for you. In my experience, this would be Critical Rate.

Whatever the number in the character stat screen is, it's not your actual critical chance. It's some arbitrary number from 0 to 235 (without equips) that increases your critical chance. We know that critical chance maxes out at 40%, and we know that it's one of the main stats of Archers, and plays a key role in a handful of Archer skills. So the question is, how does Critical Rate work?

I did a test where I shot 500 arrows with my Critical Rate at the minimum and maximum stats I could give it. In this case, those numbers were 55 and 151. Then I pumped up Sharp Eyes and took it all the way to 256. Be aware that 500 hits is still a small sample size, and without some insight from Nexon, a lot more tests would be necessary to get a good understanding of this variable.

With 55 Critical Rate, 7% of my shots were crits.
With 151 Critical Rate, 13% of my shots were crits.
With 256 Critical Rate, 20% of my shots were crits.

This could mean a few things. It could mean the numbers are way off margin, and more tests need to be done, or it could mean that Critical Rate is non-linear, and has diminishing returns. I would assume it's the former.

Once the level cap is increased, higher-level equips are available, and we can start putting our Critical Rate into the 300s, it will be worth trying again.

Elopat

Comments

  • lolzfaillolzfail
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    edited 11:49PM May 11, 2018
    Hi, Based on your data I would assume the simplest formula would be the square root of critical rate, which would be similar to the accuracy formula in Maplestory 1. This achieves diminishing marginal return, while being reasonably accurate for 55 and 151. I do not see a reason to have a complicated crit rate formula so this is possibly the correct formula. As for how you got 20% in 500 hits for 256 crit, there is a 98.53% chance that it would lie between 12 and 20%, so 20% still lies in the 99% confidence interval. However, I would recommend re-testing it to be sure.

    In addition to your suggested test, I would like to add that it would be good to test crit rate on low level and high level monsters to see if monster level affects crit rate.

    Also, I would like to add minimum number of HITS to achieve certain confidence intervals:

    55 crit, predicted 7.4162%, 95% confidence interval, ±0.74162 allowance (10%) : 4796 hits minimum
    55 crit, predicted 7.4162%, 95% confidence interval, ±0.0074162 allowance (1%): 479586 hits minimum
    55 crit, predicted 7.4162%, 95% confidence interval, ±0.5 allowance : 10551 hits minimum

    151 crit, predicted 12.2882%, 95% confidence interval, ±1.22882 allowance (10%) : 2743 hits minimum
    151 crit, predicted 12.2882%, 95% confidence interval, ±0.122882 allowance (1%) : 274209 hits minimum
    151 crit, predicted 12.2882%, 95% confidence interval, ±1 allowance : 4141 hits minimum
    151 crit, predicted 12.2882%, 95% confidence interval, ±0.5 allowance (10%) : 16563 hits minimum

    256 crit, predicted 16%, 95% confidence interval, ±1.6 allowance (10%) : 2017 hits minimum
    256 crit, predicted 16%, 95% confidence interval, ±0.16 allowance (1%) : 201684 hits minimum
    256 crit, predicted 16%, 95% confidence interval, ±1 allowance : 5164 hits minimum
    256 crit, predicted 16%, 95% confidence interval, ±0.5 allowance : 20653 hits minimum

    55 crit, predicted 7.4162%, 95% confidence interval, ±0.74162 allowance (10%) : 8285 hits minimum
    55 crit, predicted 7.4162%, 95% confidence interval, ±0.0074162 allowance (1%): 828411 hits minimum
    55 crit, predicted 7.4162%, 95% confidence interval, ±0.5 allowance : 18226 hits minimum

    151 crit, predicted 12.2882%, 99% confidence interval, ±1.22882 allowance (10%) : 4737 hits minimum
    151 crit, predicted 12.2882%, 99% confidence interval, ±0.122882 allowance (1%) : 473655 hits minimum
    151 crit, predicted 12.2882%, 99% confidence interval, ±1 allowance : 7153 hits minimum
    151 crit, predicted 12.2882%, 99% confidence interval, ±0.5 allowance (10%) : 28609 hits minimum

    256 crit, predicted 16%, 99% confidence interval, ±1.6 allowance (10%) : 3484 hits minimum
    256 crit, predicted 16%, 99% confidence interval, ±0.16 allowance (1%) : 348379 hits minimum
    256 crit, predicted 16%, 99% confidence interval, ±1 allowance : 8919 hits minimum
    256 crit, predicted 16%, 99% confidence interval, ±0.5 allowance : 35673 hits minimum

    Basically if you test at least the minimum number of hits, and it is outside the allowance, there is a (100-x)% chance of it happening if my formula is correct, which means it's quite unlikely for my formula to be correct. On the other hand, if it lies within the allowance my formula is reasonable. For example, if you have 256 crit, and hit 9001 times, and 14.8% of them were crits, then there is less than 1% chance that it would happen based on my formula(99% confidence interval with ±1 allowance is 8919 minimum hits), so my formula is likely inaccurate. However, if 15.5% of them were crits it lies in 16±1%, so my formula is reasonably accurate.

    Another theory I have is that we have positive base crit rate at 0 crit, say 3%, and crit linearly contributes to crit rate. Considering there is a cap on crit rate(40%) and points added to crit(60?), the disadvantage of a linear system is somewhat negated. Also, we would need 1600 crit in order to hit 40% by the square root model, which is ridiculous considering we can only get 180 crit from AP, and it does not grow with level(?). On the other hand, the linear model would achieve cap crit at around 475 crit.

    Sadly I have no beta key so I can't test it out myself, but do try to do as many hits as possible, and do more hits the lower your crit is. 1 million hits is ideal, but not realistic, unless we have many people testing with the same crit. However, having different values for crit also help significantly by letting us know whether the crit rate growth is linear or diminishing. Try to do at least 5000 hits, since 55 is the base crit for archers, I assume. 10k to 100k would be nice.
    DrYoshiyahu
  • lolzfaillolzfail
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    Leave a comment below with your Critical Rate and often you got a crit.

    Also include the number of hits please, to ensure accuracy of the cumulative data.
    DrYoshiyahu
  • DrYoshiyahuDrYoshiyahu
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    @lolzfail Thanks for all that. That's awesome!

    If you're correct though, that means 1600 Critical Rate would be necessary to get 40% critical chance, which is so much higher than I think anyone was expecting (or wanting).

    Then again, I was pretty shocked at how low the rates were with Sharp Eyes. Never mind how low the maximum is. I mean, I actually did get to 100% on my Bowmaster in GMS1, so being relegated to a max of 40% is a pretty harsh blow.
    lolzfail
  • lolzfaillolzfail
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    No problem! I happened to just have done statistics homework last week so all the calculations I did were already on my worksheets haha. And yes, MS1 crit rates were so high, I got 100% on a f/p xD.
    Also, I added in the possibility of a linear model that does not start at 0, which would allow for crit chance to be capped at a more realistic 475. We would need more data from different crit stats to decide which is more plausible though. Maybe adding 1 point in se at a time, and testing with and without weapons for each level of se. This would be rather time consuming though, but with more data points we can see the shape of the graph better.
    DrYoshiyahu
  • lolzfaillolzfail
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    edited 8:30AM May 13, 2018
    It seems, according to 2 of your screenshots, that luk affects crit rate invisibly:
    1kjAkof.jpg
    okjXWau.jpg

    Assuming my formula for the displayed attack is correct:

    VA = (WA + A * PA) * (1 + (CD - 1) * CR)

    Then assuming you do have 7% crit rate at 55 crit and 27 luk, we have

    227 = A * 230 * (1 + 0.25 * 0.07)
    229 = A * 230 * (1 + 0.25 * X)

    where X is the new crit rate at 61 luk. Bearing in mind that attack is most probably rounded down, we have that 8.8% < X < 10.6%.

    I have a suspicion that luk gives 0.1% crit rate per point, and crit gives 0.2% per 3 points(1 AP). This should fit your observations rather well. However, as usual, more data is needed to reach a conclusive decision.

    If possible, could you add luk until your displayed attack increases by 1, note down that amount, then add it until it increases by 2 and note down that new amount. This would help in narrowing down the range for the contribution of luk as it minimises the rounding error, also same for crit.
    DrYoshiyahu
  • DrYoshiyahuDrYoshiyahu
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    edited 5:00AM May 13, 2018
    Yeah, so, it seems that LUK is influencing the critical hit chance (actual liklihood of critting) but not the Critical Rate (arbitrary variable used for calculations). Unfortunately, the wording used is "Affects critical rate", so that's just misleading. But yeah, let me do your test and get back to you.

    227 ATT: 27 LUK ~ 40 LUK
    228 ATT: 41 LUK ~ 58 LUK
    229 ATT: 59 LUK ~ ?

    227 ATT: 55 CRIT ~ 70 CRIT
    228 ATT: 73 CRIT ~ 91 CRIT
    229 ATT: 94 CRIT ~ 115 CRIT
    230 ATT: 118 CRIT ~ 136 CRIT
    231 ATT: 139 CRIT ~ ?
  • lolzfaillolzfail
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    edited 8:31AM May 13, 2018
    I got a formula which fits your attack data perfectly. However, it doesn't seem to agree that well with the data you provided at the start of this thread.

    Getting from 228 to 229 attack requires between 17(assuming 229 attack is achieved very close to 58 luk) and 19 luk(assuming 228 attack is achieved very close to 40 luk). Similarly, getting from 228 to 229 attack requires between 18 and 24 crit, while from 229 to 230 requires between 21 to 27 crit. Combining both inequalities, we need 21 to 24 crit to raise 1 attack. All these are assuming a linear model, so 1 attack requires the same amount as long as your base attack is unchanged.

    We have that 58 luk 55 crit gives 228 attack, and 27 luk 94 crit gives 229, so 39 crit > 31 luk
    Also, 59 luk 55 crit gives 229 attack, and 27 luk 91 crit is 228, so 32 luk > 36 crit.

    Taking least common multiples, we can combine both inequalities to obtain 416 luk > 468 crit > 372 luk. We see that 0.8 * 468 = 374.4 falls within that range(I tried other values near the middle and other end but none of them worked). As such, we shall estimate 1 crit to be 0.8 luk. (keep in mind that 1 AP of crit is work 2.4 AP of luk)


    crit chance = A * luk + (0.8 A) * crit

    Using the assumption above, we can convert everything into luk equivalent, so

    227 ATT: <71 LUK ~ 84 LUK
    228 ATT: 85 LUK ~ 102 LUK
    229 ATT: 102.2 LUK ~ 119 LUK
    230 ATT: 121.4 LUK ~ 135.8 LUK
    231 ATT: >138.2 LUK

    We need at most 17.2 luk for 1 attack it seems (from 228 to 229) , so we can assume it is 17.2, since it is very close to the lower bound of 17 above. 0 luk 0 crit is around 223 attack. That would be 55.75 attack for 100% crit rate, so 1% crit rate is approximately 10 luk, or 12.5 crit. This leads us to the final formula:

    crit chance = luk/1000 + crit/1250

    This formula seems to agree with all the data on attack provided. As we still do not know what your exact base attack is, for verification I assumed 229 attack is obtained at 102.2 LUK since it is extremely close to 102 LUK which stays at 228 attack.

    The next problem would be the data you provided at the start of this thread. According to this formula, which I believe to be correct since it is very simple and the numbers are nice, you should have 23.18% crit rate at 256 dex, and 14.78% crit rate at 151 dex, which is rather far from your observed values. However, they both still lie in the 95% confidence interval, so it might be possible you were just unlucky with crits. There are also other reasons why I chose my formula over that set of data.

    1) I have explained above that 500 is way too few hits, especially at higher crit rates, so that set of data may not be very accurate.
    2) The formula I came out with has nice numbers and fits your attack data very well.
    3) Attack does not lie. Unless I got the formula for attack wrong, there is no reason for my formula to contradict the attack data. Since I derived the formula with minimal assumptions, it is highly unlikely for the actual formula to differ much from mine, while the set of data at the start of the thread differs by way too much to be fixed with small adjustments.

    Therefore, I believe my formula to be correct. By the way, is sharp eyes reflected in your attack? If it is, then 256 crit should be 236 attack, and 151 crit should be 231 attack.
    DrYoshiyahu
  • DrYoshiyahuDrYoshiyahu
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    No, unless I state otherwise, Sharp Eyes isn't active.

    And yeah, the stats in the original post were never going to be very reliable. It's a nice general sense, but I'd go with your stats over mine any day.
  • lolzfaillolzfail
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    No, unless I state otherwise, Sharp Eyes isn't active.

    What I meant to ask was whether casting sharp eyes will change your attack.

  • AndromedyAndromedy
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    edited 1:29PM May 13, 2018
    lolzfail wrote: »
    No, unless I state otherwise, Sharp Eyes isn't active.

    What I meant to ask was whether casting sharp eyes will change your attack.

    So I was rather curious about this myself so I hopped on as soon as I saw and tried this out for myself.

    My base ATT with my current gear/stats is 1464.

    After casting my level 9(max possible level) sharp eyes my attack was boosted by 23 point to 1493.

    So yes Sharp Eyes does give you attack as well from adding on to the crit stat.
  • lolzfaillolzfail
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    Andromedy wrote: »
    lolzfail wrote: »
    No, unless I state otherwise, Sharp Eyes isn't active.

    What I meant to ask was whether casting sharp eyes will change your attack.

    So I was rather curious about this myself so I hopped on as soon as I saw and tried this out for myself.

    My base ATT with my current gear/stats is 1464.

    After casting my level 9(max possible level) sharp eyes my attack was boosted by 23 point to 1493.

    So yes Sharp Eyes does give you attack as well from adding on to the crit stat.

    That would sound about right. level 9 sharp eyes is 105 crit, which would equate to 8.4% crit rate, which is 2.1% attack additively. And your attack increased by 29(You said 23 though) which is around 2% so it should be accurate.
    DrYoshiyahu
  • DrYoshiyahuDrYoshiyahu
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    edited 6:25AM May 14, 2018
    @lolzfail So are we now at the point where we (you) know exactly what the damage formula for Archers is?
  • lolzfaillolzfail
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    edited 6:53AM May 14, 2018
    @lolzfail So are we now at the point where we (you) know exactly what the damage formula for Archers is?

    Nope, we don't but everything so far agrees with the formula I hypothesized so we can assume it to be true to a large extent, until someone disproves it. We will never know the exact formula unless Nexon gives it to us, or someone datamines it, but there is no reason to doubt this formula as the data fits perfectly, and it is a formula that looks legit. So basically, we are at the point where this formula is suitable to be used as if it was the exact formula, until someone finds information to disprove it.

    EDIT: I would also like to add that my formula for displayed attack, VA = (WA + A * PA) * (1 + (CD - 1) * CR) is almost certainly correct(just that we do not yet know what A is). If I'm not wrong, the tooltip for displayed attack says it is affected by your physical attack, crit chance and weapon attack(do correct me if I'm wrong). This is the only formula that makes sense given those stats, in addition to my observation of weapon attack giving very slightly more than 1:1 attack(due to base crit chance).
    DrYoshiyahu
  • OlujiwanOlujiwan
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    If you're correct though, that means 1600 Critical Rate would be necessary to get 40% critical chance, which is so much higher than I think anyone was expecting (or wanting).

    Yes and no. While I'm an assassin, this makes sense.

    Assassins get a level 50 set bonus on their gear with 2 options;

    1. 20% increased attack speed when wearing the full set (proc chance on hit)
    2. 1000 crit rate when wearing full set (proc chance on hit)

    Last night, because of lack of data, I was thinking those were ridiculously high numbers. Now that I think about it, it makes sense. Why would developers add a set that is pretty much useless? On the other hand, I definitely feel like I crit way more than just 20% atm (my stats are similar to yours).
  • DrYoshiyahuDrYoshiyahu
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    lolzfail wrote: »
    If I'm not wrong, the tooltip for displayed attack says it is affected by your physical attack, crit chance and weapon attack(do correct me if I'm wrong).

    Actually, (and this is partly my fault) magic attack affects it, which means the character's base INT stat affects it.

    I can give you any transcriptions of any descriptions you need.
    Attack is "This stat represents how effective your character will be in combat. Affected by weapon attack, physical attack, magic attack, critical rate, and critical damage.

    Weapon Attack is "The attack stat of the equipped gear. Increases the damage dealt to enemies."

    Bonus Attack is "Your bonus attack power. Increases damage dealt to enemies. Its efficiency increases with the rank of your weapon. Has no effect when no weapon is equipped.

    Physical Attack is "Increases the damage dealt to enemies with physical attacks.

    Magic Attack is " Increases the damage dealt to enemies from magic attacks."

    Strength is "The main stat for Knights, Berserkers, and Runeblades. Increases the physical attack of Knights, Berserkers, Wizards, Priests, Archers, Thieves, Runeblades, and Soul Binders. Affects physical resistance."

    Dexterity is "The main stat for Archers, Heavy Gunners, and Strikers. Increases the physical attack of Knights, Berserkers, Priests, Archers, Heavy Gunners, Assassins, Runeblades, Strikers, and Soul Binders. Affects physical resistance and accuracy."

    Intelligence is "The main stat for Wizards, Priests, and Soul Binders. Increases magic attack and affects magic resistance."

    Luck is "The main stat for Thieves and Assassins. Increases the physical attack of Wizards, Thieves, Assassins, and Heavy Gunners. Affects critical rate."
  • lolzfaillolzfail
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    edited 3:20AM May 15, 2018
    Actually, (and this is partly my fault) magic attack affects it, which means the character's base INT stat affects it.

    In that case, we may have to test out adding int and seeing whether it affects attack.(EDIT: never mind, I just saw your other comment) As far as I know wizards get around 1.1 attack per 1 magic attack, so if the attack doesn't increase from magic attack for archers then we might be able to exclude it from the formula. However, that should not change anything related to crit rate as my calculations were percentage based, so my formula still works.
  • lolzfaillolzfail
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    Olujiwan wrote: »
    If you're correct though, that means 1600 Critical Rate would be necessary to get 40% critical chance, which is so much higher than I think anyone was expecting (or wanting).

    Yes and no. While I'm an assassin, this makes sense.

    Assassins get a level 50 set bonus on their gear with 2 options;

    1. 20% increased attack speed when wearing the full set (proc chance on hit)
    2. 1000 crit rate when wearing full set (proc chance on hit)

    Last night, because of lack of data, I was thinking those were ridiculously high numbers. Now that I think about it, it makes sense. Why would developers add a set that is pretty much useless? On the other hand, I definitely feel like I crit way more than just 20% atm (my stats are similar to yours).

    Interesting... according to a linear model you would only need less than 500 crit to cap crit rate. Do you happen to have the set? If so, do you mind proccing the effect and seeing how much your attack score increases by? I would need to know your attack score, crit and luk before and after the effect.
  • SubglaciousSubglacious
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    Method for testing crit rate:
    Use the Personal Performance Meter to ensure accuracy. You can enable it under Options->Game->Interface
    Go to the guild hall (creating one costs 2k or something), on the right side there's a 3x3 square with 3 dummies. If you stand in the middle, Arrow Storm will hit all 3. I'm calling one "rotation" the combo of using Arrow Storm and then Arrow Stream until Arrow Storm is off cooldown. Arrow Stream fires approx 22 times in 13 seconds, so a conservative estimate for total hits from one rotation is 60 from Arrow Storm and 20*5 from Arrow Stream. Doing 20 rotations is at least 3200 hits and takes about 260 seconds.
    I'm sure there are ways to get more hits per second, something like the priority list being Arrow Storm > Eagle's Majesty > Rapid Shot > Arrow Stream where you use the first one in the list that's available, but that requires more effort so meh
    You can set up a macro with just Arrow Storm and Arrow Stream, that way you only have to hold down one button.

    When I say crit rate, I mean the rate that's shown on the Character page, not the Stat Sheet. Presumably that one is just bugged and doesn't display the value after adding in Luck >.>

    20 rotations, 27 luck, 73 crit rate: 8.41% crit
    20 rotations, 56 luck, 73 crit rate: 11.53% crit
    20 rotations, 27 luck, 160 crit rate: 14.73% crit
    20 rotations, 27 luck, 265 crit rate: 23% crit

    I think it's safe to say that crit chance increases linearly or close to linearly with crit rate. One MAJOR factor missing from this discussion is the enemy's Critical Evasion! Seems like the only proper way to test crit rate is to grab another player and go to a hostile zone. I would be up for this during the next beta (if there is one), but right now I'm more interested in figuring out the damage formula. IMO Crit Rate is sort of a dump stat until you can get gear that significantly increases critical damage, so there's still some time left before it becomes super important to know the formula :p
  • DrYoshiyahuDrYoshiyahu
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    edited 4:49PM May 15, 2018
    I was wondering if they'd included any of the training dummies in the CBT. Thanks for the heads up. I was hoping to see one in the furniture lists. I'm 100% sure that the training dummies don't have any critical evasion.

    You can get a pretty good idea of whether CRIT or DEX is better for you by just checking your ATT stat on the character page, but otherwise, use the performance meter. In my experience, DEX is a stronger option at lower levels, but CRIT becomes stronger later on. Your skill points make the biggest difference, because Bronze Eagle and Sharp Eyes are both going to have impacts on what you should be building.

    As for a damage formula, it seems to be solved now. Check out that thread here.
  • SubglaciousSubglacious
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    What's making you sure they don't have crit evasion? I think there's a chance they might since players have it too.

    That thread is for Attack which doesn't seem to be a great indicator of how much damage you actually do. In the Attack formula physical attack is just added on but I'm fairly certain that physical attack is a multiplier in the actual damage formula.