How precise is too precise?

[Brother Oni]

That's some badly designed equipment.

It is indeed, and we can't monkey around with the report template as it's a validated piece of kit, so changing that single parameter involves enough paperwork to make a Futurama bureaucrat wince.

[Kato]

So... you're asking: if I'm programming some sort of computer application, what level of precision does said application need to allow for?

No, (pure) computing is probably farthest from what I was thinking about. I'm looking at "real" applications, as in when the numbers actually go into building something or such.

I have worked on a pension application where the use of the single data type for percentage rates caused errors in the amounts shown to recipients (as it happened, it did not cause errors in the amounts paid, just those shown). (This is because multiplying a double by a reduces the precision to that of a single.) So yes, we needed an accuracy of at least 7 significant figures (it wasn't in $ or £ which pushed the values up).

I have also worked on a mathematical modelling engine where the developer's use of the single data type caused the numbers calculated to be different from our test values, however because of the nature of the model the error was completely irrelevant (this was a case of calculate down to exact integers rounding up).

So, again, the amont of precision required depends on what you are doing. An example of where you want 5 or more significant figures is most financial applications, which are futher complicated by the problem of intermediate rounding.

I feel stupid but I don't get it. I mean, I get the part about doubles turning into singles etc but I don't see how this caused the error described. Though I guess if we actually talk really large numbersof pure money being moved I can kind of see it but where does a percentage with seven decimals come from?

Also, I don't see how more decimals help with repeated rounding...

Hey OP.

The amount of decimals are based on your Uncertainty and number of significant figures, as a rule of thumb, you reduce your uncertainty to one significant figure and match your result in decimals.

As I tried to explain, measuring is rather the opposite of what I care for. I'm interested in when such precise numbers come into play when being applied to "making somethin".

[Khedrac]

I feel stupid but I don't get it. I mean, I get the part about doubles turning into singles etc but I don't see how this caused the error described. Though I guess if we actually talk really large numbersof pure money being moved I can kind of see it but where does a percentage with seven decimals come from?

OK, take "1.3". If this is stored as a single it = 1.3 - which is fine
If this is stored as a double it still equals 1.3 - still fine
But if it is converted from a single to a double Visual Basic simply adds 2 bytes of zeroes to the end of the number and it now equals 1.29999995231628 (sorry 9th significant figure not 7th).
Now suppose you multiply 1 million by 1.3 where the 10 million is stored as a double and the 1.3 as a single - VB converts the single to a double for the calculation and the error appears:
You get 1,299,999.95231628 which rounds to 1,299,999.95 - and you now have a visible error in an amount of money - something people tend to get techy about.
(There is a way round this, convert it to a decimal - a number with even more precision but held differently - before multiplying the double.)

Also, I don't see how more decimals help with repeated rounding...

Sorry, this one is nothing to do with extra decimal places, it was an explanation of why some systems use "round .5 to even" rather than the mathematical "round .5 up". What it does is reduce the error that creeps in when intermediate rounding happens too often, something that has to happen when dealing with money that needs to belong to someone in discrete amounts.
It works thus:

	round at end	mathematical rounding	financial rounding
Start	1	1	1
× 1.5	1.5	1.5	1.5
rounds to	1.5	1.5	1.5
× 1.5	2.25	2.25	2.25
rounds to	2.25	2.25	2.25
× 1.5	3.375	3.375	3.375
rounds to	3.375	3.38	3.38
× 1.5	5.0625	5.07	5.07
rounds to	5.0625	5.07	5.07
× 1.5	7.59375	7.605	7.605
rounds to	7.59375	7.61	7.62
× 1.5	11.390625	11.415	11.4
final round	11.39	11.42	11.40

In both cases usuing intermediate rounding gives the "wrong" answer, but using financial rounding decreases the error.

[jayem]

No, (pure) computing is probably farthest from what I was thinking about. I'm looking at "real" applications, as in when the numbers actually go into building something or such.

As I tried to explain, measuring is rather the opposite of what I care for. I'm interested in when such precise numbers come into play when being applied to "making somethin".

Oh ok, I think I understand the question.
I think the answers still remains basically the same. Broadly speaking cut it to a bit less precision than you need to measure it to. And if it doesn't work, you should have cut to more precision.*

When writing them for someone else the same applies. Work out what's important, and specify them so that they do their job to the required tolerance, but not too much more.
Ideally separately specify the things that probably aren't important so if there is any doubt, and if you can actually get ludicrous precision for free then you may as well note it down for posterity.

*NB the context aspect still applies, a stake will work (even aesthetically) if it's a cm higher or lower. A vacuum chamber things need to be pretty damn close (although as materials swell/shrink you probably want something that will mask that anyway).

[Lvl 2 Expert]

Sorry, this one is nothing to do with extra decimal places, it was an explanation of why some systems use "round .5 to even" rather than the mathematical "round .5 up".

In both cases usuing intermediate rounding gives the "wrong" answer, but using financial rounding decreases the error.

To even? What does that mean? First time up second time down?

(Also I'm assuming the 7.62 is supposed to be 7.60.)

[Knaight]

To even? What does that mean? First time up second time down?

To the even previous digit instead of the odd one. 0.5 rounds to 0, 1.5 rounds to 2, 2.5 rounds to 2, 3.5 rounds to 4, 4.5 rounds to 4, 5.5 rounds to 6, etc. Similarly 0.55 rounds to 0.6, as does 0.65.

interested in when such precise numbers come into play when being applied to "making somethin".

The necessary precision depends on what you're making. Large scale construction (skyscrapers, bridges) rarely need accuracy beyond the nearest millimeter, mechanical components in smaller systems can easily need micrometer accuracy, specialized electronic components can require exacting placement of individual atoms. The same thing applies to other units of measurement. You don't necessarily need the same sort of exacting masses or volumes when mixing concrete as for sophisticated many step chemical processes. Temperature control good enough for bulk steel or aluminum production isn't necessarily good enough for operating a fractional distillation column at a petrochemical plant (or candy making, to use a more traditional craft).

[veti]

When calculating, use as many digits as you reasonably can. (Because otherwise you're introducing avoidable, systemic errors that will compound through the calculation, that's why.)

Measurement errors, barring faulty equipment, are essentially random, which means they'll tend to smooth out over time. But systemic errors compound, and can easily have a surprisingly large effect on the final result.

When reporting the results, that's when you round.

My work involves dealing with several millions of data points every month. I boil these down to a couple of hundred numbers, which are each reported to 6 or 7 significant figures. The reports are monitored, and if anyone thinks I'm shirking on the calculations, I'd be up to my neck in auditors.

But when reporting to my own management, they don't want anything like that precision. For them I just need about half a dozen numbers, and 3 or 4 significant figures are plenty. That's because, for them, all that really matters is "are we performing above or below our expectations?", and 3 sf is quite enough to answer that.

[Kaytara]

If you're using experimental data, your decimals should only be as precise as your measuring equipment - no more and no less. If you only have the tech to measure weight down with a sensitivity down to the gram, then calculating with numbers down to the mg won't make much sense, since at that point, the difference between, say, 600.5 g and 600.45 g won't reflect any real measurable difference in experimental values anyway.

[FreddyNoNose]

Or
How many decimals matter?

Ideally from personal / professional experience, how many digits of a given number are actually relevant for real applications? And just in case it's not obvious, I'm not talking about cases of 1.0000246 because clearly then your number should be 2.46 10^-5 anyway.

I've had a discussion with a few people about this and we couldn't really agree. I feel four or five at most would be the point where you would stop caring and your safety margin or if you can allow it cheapness kicks in and rounds up or down depending on what you're doing anyway.

What is the cost of precision? Is there some loss that can be measured when comparing the different levels of precision? Comparing those might be a good starting place. Other factors might come into play like timing to get those extra bits in some critical real time/ephemeral situation.

[Kato]

... I really Need to not always Forget about my own threads... Sorry.

OK, take "1.3". If this is stored as a single it = 1.3 - which is fine
If this is stored as a double it still equals 1.3 - still fine
But if it is converted from a single to a double Visual Basic simply adds 2 bytes of zeroes to the end of the number and it now equals 1.29999995231628 (sorry 9th significant figure not 7th).
Now suppose you multiply 1 million by 1.3 where the 10 million is stored as a double and the 1.3 as a single - VB converts the single to a double for the calculation and the error appears:
You get 1,299,999.95231628 which rounds to 1,299,999.95 - and you now have a visible error in an amount of money - something people tend to get techy about.
(There is a way round this, convert it to a decimal - a number with even more precision but held differently - before multiplying the double.)

[....]
In both cases usuing intermediate rounding gives the "wrong" answer, but using financial rounding decreases the error.

I guess I get the first one... except it seems relly weird why VB would do that. Then again, I'm by far not qualified to talk about the quirks of computing, be it by VB or anything else.


As for the second part, I also see your point, for the given example anyway. My gut tells me there should be cases when mathematical rounding would be more accurate but then again, not my field, at least not that deep, so I'll assume whoever spent decades thinking about it properly.

Oh ok, I think I understand the question.
I think the answers still remains basically the same. Broadly speaking cut it to a bit less precision than you need to measure it to. And if it doesn't work, you should have cut to more precision.*

When writing them for someone else the same applies. Work out what's important, and specify them so that they do their job to the required tolerance, but not too much more.
Ideally separately specify the things that probably aren't important so if there is any doubt, and if you can actually get ludicrous precision for free then you may as well note it down for posterity.

*NB the context aspect still applies, a stake will work (even aesthetically) if it's a cm higher or lower. A vacuum chamber things need to be pretty damn close (although as materials swell/shrink you probably want something that will mask that anyway).

I feel we're getting closer to my problem. But we're still too deep in the realm of theoretical application where a measurement as precise as possible is useful (which clearly it is) Of course if I could and it was for free and it wouldn't require more time, I'd measure it down to the tenth decimal but that is never the case.

I'm wondering about concrete examples where you want to be really precise. (I should really edit that Initial post, I hope I can remember) The vacuum chamber is a pretty good example but as you say, due to shrinking and swelling you won't go that extrem because you'll include some material anyway that compensates for that, so no need to work more precise than what this will affect anyway.

The necessary precision depends on what you're making. Large scale construction (skyscrapers, bridges) rarely need accuracy beyond the nearest millimeter, mechanical components in smaller systems can easily need micrometer accuracy, specialized electronic components can require exacting placement of individual atoms. The same thing applies to other units of measurement. You don't necessarily need the same sort of exacting masses or volumes when mixing concrete as for sophisticated many step chemical processes. Temperature control good enough for bulk steel or aluminum production isn't necessarily good enough for operating a fractional distillation column at a petrochemical plant (or candy making, to use a more traditional craft).

Yes, you nned micrometer accuracy but the whole Thing is, as you say, smaller. So the precision (by scale) is not that much higher. (Also, I think the cases where "precision by the atom" are if applicable so rare it's negligible). The same goes for the chemical processes. Of course if you do a small sample you're going to measure by the ml or finer, but if you go back and do industrial sized amount, I think it's unlikely your 1000l l need to be measured down to that precision (?). And even for candy making or other chemical temperature control, it might not be the "who cares if it's 3000 or 3200K" case, but nobody is going to say "no, not 60.01°C, exactly 60°C!" because qute honestly, it seems impossible to do.

[georgie_leech]

Okay, so you're looking for specific examples where more precision is needed? Or are you looking for a general rule for how precise things should be from a percentage standpoint? Because either way, this is sounding more and more like "how long should a document be."

Also, as a point of interest, we totally can get temperature measurements that precise.

[jayem]

Okay, so you're looking for specific examples where more precision is needed? Or are you looking for a general rule for how precise things should be from a percentage standpoint? Because either way, this is sounding more and more like "how long should a document be."

Also, as a point of interest, we totally can get temperature measurements that precise.

As a case study consider making a micrometer:
You need to use the device to measure to 5 micrometers afterwards, so you need the actual position of the thread to be within 2/3 micrometers of the theoretical position. One revolution takes to about 500 micrometers So each pitch of the screw has to be at worst 498-502 micrometer (with the path straightened out).
In practice you also need the errors to not add up over say the 50,000 micrometers (50mm, 100 revolutions) of the instrument.
I believe a fairly standard technique for doing a screw thread is to have a locally key shaped cylindrical 'die head' (which guides the cutting edge based on the previous thread), so any errors here will add up. So really you need the die head for machining the screw to be accurate to 0.02 micrometers in the 500, or as it were you need it to be 500.00 micrometers long.

Screw machines (no relation to screw threads, found while googling), Oregon claim 0.0002 inches tolerances for objects 0.125-1.0 inches wide as being typical for that class of machine and offer moderate sized runs (less than 100,000). On the basis that they presumably plan to do business. That suggests that there is a decent range of products where 4-5 significant figures is about right. But they also 'boast' about it which suggests that most things don't need that tolerance. And they need to make the things which suggest at least something else needs another order of magnitude.

For casual use, I'd say there's a reason the SI scale goes G,M,k, ,m,u. Or our use of cm.
So if you really have no idea I'd see how aiming for between 0.2% and 2% looks (I.E using to the nearest mm if small measurements are 50mm-500mm). And then refine for consistency, feasibility and necessity. But that's very much a subjective value (see the commend about document length).

[Kato]

Okay, so you're looking for specific examples where more precision is needed? Or are you looking for a general rule for how precise things should be from a percentage standpoint? Because either way, this is sounding more and more like "how long should a document be."

Also, as a point of interest, we totally can get temperature measurements that precise.

Firmly the former. Because obviously for the second the answer will be "it depends" most of the time. And while you can clearly make this about just extending a documents length, I guess there must be some cases where it matters.

Also, I'm not questioning if we can measure temperature that precisely, but if you can guarantee a consistent temperature in a large-ish Container of a substance, especially if that substance is constantly reacting and thus absorbing / radiating energy. (And therefore if it matters to be that precise at a given point where you would measure)

As a case study consider making a micrometer:
You need to use the device to measure to 5 micrometers afterwards, so you need the actual position of the thread to be within 2/3 micrometers of the theoretical position. One revolution takes to about 500 micrometers So each pitch of the screw has to be at worst 498-502 micrometer (with the path straightened out).
In practice you also need the errors to not add up over say the 50,000 micrometers (50mm, 100 revolutions) of the instrument.
I believe a fairly standard technique for doing a screw thread is to have a locally key shaped cylindrical 'die head' (which guides the cutting edge based on the previous thread), so any errors here will add up. So really you need the die head for machining the screw to be accurate to 0.02 micrometers in the 500, or as it were you need it to be 500.00 micrometers long.

Screw machines (no relation to screw threads, found while googling), Oregon claim 0.0002 inches tolerances for objects 0.125-1.0 inches wide as being typical for that class of machine and offer moderate sized runs (less than 100,000). On the basis that they presumably plan to do business. That suggests that there is a decent range of products where 4-5 significant figures is about right. But they also 'boast' about it which suggests that most things don't need that tolerance. And they need to make the things which suggest at least something else needs another order of magnitude.

Hm... I guess I see the micrometer as a decent example. I mean, it's just a measuring tool but in the end that doesn't qualify it from being "something actually built".

With the screws I feel more like it's boasting than actual necessity but without more solid Information I'm not going to dismiss it entirely.

[Brother Oni]

Hm... I guess I see the micrometer as a decent example. I mean, it's just a measuring tool but in the end that doesn't qualify it from being "something actually built".

With the screws I feel more like it's boasting than actual necessity but without more solid Information I'm not going to dismiss it entirely.

Would you accept the tolerances for the injection moulds of Lego pieces? This mould was retired after manufacturing 120 million pieces so as to retain the 2µm tolerances on the parts.

The result of this feat of over-engineering are high quality items that cost an arm and a leg, but pieces from 40 years ago will fit new pieces made today (I can personally attest to ~30 year old pieces still fitting) and are on average, good for over 37,000 attach/unattach operations.

Compare this to cheap lego knockoffs where pieces within a set often don't fit properly.

[Kato]

Would you accept the tolerances for the injection moulds of Lego pieces? This mould was retired after manufacturing 120 million pieces so as to retain the 2µm tolerances on the parts.

Hm.. While I see the benefits in making a good mould, my gut says overengineering in this case. While I appreciate a good fit and long lasting pieces, I don't think this requires micrometer accuracy. I really don't know but I feel the good fit is still achieved by precision in the range of 1 or 0.1% and durability is likely a matter of material more than mould?

[Xyril]

I feel we're getting closer to my problem. But we're still too deep in the realm of theoretical application where a measurement as precise as possible is useful (which clearly it is) Of course if I could and it was for free and it wouldn't require more time, I'd measure it down to the tenth decimal but that is never the case.

This has been a thoroughly entertaining and fairly informative thread, and it wouldn't have happened without your very open-ended question, so please don't take this criticism as overly, well, critical, I suppose, but if people have spent over a page trying to give you useful information and generally circling what you're looking for, and you've had time to respond, but only to either engage in a brief debate over minutiae or to say "nah, this isn't quite what I was asking for," then I would suggest that maybe the problem is that you need to take a breath, ask yourself precisely what it is you want to know, and then figure out how to precisely articulate what it is you're after? Just my two cents: The importance of precision isn't limited to quantitative concepts.

From what I can gather, "It depends" is basically the answer you're looking for, since this last post suggests that you're concerned about balancing cost and usefulness. If you're looking for real life examples, once again the answer is "it depends." When you're building a bolt-action rifle for target shooters and hunters, and 90% of them can't shoot more accurately than 0.5 MOA or so, and 99% don't have essentially unlimited funds, then you're going to mill everything to a certain amount of precision, and that's a real life example. When you're building a rifle for the top marksmen in the military, then you're getting into a situation where your customer can afford to spend more, and your end user's inaccuracy won't be the limiting factor, then you're going to have to be much more precise, and that is an equally valid real life example.

The only thing I can think of--until you give us more information of course--that would satisfy you is if everyone just kept saying "Well, I know X application where we find it worth the cost to go to N significant figures," until we reach a point where somebody wins. Admittedly, this would be a very cool bit of trivia to discover, but I suspect that the answer we get wouldn't really shed much light on what is typical.

[jayem]

With the screws I feel more like it's boasting than actual necessity but without more solid Information I'm not going to dismiss it entirely.

They're 'screw machines'. I don't fully understand what they are*, but they're flexible programmable cutting devices. The lathes and milling machines offered a similar tolerance.

I wouldn't have thought of it but lego does rely on pretty near exact contact.

*The example pictures did have screw threads on various types of attachment, so that could be the source of the name. Or the mechanism it uses to get the drills? in the right place might use screws.

[Xyril]

I wouldn't have thought of it but lego does rely on pretty near exact contact.

Thus answering the question of why Megabloks sucked.

[Rockphed]

Thus answering the question of why Megabloks sucked.

I always thought it was because they used weird colors (e.g. super bright neon green and a white that almost glowed in the dark).

[Brother Oni]

Hm.. While I see the benefits in making a good mould, my gut says overengineering in this case. While I appreciate a good fit and long lasting pieces, I don't think this requires micrometer accuracy. I really don't know but I feel the good fit is still achieved by precision in the range of 1 or 0.1% and durability is likely a matter of material more than mould?

From what I understand from other people more familiar with manufacturing, a tolerance of 2µm is ridiculously tight on plastic materials. Durability is significantly affected by the mould - plastic is comparatively soft, so if the parts don't fit properly, they'll deform and hence degrade more rapidly.

I would also think the level of precision would depend on the likelihood of a part falling outside tolerances, the detectability of such a fault and the consequences of any failures. A lego piece is fairly minor while something not quite right on a space shuttle could potentially be catastrophic.

If you really want to delve down this rabbit hole, it's called a risk assessment and this conversation is getting far too close to my day job for my liking.

I always thought it was because they used weird colors (e.g. super bright neon green and a white that almost glowed in the dark).

As mentioned plastic isn't that good for such precision materials, so if they start using weird dyes or materials and don't have good quality control/wide tolerance limits, I can see it resulting in crappy sets.

[Kato]

[...]
then I would suggest that maybe the problem is that you need to take a breath, ask yourself precisely what it is you want to know, and then figure out how to precisely articulate what it is you're after? Just my two cents: The importance of precision isn't limited to quantitative concepts.

Sorry, I'm aware my initial question was too open ended and apparently neither my edit nor my attempts to explain it were very helpful. So.. Since I don't think another try will work I might have to give up.

They're 'screw machines'. I don't fully understand what they are*, but they're flexible programmable cutting devices. The lathes and milling machines offered a similar tolerance.

To be honest, my initial thought was it's a machine that makes screws. But it seems it's what is called a Drechsler in German (I'm sure there is a more modern term but I fail to remember it)

If you really want to delve down this rabbit hole, it's called a risk assessment and this conversation is getting far too close to my day job for my liking. :smallish: .

Aw, too bad. It might be your work would fit quite well to what I'm looking for but I very much understand drawing a line between this and that.

[Xyril]

I always thought it was because they used weird colors (e.g. super bright neon green and a white that almost glowed in the dark).

That probably didn't help, but I distinctly remember a non-negligible number of them clinging poorly or taking a lot of force to put together or pull apart (both when used with LEGOs and with other MegaBloks)

[Mando Knight]

From what I understand from other people more familiar with manufacturing, a tolerance of 2µm is ridiculously tight on plastic materials. Durability is significantly affected by the mould - plastic is comparatively soft, so if the parts don't fit properly, they'll deform and hence degrade more rapidly.

I would also think the level of precision would depend on the likelihood of a part falling outside tolerances, the detectability of such a fault and the consequences of any failures. A lego piece is fairly minor while something not quite right on a space shuttle could potentially be catastrophic.

If you really want to delve down this rabbit hole, it's called a risk assessment and this conversation is getting far too close to my day job for my liking.

There's also the founder's motto, "det bedste er ikke for godt" (officially rendered "only the best is good enough", but a bit more literally "the best is never too good"). Apparently-ridiculously-stringent standards led to Lego grabbing hold of a reputation for high quality product, one that (alongside the brick's incredible versatility and some back-to-basics company reforms after a financial tumble in the late 90s/early 2000s) helped make the company one of the largest toy manufacturers in the world (depending on year-to-year sales, it's roughly on par with Mattel, with Namco-Bandai and Hasbro not too far behind).

As mentioned plastic isn't that good for such precision materials, so if they start using weird dyes or materials and don't have good quality control/wide tolerance limits, I can see it resulting in crappy sets.

That probably didn't help, but I distinctly remember a non-negligible number of them clinging poorly or taking a lot of force to put together or pull apart (both when used with LEGOs and with other MegaBloks)

On a similar note, Lego uses a different plastic (polycarbonate rather than Acrylonitrile butadiene styrene, or ABS) for their translucent elements (and anything where more strength is needed than what ABS can provide). Polycarbonate also generates greater friction than ABS, to the point where the official design guidelines forbid Lego from making a set where the "lightsaber blade" rod is supposed to be inserted into a translucent cone piece: a child might not be able to pull the two pieces apart safely without damaging the elements.

[georgie_leech]

There's also the founder's motto, "det bedste er ikke for godt" (officially rendered "only the best is good enough", but a bit more literally "the best is never too good"). Apparently-ridiculously-stringent standards led to Lego grabbing hold of a reputation for high quality product, one that (alongside the brick's incredible versatility and some back-to-basics company reforms after a financial tumble in the late 90s/early 2000s) helped make the company one of the largest toy manufacturers in the world (depending on year-to-year sales, it's roughly on par with Mattel, with Namco-Bandai and Hasbro not too far behind).

Amusingly, they're also the world's largest tire manufacturer. Of tiny tires sure, but they still count

[ereinion]

A presentation showing some guidelines for legal and illegal connections can be found here. Mando Knight's example is covered there, and you can also learn that the logo on top of Lego-bricks are 0.14 mm tall, making the build on the left legal (it's a Technics brick without the logo on top), while the one on the right is illegal.

Spoiler: Largish image

Show

[tiornys]

Amusingly, they're also the world's largest tire manufacturer. Of tiny tires sure, but they still count

Hmm. Out of whimsical curiosity, is that by tire count, volume, or both (volume would almost certainly imply both)?

[Rockphed]

Hmm. Out of whimsical curiosity, is that by tire count, volume, or both (volume would almost certainly imply both)?

Considering how much Legos cost, it could be by price.

[georgie_leech]

Hmm. Out of whimsical curiosity, is that by tire count, volume, or both (volume would almost certainly imply both)?

Mostly by quantity, alas. Still, I will use my slightly misleading trivia knowledge when the opportunity arises.

Considering how much Legos cost, it could be by price.

Heh.