I've have several physics and chem books which attempt define the term "significant figures".

Unfortunately, the rules seem to be rather vague (they read like the IRS tax code) and I'm still not completely clear what actually constitutes a significant digit. My understanding is that significant figures refer to the known accuracy of each digit to the right of the decimal point.

For example, if a caliper in a machine shop can accurately measure to 1/1000 of an inch and it measures the diameter of a shaft as being 2.987 inches, then 9,8, and 7 are to the right of the decimal point and considered as significant figures. However if the same shaft is measured with caliper accurate to 1/10,000 of an inch and it indicates 2.9870 inches, then the last zero is also a significant figure.

So can anyone provide an answer to my question (in their own words rather than posting a link)?

 

Glenn...I believe in that context you could say it means precision........how fine you need/want to measure.

 

Your interpretation is the one I use.

 

This is what I found on line:
https://en.wikipedia.org/wiki/Significant_figures

It's seems like something from the IRS tax code and it's got a lot of exceptions. One of the main applications of significant figures is scientific notation which has a single digit number with two digits to the right of the decimal point multiplied by a power of 10.

 

Signifigicant figures is different thing than accuracy and precision and it doesn;t mean after the decimal point.

Example: 1.4567 is 5 significant figures. 1.1111100 is 7 significant figures.

It's 8 sig figs, not 7. SLK001 correction in post #18. I can't count.

You cant ad these numbers and get 1.567800. The answer MUST be 1.5678 because one of the measurements doesn't have the same number of significant digits.

If you have PI*r^2 and measure r to 0.001, you don;t use PI = 3.1415926 because it's meaningless.
r^2 would be significant to 0.000001, and 3.141592 would be the correct number to use PI is probably a bad example.

Precision
Is how something compares to the standard. e.g. 1.000000000000 Volt.

Accuracy is if you have to pieces of something that's cut:
10" +-1/4 and another that's cut 10+-1/8 , then the pieces added together is 20+-3/8

 

One piece of weirdness is that the numbers .001 and 0.001. Inthe second number the first zero is significant.
The first number could be 0.001, 1.001, 2.001 ....9.001, if you use rounding in reverse.

 

"significant figures" sets the bandwidth of precision.

 

Signifigicant figures is different thing than accuracy and precision and it doesn;t mean after the decimal point.

Example: 1.4567 is 5 significant figures. 1.1111100 is 7 significant figures.

You cant ad these numbers and get 1.567800. The answer MUST be 1.5678 because one of the measurements doesn't have the same number of significant digits.

If you have PI*r^2 and measure r to 0.001, you don;t use PI = 3.1415926 because it's meaningless.
r^2 would be significant to 0.000001, and 3.141592 would be the correct number to use PI is probably a bad example.

Precision
Is how something compares to the standard. e.g. 1.000000000000 Volt.

Accuracy is if you have to pieces of something that's cut:
10" +-1/4 and another that's cut 10+-1/8 , then the pieces added together is 20+-3/8

I have never before seen it explained like that.

To me,

resolution refers to resolving power - what is the finest measurement one can make.

precision refers to repeatability - the ability to repeat the measurement and get the same result.

accuracy refers to the exactness of the measurement - how closely the measurement meets the correct value.

significant figures refers to the usefulness of the quoted digits to the right, including trailing zeros - how meaningful are the rightmost digits (including trailing zeros) when resolution, precision and accuracy are all taken into account (i.e. the worst case of resolution, precision and accuracy).

 

that was the best one yet.

 

I've have several physics and chem books which attempt define the term "significant figures".

Unfortunately, the rules seem to be rather vague (they read like the IRS tax code) and I'm still not completely clear what actually constitutes a significant digit. My understanding is that significant figures refer to the known accuracy of each digit to the right of the decimal point.

For example, if a caliper in a machine shop can accurately measure to 1/1000 of an inch and it measures the diameter of a shaft as being 2.987 inches, then 9,8, and 7 are to the right of the decimal point and considered as significant figures. However if the same shaft is measured with caliper accurate to 1/10,000 of an inch and it indicates 2.9870 inches, then the last zero is also a significant figure.

So can anyone provide an answer to my question (in their own words rather than posting a link)?

The decimal point has nothing to do with it. If it did then 0.25 gallons and 32 fluid oz would not have the same number of sig figs even though the two are identical measurements.

"Significant figures" is a simple but approximate shorthand way of expressing the level of confidence in the correctness of a value.

First, you need to understand that accuracy and precision are two different concepts. Accuracy refers to how close the measurement is to the correct value. Precision refers to how reproducible the measurement is. You can have one without the other, though usually any effort to get high performance in one entails getting a comparable performance in the other.

Take as an example your car odometer. Let's say that you reset your trip odometer today when you are stopped with your front bumper even with a stake that you have put in the ground on the side of a long, straight highway. You then drive down the highway, always staying in the same lane, and stop just as the trip odomenter reads 100.0 miles and put another stake in the ground even with your front bumper. You repeat this every day for a hundred days, each time putting a new stake at the ending point. At the end of that time there are a couple of different questions that can be asked. First, what is the actual distance between the starting stake and each of the ending stakes? If you car has tires on it that are 10% taller than they should be, the stakes will be about 110 miles apart instead of 100 miles. That's a measure of how accurate your odometer is. Your accuracy is only within 10%. But second you can focus on how close the ending stakes are to each other. It's possible that the nearest stake and the furthest stake are only 200 feet apart and that the vast majority of them are within 50 feet of the average. That's a measure of the precision of your odometer. Your precision is on the order of 100 ppm.

Which is better? It depends. If you have high-precision in your measurements, then you can often obtain high accuracy via calibration. But sometimes this isn't possible, for instance if the precision is limited by noise. In that case, if you can make the measurements sufficiently accurate (meaning that, on average, they are close to the correct value), then you can often obtain high precision by making an ensemble measurement and using the average.

There are lots of very fine nuances in expressing the uncertainty in a measurements, so seeing something like

distance = 401.53 ± 0.02 miles

just doesn't really tell the story. Does it mean that we are absolutely certain that the correct distance is somewhere between 401.51 miles and 401.55 miles? Maybe. But it might also mean that the standard deviation of our uncertainty is 0.02 miles so while it is highly likely that the correct value is within that range, there is some chance that it is not. Also, does that uncertainty represent the accuracy of the measurement, or the precision? It could be either. It also says nothing about what the dominant source of the error is. It could be random errors due to the resolution of the measuring instrument, or it could be due to environmental factors such as temperature or fundamental noise. It could be different kinds of systematic errors such as offset, gain, or higher order terms. In many cases we really don't care, we just need a qualitative warm fuzzy with regards to how "good" the measurement is. In other cases it is critically important and page after page is devoted to a comprehensive, quantitative assessment of multiple error sources.

So what do we mean when we say that a distance measurement, say 401.53 miles, is good to five sig figs? What we mean is that, roughly, we are confident that the reported value is within half of the fifth digit, from the left most, of the correct value. Like the above, sig figs, by itself, leaves unanswered the question of whether we are talking accuracy or precision and whether we are talking about hard or statistical limits. Why half of the last digit? In the absence of better information we are saying that if the actual value was half of that last digit higher or half of that last digit lower one of those would have resulted in a different reported value by one increment of the least significant digit.

So if we claim a distance measurement of 401.53 miles, we are implying five sig figs and that it should be considered as 401.53 ± 0.005 miles. But that could be optimistic and it could be as bad as 401.53 ± 0.05 miles and still be justifiable as being "five sig figs". That's why "sig figs" are only an approximate indicator of the confidence in the measurement.

Another way of looking at sig figs quantitatively is that the base-10 log of the measurement divided by the span of the uncertainty is the number of sig figs. So for 401.53 ± 0.005 miles this is

sig figs = log(401.53 miles / 0.01 miles) = 4.6 sig figs.

For normal use, we take the ceiling and express the answer using that number of digits (in base 10). Notice how this gives us a very natural way of knowing how many sig figs we would have in some other number base or when doing units conversions.

For instance, if we have 128 fluid ounces but we need to report it in U.S. gallons, how should we write it? If we just say 1 gallon, we are only claiming to know the value to within half a gallon, whereas our original value is claiming to know it within a half an ounce. Our original value had

s.f. = log(128 oz / 1 oz) = 2.1 sig figs.

So our new value should be expressed with 3 sig figs (assuming we aren't going to be more explicity with the uncertainties) as 1.00 gallons.

 

I was taught it was the number of decimal places. Of course I was also taught that 2 significant figures on a slide rule could solve anything.

 

I was taught it was the number of decimal places. Of course I was also taught that 2 significant figures on a slide rule could solve anything.

That would depend on what you mean by "decimal places". There's more than one interpretation. The most common would say that the value

x = 123456.78

is reported to two decimal places.

But this value is NOT written with two sig figs.

Another interpretation is that it is the total number of digits in the value, which in this case would be eight and, indeed, this value is reported to contain eight sig figs.

But what about

x = 123000

How many sig figs does it have?

If it is the previous number rounded to the nearest thousand, then it has three sig figs. But what if it is a value rounded to the nearest ten dollars? Or the nearest dollar?

As written, we don't know. We might be able to determine from the context, or we might not. There are also various ways of indicating where the significant digits end and place-holding trailing zeros begin.

 

In that case, I would say the number of significant figures is the number of digits following the decimal point after the number has been normalized to scientific notation.

Hence \(1.23 \times 10^5\) has two significant figures while \(1.2300 \times 10^5\) has four significant figures.

 

Example: 1.4567 is 5 significant figures. 1.1111100 is 7 significant figures.

You cant ad these numbers and get 1.567800. The answer MUST be 1.5678 because one of the measurements doesn't have the same number of significant digits.

If I was adding them, I would get 2.5678. And 1.4567 is 4 significant digits.

 

Hello,

Significant figures (or digits) is just that. It says it in the name itself. Significant means significant to the mathematics or application at hand, which means the number of digits that express the number to the desired accuracy.

This is simple when you think about it. If you have two numbers:
123
1234

you assume that the first one requires all three digits in order to complete a calculation correctly, while the second number requires four.

Thus the following all have 3 significant figures:
123
12.3
1.23
123e-3
1.23e-4

The number:
0.0123

has interpretation of either 3 or 4, depending on the application. If you create a table of numbers:
0.0123
0.1234
0.2345

it might really mean this instead:
0.01230
0.1234
0.2345

but that last zero in the first entry is not shown in order to keep all the numbers lined up.

So there is a little variability here, and usually there is some other reference as to what they are such as:
"Here is the data measured over two days. The voltages are shown to three significant figures"
0.123
0.045
0.523
1.220

note the last entry requires a zero at the end. The number 1.22 may have been rounded in order to achieve that. The zero may or may not be displayed.

You can use log base 10 and rounding to create numbers that all have the same significant figures.

 

In that case, I would say the number of significant figures is the number of digits following the decimal point after the number has been normalized to scientific notation.

Hence \(1.23 \times 10^5\) has two significant figures while \(1.2300 \times 10^5\) has four significant figures.

So \(9 \times 10^3\) has zero significant figures? That '9' has no significance?

 

So \(9 \times 10^3\) has zero significant figures? That '9' has no significance?

Ok, just add one more. I'm not picky.

 

Ok, just add one more. I'm not picky.

Yeah, I fell into that hole too, following the thread.

In KISS's post above, "Example: 1.4567 is 5 significant figures. 1.1111100 is 7 significant figures", the first is correct, but the second has 8 significant figures.

 

Yep. Good catch.

 

The main takeaway is that which side of the decimal point a digit is on has absolutely no bearing on the number of significant digits. It's simply how many digits (figures) have significance. Where people get messed up with this is that we need to treat trailing zeros a bit differently depending on which side of the decimal point they are on. But this is because we need them whether they are significant or not if they are to the left, while we only need them if they are significant when they are on the right.