SigFigs in this simple example

[RedKnight]

Hi folks,

 

Here's an armchair question...

 

A significant figure (sigfig) is the number of figures (a.k.a. digits) to which a number is meaningful, even if the numbers are zero. For example, 10.000 has 5 sigfigs if indeed one has that much precision.

 

But how about in the instance of degrees of a compass, when measuring it as a full 360 degrees... if the current measure is 5 degrees, how many sigfigs does this have?

 

I would think it effectively has three. Or maybe it's simply a question of semantics.

 

Your thoughts?

 

Mike

P.S. My first post here! ScienceForum.net looks like a nice active place.

 

[John Cuthber]

As specified, it has 1 sf.

However 5.00 degrees has 3sf.

[Bignose]

I think it is also important not to confuse significant figures of a number and the accuracy of a measuring device. The accuracy of a measuring device ought to drive the significant figures of the reported value, but so, very, very often it isn't done correctly.

[imatfaal]

Leading zeros are not significant. 005 degrees 07 mins and 04 seconds as some in my industry would put it - is the same as 5 deg 7 mins and 4 seconds; there is no change in accuracy or implied accuracy. The use of leading zeros in time and degrees (eg 0200 hrs) is to show that there hasn't been a mistake in the number of trailing zeros I believe; if all times are 4 digits and all bearings are three then mistakes are less likely. But these are not significant digits

[tar]

RedKnight,

 

I see your question. I had the same kind of misunderstanding of the principle, when I learned about measuring practices in Physics, at NJIT many years ago.

 

You are thinking that a measure of 355 degrees is no more or less accurate, and no more or less significant than a measure of 5 degrees, so a measure of 5, when it comes to degrees, should be just as significant, of equal significance, so to speak, as a measure of 355 degrees, that is ranked as having 3 significant digits. So "effectively" you figure that 5, by itself, is good to, or is "as if" it has 3 significant digits. Well, as it turns out, as everyone else has already explained, it does not mean that a particular number has a significance rank, it means that if you cannot accurately measure to a particular precision, any numbers guessed at, averaged, interpolated or otherwise arrived at signifying that you have measured something to an accuracy greater than you are capable of measuring, are not significant of anything, and should be discarded.

 

So 5 degrees has one significant digit, 12 degrees has 2 significant digits and 355 has three significant digits, and in all cases you are promising you have the ability to measure in whole degrees. If you had reported measuring 5 degrees and said to someone that was good to three significant digits, you would be insinuating that you had measured it as 5.00 degrees and had the ability to measure with no question to hundredth of a degree, and with some lesser degree of certainty, to 1000ths of a degree.

 

The general rule, as I understand it, is to report only significant digits, and when mixing data where one set of data is reported to lets say tenths of a gram and another set is reported to hundreths of a gram, you should round your more accurate data to the significant digits available in the set with fewer significant digits before merging the data.

 

Example. If I say I wheigh 200 pounds and Jim says he wheighs 200.00865 pounds, the two of us together, would wheigh 400 pounds, not 400.00865 pounds. If however, I reported my wheight as 200.00 pounds, I believe you could report our combined wheight as 400.01 pounds. But it was a long time ago, I took physics at NJIT. The correct reported combined wheight may be 400.0. I will yield to BigNose for the actual conventional treatment of those two measurements, added together.

 

Regards, TAR

Edited April 24, 2014 by tar

[RedKnight]

Hi, thanks, tar...

 

I pondered the replies here for a while (sorry for not responding!) and came to see that, indeed, this "5" really is just one sig fig.

 

But, still, there is something going on that I have not yet been able to enunciate well. Or maybe there simply is no specific term for it. Bignose (and you?) has called it measurement accuracy, which is of course entirely accurate. But is there not some more specific term?

 

Mentally, I think of a directional reading of, say, 270 degrees from a compass with a full 360 degree accuracy, with that of some very crude wind vane which, for sake of argument, can only return the general direction and thus only ever returns 0 for north, 90 for east, 180 for south, and 270 for west. Yes, of course this is a contrived example - but then, there are many real-world examples of one device measuring much more accurately than another.

 

In this crude example, a measurement of "270" on the one is a lot different than the other, in terms of the percent circumference of a circle it might possibly describe (1/360 for the one, 1/4 for the other). One can even argue the windvane isn't even really returning 270.

 

A more realistic example would be e.g. comparing an old analog dial-type resistometer (volume knob) with 10 settings, to a modern digital dial knob.

 

But is "measurement accuracy" the best term for this? Isn't there something a little more specific? Such as ... I don't know... instrument-specific inherent precision, or something?

 

Thanks if anyone has ideas!

[John Cuthber]

This might help

http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf

[tar]

RedKnight,

 

I think it is possible to overthink a thing. I do it all the time.

 

Yes there is a "signifigance" issue going on between analogue and digital measurements. For instance, a sweep second hand, spends more time being precisely the right time, than a secound hand that "ticks" between the minute marks. And on a digital clock, the darn thing shows 2:02 for the whole minute between when it was 2:01 and 2:03. You can not tell, just looking at it, whether it just turned 2:02 or is about to change to 2:03.

 

Looking at the brass rooster on the barn roof, you actually see exactly which direction the wind is blowing at the moment, and every little wiggle and angle tells you significantly, something about the wind direction, as that rigidly attached to the roof, nearby the pivoting rooster, is an indicator pointing North, which gives you South and East and West and NW and SSE and so on to judge the wind, or measure the wind as coming from.

 

You cannot though, from the ground say the wind is coming from 271 degrees, 12 minutes, and 42 seconds from North, counting clockwise from the top, because the rooster setup was not designed to judge the wind direction that precisely. The N on the appartus might not even be alligned exactly toward the North Pole, as someone could have positioned the thing using a compass, that was pointing toward magnetic North.

 

In anycase, all these factors are significant and therefore judging the wind direction from the ground the 42 seconds has no meaning, the 12 minutes has no meaning and the 271 has no meaning. You can just say the wind is coming from the West.

 

So us overthinkers, you and me both, have to learn to use the terms like significant digits, exactly in the way they were designed to be used, for the purposes they were designed to be used, and excercise good judgement in applying them, not taking something too literally if it would be silly to do such, or too liberally if it would be silly to do such, but follow the letter of and the principle behind the rule.

 

All the other aspects of measurement and significance you are referring to are still real and meaningful, and must be considered when taking a measurement and reporting it, but a measurement of 5 degrees has but one significant digit. It is NOT effectively three.

 

Regards, TAR

[RedKnight]

Thanks folks, I appreciate it. John, it's interesting to see that they "standardize" on using standard deviation (and similiar stats) for expressing uncertainty.

 

Still, I think my question focuses more on precision than uncertainty or accuracy.

 

Tar, I'm sure you're right... we two overthink things, perhaps. But then sometimes concepts are in usage enough that words are generated or borrowed. Like weltgeist. Or menage a trois. There once were times when some people thought of such ideas or in such ways, but there wasn't a word for it (yet).

 

However, apparently my idea isn't sufficiently "popular" to have it's own word or line of study, as it were. At least not for folks who've viewed this.

 

Still... there we are. Thanks for helping, all!

 

Mike