Very useful link - thanks WBahn.

 

WBahn,

Is the NISTs opening line going to be a tag in your signature line?

The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit.

 

WBahn,

Is the NISTs opening line going to be a tag in your signature line?

The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit.

It should be. He states it so often.

 

WBahn,

Is the NISTs opening line going to be a tag in your signature line?

The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit.

If I ever set up a sig, it will almost certainly be something like this. After being a member of the forum for just a few weeks, I was really thinking I should have made by user name "UnitsNazi".

 

...
P = 38A * 235V = 8930W = 8.39kW

Shouldn't that be 8.93 kW?

If you are going to be the forum "units nazi" can I be the "dyslexia nazi"?

...
Maybe the answer is wrong because of Decimal Place Accracy
... Should have been 8.93 KW instead of 8.9 KW

That would be my call. I was taught in electronics exam calcs if no precision is specified to always use 3 decimal places as a classroom standard.

If the examiner specifically asked for the answer in kW, the two correct answers would be 8.930 and 8.930kW.

 

Shouldn't that be 8.93 kW?

If you are going to be the forum "units nazi" can I be the "dyslexia nazi"?

Sure. Hopefully after my eye surgery on Thursday I will be able to actually see what I've typed!

That would be my call. I was taught in electronics exam calcs if no precision is specified to always use 3 decimal places as a classroom standard.

The general rule is to use three sig figs if the proper number of sig figs can't be readily determined. You should never use two or less or more than four unless you have good reason. For reporting purposes, a common rule when I was starting out was to not count a leading 1 as one of the three sig figs. Arguably, a leading 2 or even 3 shouldn't be counted, but I'm not aware of anyone asserting that.

But using "3 decimal places" doesn't work. Take this problem, for instance. Would someone have to put

P = 8390.000 W

if reporting the answer in watts?

Or put

P = 0.008 MW

if reporting the answer in megawatts?

Does it make sense that the first answer is known down to the milliwatt but the second only to the nearest kilowatt just because of the units that the answer is reported in?

If the examiner specifically asked for the answer in kW, the two correct answers would be 8.930 and 8.930kW.

If specifically asked to report the answer in a given unit, then technically the units have been provided and the answer would just be the numerical portion, as providing the units in the answer would effectively square the units. But I would personally accept the answer if the units were provided and would argue with an examiner that didn't that they were being unreasonable.

The proper way -- but seldom worth the effort -- is to do a propagation of errors.

Based on the values given, and lacking any other information to the contrary, you have to assume that the data values are

I = 38±0.5 A
V = 235±1 V

As fractional uncertainties, these are

ΔI/I = 0.5A/38A = 1.316%
ΔV/V = 0.5V/235V = 0.2128%

Assuming the uncertainties (i.e., measurement errors) are independent, the uncertainty in the result would be

ΔP/P = √[(ΔI/I)^2 + (ΔV/V)^2] = 1.333%

Given the nominal value of P = 8930 W, ΔP = 119 W.

Thus the best answer would be either

P = 8.9±0.1 kW

though

P = 8.93±0.12 kW

would not be unreasonable.

If just reporting a value and no uncertainty level, you would generally round the uncertainty down to the next smaller half digit, thus 123 W gets rounded down to 50 W, and the result reported such that it is consistent with that. So that would be

P = 8.9 kW

Since that has an implied uncertainty of ±0.05 kW = ±50 W.

The tricky part comes in interpreting trailing zeroes that are to the right of the decimal point. Does I = 400 A mean that you know that the current is within half an amp of 400 A, or does it mean that you only know it is within 50 A of 400 A? You can't tell without some other contextual information.

Fortunately, it seldom matters in practice because assuming a single value has more accuracy than it really does will not result in an excessive number of sig figs in the result because the final uncertainty is bounded by whichever measurement has the worst uncertainty. The above example is a case in point. Do we really know that the votlage is withing 0.5V of 235V? Probably not. But notice that it is the uncertainty in the current that completely dominates the uncertainty in the result.

Which is why you can look at the original numbers, say that one of them only has two sig figs and therefore assert that the answer is justifiable to only two sig figs. No one will bat an eye if it is reported to three sig figs, in accordance with normal convention, but reporting it to four sig figs, though generally accepted without much comment, really isn't justifiable in this case.

 

...
But using "3 decimal places" doesn't work. Take this problem, for instance. Would someone have to put
P = 8390.000 W
if reporting the answer in watts?

Yes.

...Or put
P = 0.008 MW
if reporting the answer in megawatts?

Again, yes. Results to 3 decimal places were required in exams.

General rules are not always perfect. "3 decimal places" is a perfectly good standard for many cases, and is better than no standard.

Take for example this situation;
R = 2v / 3A = 0.666 (repeating) ohms

Going with a standard of 3 decimal places gives a very reasonable 0.667 ohm(assuming rounding was done).

Going with the initial precision of <1 significant digit gives you two fairly unreasonable answers of 0 ohm or 1 ohm.

...
If specifically asked to report the answer in a given unit, then technically the units have been provided and the answer would just be the numerical portion, as providing the units in the answer would effectively square the units.
...

I think you are wrong there. English semantics allows for some redundancy where it would be logical or "common sense".

For example If I asked how many apples are there in 2 rows of 3 apples per row, the result is 6 apples, and could be expressed as "6" (because the units are already given) or expressed equally correctly as "6 apples". There's no reason to believe that the answer is squared or requires squaring in any way.

...
The proper way -- but seldom worth the effort -- is to do a propagation of errors.
...

Nice work, thanks! I haven't seen that in many years.

It will be interesting to see what the "error" in the O.P's question actually was, but I'm pretty sure we already found it.

 

Again, yes. Results to 3 decimal places were required in exams.

Then that makes it actually pretty meaningless. I think we can agree that the level to which a quantity is known is independent of the prefix one attaches to the unit when the quantity is reported.

General rules are not always perfect. "3 decimal places" is a perfectly good standard for many cases, and is better than no standard.

I never said that it was "3 decimal places" or nothing. That's a false dichotemy. In fact I specifically recommended three significant figures when lacking a good reason to do otherwise, and I think that the three sig fig rule is far superior to a blind 3 decimal places rule. The primary thing is that it maintains the level of reported accuracy independent of the prefix or even units that are used.

Consider this:

Something is measured as being 0.502 yards. So one person reports that as being 1.506 feet, another person reports it as being 18.072 inches, and another person reports it as being 459.029 millimeters.

Now, you only see the 459.029 millimaters. Are you going to really believe that this measurement of something, whatever it is, that is nearly half a meter long is really known to within a micron? Or are you going to question the legitimacy of reporting a result to that many sig figs?

Wouldn't it be better to report results in such a way that they remain consistent as far as the degree of accuracy they imply? So these values would be reported as 1.51 feet (or possibly 1.506 feet if you use the rule that a leading one doesn't count as one of the three sig figs), 18.1 inches (or, again, possibly 18.07 inches), and 459 millimeters?

This last implies that the measurement is known to roughly 0.5 mm, which is 0.0016 feet which is consistent with the 0.0005 feet implied by the original value.

Take for example this situation;
R = 2v / 3A = 0.666 (repeating) ohms

Going with a standard of 3 decimal places gives a very reasonable 0.667 ohm(assuming rounding was done).

Going with the initial precision of <1 significant digit gives you two fairly unreasonable answers of 0 ohm or 1 ohm.

Three points:

1) Going with the initial precision as the guide would not yield either 0 Ω or 1 Ω, it would yield 0.7 Ω, which is very possibly all the better that the resistance is actually known to. In practice, results such as these often ARE only known to one or two sig figs because the tendency, if they are known to three or more, would be to report at least one of them after the decimal point. Think about it. If you measure voltage and your meter displays 5.0 V, you might right that as just 5 V, but if it said 5.027 V, aren't you very likely to at least write 5.0 V?

2) Small whole numbers are often considered either exact or are considered to be known to three sig figs, barring information to the contrary. So, by that convention, that would be reported as 0.667 Ω.

3) What about 2 V / 3000 A. Going with a standard of 3 decimal places that would have to be reported at 0.001 Ω. Going with three sig figs would make it 0.000667 Ω. Which is more reasonable?

I think you are wrong there. English semantics allows for some redundancy where it would be logical or "common sense".

The boundaries between scientific rigor and prose semantics is blurry, to be sure. I don't think there is a concrete right or wrong answer on this one. Which is why I indicated that I would not only accept either, but I would take issue with someone that didn't.

A more likely example of this would be in a table in which the column heading indicated units, such as "Length (cm)". The numbers in the column should NOT include the units. But occasionally you will see an entry in such a column that is outside the range conveniently reported in the indicated unit. Technically, it should be given using scientific notation in that unit, but it would not be unreasonable to override the indicated units by supplying a different unit explicitly, such as saying "2.1 m". You might also have an entry that is naturally expressed in another unit, so you might put "1 in", particularly if it was useful to indicated that this were someone exact or some definition. I don't know how the editors of a prestigious journal would treat such a practice. My guess is they would insist that the entries be 2100 "(2.1 m)" and "2.54 (1 in)" since putting alternate measurements in other units in parentheses after the main measurement is the "official" way.

It will be interesting to see what the "error" in the O.P's question actually was, but I'm pretty sure we already found it.

It would be nice to know.

 

Consider this analogy:

.......................

Remember a number of years ago the Mar's Climate Orbiter crashed because "....the flight system software on the Mars Climate Orbiter was written to take thrust instructions using the metric unit newtons (N), while the software on the ground that generated those instructions used the Imperial measure pound-force (lbf). This error has since been known as the "metric mixup" and has been carefully avoided in all missions since by NASA..." (from Wikipedia).

 

I use the Mars orbiter example to show that things really do happen, but I've found that there is just a visceral reaction when you use a medical example, even a hypothetical one, because people can easily relate to the idea of them or a loved one dieing because of a doctor's careless mistake. After instilling that idea firmly in their mind, they then can sometimes make the transition to understanding how much more costly an engineer's careless mistake could be for them and their family.

 

...
Something is measured as being 0.502 yards. So one person reports that as being 1.506 feet, another person reports it as being 18.072 inches, and another person reports it as being 459.029 millimeters.

Now, you only see the 459.029 millimaters. Are you going to really believe that this measurement of something, whatever it is, that is nearly half a meter long is really known to within a micron? Or are you going to question the legitimacy of reporting a result to that many sig figs?
...

Me personally? I'm very happy with the answer of 459.029 mm.

The answer of a calculation is superior if it has less error induced from the calculation.

Although this may not matter in your examples, where the answer is a finished exam question result, in my (personal) experience the answer from a simple calculation will often be used later as data entry for further calculations.

If a person gets obsessive about the precision and rounding the answers to the "optimal" decimal places, then data is lost.

So yes, I would go with 459.029 mm and be happy that data could later be plugged into a "mm per second" calc or whatever and retain a great deal of the precision of the initial values in the calc who's answer was "459.029xxx".

Some of my notes when doing design work have a LOT more than 3 decimal places written on them.

 

Me personally? I'm very happy with the answer of 459.029 mm.

I think the point he's trying to make is that the significant digits imply the precision of the measurement. I can take my my yard stick, which has graduations of 1/4" and take a rough measurement of a piece of wood to be 1.5ft. Let's say this piece of wood is from a mock-up of something that my team and I are planning to build out of metal, but is still a work in progress. I provide the measurement to someone else who casually asks for it, and they convert it to 457.200mm and provide figure that to the person who does the CAD work. So the CAD guy sees 457.200mm, which implies a high degree of accuracy from a precision tool, and he specifies in a tolerance to the machinists of 457.200mm ± .05mm, which is a very tight tolerance and hard to machine. They waste a bunch of time and money machining this super tight tolerance, and in the end, it's off by a full 1/8" because of the precision of the original measuring device.

Sure, you could chock that example up to poor team communication, and it is true to a degree, but exercising proper use of significant digits might have clued someone in along the way.

BTW what WBahn is saying about significant digits is what I was taught in college chemistry. I wish I could scan it and post it; it was chapter one. It explained it really well, with plenty of examples. In lieu of that, here is what Wikipedia has to say about it: http://en.wikipedia.org/wiki/Significant_figures

 

Intermediate results should generally retain at least two sig figs more than is justified specifically to prevent unintended errors due to roundoff creeping in and affecting the least significant digit (or more) in the final result. Where possible, it's generally best just to retain intermediate results in memory of some kind so that roundoff occurs well down the sig fig chain.

Of course, there are cases where treating sig figs is particularly tricky and extra care must be exercises. This often happens when taking the differences of two numbers that are close to the same value or working with logarithms, which are highly compressive in nature. In both of those situations it is very easy to lose all of your significant digits and end up with a completely meaningless number without realizing it. How this would manifest itself is that, were a careful propagation of errors to be done, the uncertainty in the final result might be an order of magnitude or more larger than the result itself. In essence, you would have a negative number of sig figs!

As an aside, how many sig figs are in a number? The "rules" are really nothing more than useful conventions and are somewhat arbitrary and represent a compromise between convenience and rigor. But the concept itself is not at all arbitrary, yet is almost never discussed or presented; only the rules are.

The concept is actually fairly easy to get at. The idea of "how many sig figs are in x" stem from the notion of counting the number of digits that convey meaningful information. A simple way to count the number of digits in a value is to take the logarithm of the magnitude of the number in the number base being used. For for most purposes, that's base-10. So how many digits in the value 1000 or 9999? The base-10 log of each are 3.00000 and 3.99996. So if we want an integer result, we could simply round the base-10 log to the nearest whole number. So what is the smallest value that would have "4 sig figs"? This might seem simple and that the answer should be 10^3.5, or 3163, but there's a subtlety that is being ignored that I'll get to in a bit.

But let's forget about rules. There's nothing that says that you can't have 3.27 sig figs -- it doesn't have to be an integer. Having 92.25" ± 1/8" has more precision than having 14.75" ± 1/8". But how much does each actually have? The answer is that it is the base-10 logarithm of the ratio of the value to the uncertainty. So here we would have log10(92.25"/(1/8")) = 2.868 and log10(14.75"/(1/8")) = 2.072, respectively. These numbers are intuitively reasonable. With an uncertainly on the order of 0.1", we have "3 sig figs" by the rules, but looking a bit closer we would agree that the first number has closer to the full three sig figs while the second number has barely gotten better than two sig figs.

With this in mind, let's revisit what the smallest number is that would have "4 sig figs". If we have a value x reported as an integer, then the implication is that the uncertainty in that value is 0.5, making the number of sig figs log10(x/0.5). The smallest number that yields 3.5 sig figs would therefore be 1582. This is why it is frequently the case that, when reporting a number, that a leading 1 is not counted as significant, since anything between 1000 and 1581 "techncially" only has three sig figs. But, of course, anything from 1582 to 1999 technically has four. So it's a bit of a tossup, but using the "leading 1s don't count" rule biases the result in favor of keeping an additional digit that only has weak significance over discarding a digit that, after all, does have at least weak significance.

So what about when you've lost all significance and the uncertainly is larger than the reported value? Does that really have "negative" sig figs? Yes, it does. Let's say you end up reporting the difference between two lengths as 0.0125" but the uncertainty in that number turns out to be ±0.25". The number of sig figs would be log10(0.0125"/0.25") = -1.301 and, indeed, you have negative sig figs. One way to think of this is that you have, in some respects, less information than if you had no information. In the face of having no information, you would just say that you know that the answer is not zero (one length is certainly at least a tiny bit longer than the other), but that you don't know if it is positive or negative and it could be up to 1/4" either way. The big thing is that you know that you don't know which length is actually longer. But by saying that the difference is 0.0125", you are claiming that you know that the first length is actually longer than the second, because the difference is positive.

In most situations, it is better to know that you don't know something, than to think you know something when, if fact, you don't. Hence saying that the number of sig figs in this case is negative conveys a meaningful message. In that regard, the more negative the number of sig figs the more you don't know. In the case that your reported answer is 0, the number of sig figs is -∞, which means you have absolutely zero knowledge -- it's a complete roll of the die as to whether the value is actually positive or negative, even. If the magnitude reported value is equal to the uncertainty, then you have 0 sig figs, meaning that you can't say much about the magnitude, but you do at least have confidence whether the actual value is positive or negative. As the sig figs get more and more negative, it's a sign of an increasing likelihood that your knowledge of even whether the actual value is positive or negative is getting worse and worse.