# In APL calculations, does .5 round up or down?

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The 4.1 Guide on pg. 25 says when calculating APL to always round to the nearest whole number. Thing is, every way I count it, .5 is exactly between two whole numbers. Which leaves me unclear as to whether .5 rounds up or down.

This isn't just theoretical rules-lawyering, it arose from discussions about a previous session I GMed and whether playing up was legal in that situation.

Backstory:
I ran a scenario back in January (under the 4.0 Guide) where the players opted to play up to rather disastrous results. In the wake of that session I was told the group didn't qualify to play up. I've spoken to several of the players, and they all believe it was legal for them to play up.

The party was three 2nd levels PCs and three 1st level PCs. Our APL calculation was thus:

9 total levels divided by 6 PCs = 1.5
+1 to APL for 6 or more PCs = 2.5

The rounding is in question. We rounded 2.5 up to 3, thus allowing us to play up to subtier 4-5.

I did ask about APL before I ran the game. The Venture Captain who responded did specifically say to round decimals of .5 or lower down, and decimals of .6 or higher up. The VC's information is being questioned by at least one player, however, because there isn't any exact text saying that in either version 4.0 or 4.1 of the Guide.

The 4.0 Guide did say in one place to always round APL down. The example though, said to round up. That contradiction is why I asked. The 4.1 Guide simply says to "round to the nearest whole number". As mentioned previously, .5 is exactly between two whole numbers.

(To avoid any confusion, I did not purposely ignore the VC's information. In the stressful rush of prepping to run PFS for the first time, I completely misread it. Because we were later than agreed getting started, I also let the players calculate the APL without double checking it myself.)

The spoilered text probably muddies my question a bit, so to clear it up, I'm most interested in knowing whether to round .5 up or down under the current 4.1 Guide. However, it would be nice, but not strictly necessary, to also know if we miscalculated under the different wording of the 4.0 Guide.

Back in LG days, this issue came up on the old Infinite Monkeys group. Without rehashing the whole thing, this is a problem. There are lots of different approaches to rounding 0.5 that vary by geographical location, age of reader, industry of occupation, educational background, etc.

It is insufficient for the guide to say round to the nearest whole number and expect a consistent result in terms of how it's done for 0.5 in a global environment. Suggested text: "...round to the nearest whole number, rounding 0.5 [however they want it to go]."

In this case, round down.

Even if math fails you I'd say common sense should've told you there is no way a group of 3x2nd Levels and 3x1st Levels would ever be eligible to play subtier 4-5

It would be a certain TPK

EDIT:
Also, from Guite to PFSOP, p25:
"Some scenarios or
special events offer more than two subtiers. In these cases,
no PC can play at a subtier more than 1 step away from her
character level."

None of the characters was anywhere near subtier 4-5

i'm assuming the tiers were 1-2, 4-5. at that point i would have based it on what the players wanted. if they wanted to play up don't try to kill them but use the first encounter to show them this might be a bad idea.

We have to stick with RAW and it clearly says (c/p of the same part, a bit expanded) p.25:

"Within each tier, PCs should play in the
subtier in which they fall whenever possible, but they may
be allowed to play up or down, based on the average party
level at the table, as outlined below. Some scenarios or
special events offer more than two subtiers. In these cases,
no PC can play at a subtier more than 1 step away from her
character level.
"

Its a really clean cut case.

zrinka znidarcic wrote:

We have to stick with RAW and it clearly says (c/p of the same part, a bit expanded) p.25:

"Within each tier, PCs should play in the
subtier in which they fall whenever possible, but they may
be allowed to play up or down, based on the average party
level at the table, as outlined below. Some scenarios or
special events offer more than two subtiers. In these cases,
no PC can play at a subtier more than 1 step away from her
character level.
"

Its a really clean cut case.

Zrinka, With respect, your example doesn't apply because there were only two subtiers. I've expanded on your bolded text to demonstrate my point.

Further down it has exactly the same case as an example:

"If the APL of a table is between two subtiers (like APL
3rd for a Tier 1–5 scenario), the players may choose to play
up to Subtier 4–5 or down to Subtier 1–2. If, however, the
APL was calculated for six players (and thus bumped up
by +1) and this pushes a low-level table out of their subtier
and into the level between two subtiers, the players
should be strongly cautioned about playing up, as even a
party of six players may not be able to handle situations
and challenges that the higher subtier will present."

So in the case original poster wrote, I wouldn't allow the group to play up. I guess you could find a reason to let them but from experience, its usually a bad idea.

If it is 0.5, you should let the table decide with the strong encouragement to play down.

/For some reason, folks at my table always jump at the chance to play down.
//I wonder why...

 Venture-Captain, Texas—San Antonio aka Dragnmoon

Joseph Caubo wrote:

If it is 0.5, you should let the table decide with the strong encouragement to play down.

/For some reason, folks at my table always jump at the chance to play down.
//I wonder why...

Oddly my group almost always chooses to play up, even with all the deaths they have had *Few permanent*

Just keep in mind that if even one player is against playing up, you have to play down.

 Venture-Captain, Texas—San Antonio aka Dragnmoon

zrinka znidarcic wrote:
Just keep in mind that if even one player is against playing up, you have to play down.

To clear something up, that is not actually a rule, but is very good suggestion to follow, that all GMs should do.

Hasn't the general rule since the days of 3.0 been to round down any .5s? This is something I remember from my long-ago apprenticeship into the ways of d20.

That said, I would probably let a group play up if they really wanted to since I find playing up to generally be more of a white-knuckled good time .

 2 people marked this as FAQ candidate. 21 people marked this as a favorite.

If you are exactly at 0.5, just let the group decide which way they wish to go.

Regardless of the way the rule is interpreted, since it does not say otherwise, in real world math when you are rounding to the nearest whole number, .5 is always rounded up.

 RPG Superstar 2015 Top 32, RPG Superstar 2012 Top 32

Michael Brock wrote:
If you are exactly at 0.5, just let the group decide which way they wish to go.

Hooray for finality! :D

Jiggy wrote:
Michael Brock wrote:
If you are exactly at 0.5, just let the group decide which way they wish to go.
Hooray for finality! :D

Here Here! Thanks Mike :)

I believe there is also language in the guide to this effect, especially if you start at say 2.67 and would round up to 3 and then a +1 would take you to 4 because you have 6 players, that you should have the option to play down.

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Enevhar Aldarion wrote:
Regardless of the way the rule is interpreted, since it does not say otherwise, in real world math when you are rounding to the nearest whole number, .5 is always rounded up.

You always have to be careful with absolutes.

I think this came up one or two years ago and I posted at that time the wikipedia entry on rounding

You will find a total of 7 !! different ways how to round .5. It is true - the rounding up is the most common use - but this is just a convention.

For PFS games - the only bit I would like to add to the people who say - let the group decide:

Let the GM give you advice before the group decides. As GM I should have prepared the scenario. In most cases I also know the group or at least what characters are in the group.

In my view this should be decided in the best way that makes sense for survival and fun of the group. Non of the mathematical conventions above will take into account the layout of the group, experience, optimization etc.

So take advantage when once in a while when the rules don't specify which way to rule. Go with a decision that seems best.

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Thod wrote:

Go with a decision that seems best.

Like maybe the one that Mike Brock stated an hour and a half earlier? ;)

Saint Caleth wrote:
Hasn't the general rule since the days of 3.0 been to round down any .5s? This is something I remember from my long-ago apprenticeship into the ways of d20.

Yes, when rounding in d20, round down. The scope of that is within the game, not when involved in meta activities such as setting up the table, which isn't a d20 activity at all. I would hate to apply this rule for deciding how to split up the obligatory pizza when gaming, for example. :)

Enevhar Aldarion wrote:
Regardless of the way the rule is interpreted, since it does not say otherwise, in real world math when you are rounding to the nearest whole number, .5 is always rounded up.

Unfortunately, this isn't true, as I alluded to in my first post in this thread and as Jiggy has expanded on.

If the group is at 0.5 it doesn't matter. Either way will still result in a TPK. At least that's my experience...

Thanks for all of the responses. It's nice to have this ruleplaying issue cleared up, so I can get back to focussing on my roleplaying and rollplaying instead. :-)

Though clearly I need to take some GMing lessons from the Care Baird. I only managed a half-party kill.

Edric Nansen wrote:
I only managed a half-party kill.

That's a step in the right direction!

Edric Nansen wrote:

Though clearly I need to take some GMing lessons from the Care Baird. I only managed a half-party kill.

There's an easy solution..

You look at the players with a straight face and say "you realize that this is a two-parter, right?"

-James

 RPG Superstar 2015 Top 32, RPG Superstar 2012 Top 32

Howie23 wrote:
and as Jiggy has expanded on.

Wait, what?

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In Krackosia where I'm from, we are required by law to round .5 to the southwest which yeilds a result of square root of 2. Therefore, the government itself forces us to play up.

Or, we go by the old "when in doubt, don't play up" adage.

Jiggy wrote:
Howie23 wrote:
and as Jiggy has expanded on.
Wait, what?

Sorry Jiggy, my bad. It was Thod.

 Venture-Lieutenant, Kentucky—Lexington

Michael Brock wrote:
If you are exactly at 0.5, just let the group decide which way they wish to go.

Necro thread, but with season 5's new "no choice" mechanic how is this done?

There are times when you will end up with exactly .5 on the APL, and the Season 5 rule of "You should always round to the nearest whole number" is unspecific on what to do.

For example, 5.5 is equidistant from 5 (APL between) and 6 (UP tier on Tier 3-7), so there is a decision or choice to be made. Who makes the choice?

James Risner wrote:
Michael Brock wrote:
If you are exactly at 0.5, just let the group decide which way they wish to go.

Necro thread, but with season 5's new "no choice" mechanic how is this done?

There are times when you will end up with exactly .5 on the APL, and the Season 5 rule of "You should always round to the nearest whole number" is unspecific on what to do.

For example, 5.5 is equidistant from 5 (APL between) and 6 (UP tier on Tier 3-7), so there is a decision or choice to be made. Who makes the choice?

Mathematically speaking, "to the nearest whole number" means you round down at .49 or lower, or up at .50 and higher.

 RPG Superstar 2013 Top 16

Patrick Harris @ MU wrote:
Mathematically speaking, "to the nearest whole number" means you round down at .49 or lower, or up at .50 and higher.

Not necessarily. In my chemistry education, we were taught to always round .5 to the nearest even whole number. So 1.5 rounds up to 2, but 2.5 rounds down to 2.

 Venture-Lieutenant, Kentucky—Lexington

RainyDayNinja wrote:
Not necessarily. In my chemistry education

Yea there is a ton of confusion of rounding, since there is no "rounding means this" that is defined.

Other examples is computer round() follows the "round to nearest even" when equidistant.

So without verbiage in the campaign document detailing what to do when the result is exactly X.5, we can never make a "well do this" decision.

1.0-1.4x would round down. 1.5-1.9x would be up. There is more of an even distribution of numbers there. So 1.5 would round up.

RainyDayNinja wrote:
Patrick Harris @ MU wrote:
Mathematically speaking, "to the nearest whole number" means you round down at .49 or lower, or up at .50 and higher.
Not necessarily. In my chemistry education, we were taught to always round .5 to the nearest even whole number. So 1.5 rounds up to 2, but 2.5 rounds down to 2.

This isn't chemistry. But just for you, I'll revise my statement.

In non-discipline-specific mathematics, "to the nearest whole number" means you round down at .49 or lower, or up at .50 and higher.

Here is a link to a page describing the rounding function of basic mathematics, in which they clearly spell this out.

OUTSIDE of APL calculations - how are people rounding?

if my PC does half damage, with a d4 what are the damage numbers?

1 halved to 0.5 rounded to 1?
2 halved to 1.0 rounded to 1.
3 halved to 1.5 rounded to 1?
4 halved to 2.0 rounded to 2.

this has always bothered me. If you halve the damage on a d4, 75% or the answers are 1? Is that correct?

 RPG Superstar 2013 Top 16

The 0.5 should be rounded to 0, which becomes 1 non-lethal damage. Fractions in the regular game rules always round down.

RainyDayNinja wrote:
Patrick Harris @ MU wrote:
Mathematically speaking, "to the nearest whole number" means you round down at .49 or lower, or up at .50 and higher.
Not necessarily. In my chemistry education, we were taught to always round .5 to the nearest even whole number. So 1.5 rounds up to 2, but 2.5 rounds down to 2.

I didn't realize this was only in Chemistry. I was also taught this so that the number of times you round up will balance out with the number of times you round down. However, I guess this may not apply for PFS, since we aren't really concerned with multi-step calculations that need to be correct to the nearest .00001.

Off topic, but in chemistry you should be using significant figures or proper error propagation for rounding. It's never as simple as "round to the even".

David Bowles wrote:
Off topic, but in chemistry you should be using significant figures or proper error propagation for rounding. It's never as simple as "round to the even".

I think rounding to the even is part of the significant figures calculation itself. Sometimes that extra figure I need to drop is a 5. I wouldn't be surprised if this was just a stand-in rule for something better and more statistically correct for those of higher levels, but that is how it was done in my undergrad Analytical Chemistry class.

Starfinder Superscriber

Almost never that simple, but sometimes it happens. Sometimes the last digit you've got is a 5 (or the last two are 50), and you need to know how to round it. And, to use significant figures (or proper error propagation, for which significant figures is a quick approximation), you have to round....

The way I remember whether it's even numbers or odd numbers is to consider a situation where somebody is rounding multiple times. (A bad idea, by the way! Always keep several extra digits for intermediate calculations, and only round off at the end. Otherwise, you can figure out that 2+2=5. But, I use this as a way to remember the rule.) Suppose that you have the number 4.45. Rounded to one digit, that's clearly 4. However, if somebody rounded to two digits, and rounded up, you'd have 4.5. Then, round again, and you'd have 5, which is wrong for rounding 4.45 to one digit. Thus, the convention of rounding to the nearest even number would save you from ending up the wrong way. (Again, in practice rounding multiple times is frequently going to lead to errors; the better thing to do is only round once, where this is unambiguously 4.)

At any rate, it sounds like die rolls get rounded down, and Mr. Brock has stated that APL of X.5 are player decision. We just engineered a season 2 table to play in the 3-4 tier of a 1-2/2-4/6-7 because we had two level ones, but also two level 3 druids with pets at a seven player table to boot. Tier 1-2 would have been a total joke.

So my friend and I produced level 4's and we got it mathed out to 2.58 total. The level 1's got out of tier money and everyone else got 3-4. We still steamrolled the encounters as well. Season 2 just can't handle 7 players with three pets. It's really unclear if any season can handle that.

rknop wrote:

Almost never that simple, but sometimes it happens. Sometimes the last digit you've got is a 5 (or the last two are 50), and you need to know how to round it. And, to use significant figures (or proper error propagation, for which significant figures is a quick approximation), you have to round....

The way I remember whether it's even numbers or odd numbers is to consider a situation where somebody is rounding multiple times. (A bad idea, by the way! Always keep several extra digits for intermediate calculations, and only round off at the end. Otherwise, you can figure out that 2+2=5. But, I use this as a way to remember the rule.) Suppose that you have the number 4.45. Rounded to one digit, that's clearly 4. However, if somebody rounded to two digits, and rounded up, you'd have 4.5. Then, round again, and you'd have 5, which is wrong for rounding 4.45 to one digit. Thus, the convention of rounding to the nearest even number would save you from ending up the wrong way. (Again, in practice rounding multiple times is frequently going to lead to errors; the better thing to do is only round once, where this is unambiguously 4.)

actually, with rounding at the end you would be able to show that 2+2=5. because, 2.4 +2.4 =4.8 rounding to 5.

rounding before addition gives you 2.4+2.4 = 2 + 2 = 4.

;)

 Venture-Lieutenant, Arizona—Phoenix aka TriOmegaZero

Ten actions a round? Yeah, not many scenarios can challenge that.

nosig wrote:
rknop wrote:

Almost never that simple, but sometimes it happens. Sometimes the last digit you've got is a 5 (or the last two are 50), and you need to know how to round it. And, to use significant figures (or proper error propagation, for which significant figures is a quick approximation), you have to round....

The way I remember whether it's even numbers or odd numbers is to consider a situation where somebody is rounding multiple times. (A bad idea, by the way! Always keep several extra digits for intermediate calculations, and only round off at the end. Otherwise, you can figure out that 2+2=5. But, I use this as a way to remember the rule.) Suppose that you have the number 4.45. Rounded to one digit, that's clearly 4. However, if somebody rounded to two digits, and rounded up, you'd have 4.5. Then, round again, and you'd have 5, which is wrong for rounding 4.45 to one digit. Thus, the convention of rounding to the nearest even number would save you from ending up the wrong way. (Again, in practice rounding multiple times is frequently going to lead to errors; the better thing to do is only round once, where this is unambiguously 4.)

actually, with rounding at the end you would be able to show that 2+2=5. because, 2.4 +2.4 =4.8 rounding to 5.

rounding before addition gives you 2.4+2.4 = 2 + 2 = 4.

;)

That's why you use significant figures. Since 2.4 has 2 SFs, the resulting answer, 4.8, must also have 2, which it does. You wouldn't round in this situation at all.

Okay, no more of this. I thought I was done with needing these math rules. :)

 Venture-Lieutenant, Kentucky—Lexington

David Bowles wrote:
At any rate, it sounds like die rolls get rounded down, and Mr. Brock has stated that APL of X.5 are player decision.

But Mr. Brock's comment was during Season 4 where you can play up. Season 5 rules prohibit choice, so I don't think his comment of "group decide which way" response is still valid.

Oh, I didn't look at the date. My bad. So we are going with 0.5 rounds up, as that is mathematical convention?

I was under the impression that the core rulebook states to round up for APL. It's the only place where you round up in the rules.

Ah, it allows that. Didn't require

There are many different rounding methods you can use. Even within mathematics. But this isn’t mathematics (or accounting or science), this is PFS.

I actually asked Mike Brock about this at Gencon and he said we could choose up or down, which is a good solution.

Having said that, it would be appreciated if this were included in the OPG, there’s a lot of confusion around this and table variation. And it matters more than ever now.

 Venture-Lieutenant, Kentucky—Lexington

Cheapy wrote:
I was under the impression that the core rulebook states to round up for APL. It's the only place where you round up in the rules.

Not really:

Core p397 says "You should round this value to the nearest whole number (this is one of the few exceptions to the round down rule)."

This is the same rule used in the GtPFS 5.0 and doesn't help answer our question definitively.

Jason S wrote:
I actually asked Mike Brock about this at Gencon and he said we could choose up or down, which is a good solution.

But this does, so I'm going to go with Mr. Brock says it is still table choice until the GtPFS 5.0 is updated to contradict.

 Venture-Captain aka TwilightKnight

Folks this is not complicated. No reason to get all confused. The most recent comment by Brock is that the players can chose how to round 0.5
once that is determined, then you can apply the APL rules to determine which sub-tier is appropriate. No reason it needs to be any more complicated than that

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