Significant Figures: Why Leading Zeros Don't Count

[RaduAndrei]

I measure some quantity with an instrument that it is precise to two decimal points.
So maybe I get 8.84 V. Then I do some changes in my parameters and get 0.01 V.

The two measured values are precise to two decimal points. But the first one has three significant figures, while the second one has only one significant figure.

Why is that? Should not both of them have the same number of significant figures?

Why are the leading zeros not considered significant?

 

[jbriggs444]

Why is that? Should not both of them have the same number of significant figures?

Significant figures are a way of dealing with "relative error". That is, error as a fraction of the actual value.

If the actual value is smaller, the relative error is larger.

 

[Bystander]

Why are the leading zeros not considered significant?

How many "leading" zeroes can you write?

 

[RaduAndrei]

How many "leading" zeroes can you write?

For a particular unit that I chose and is Volts, only two leading zeros.

 

[RaduAndrei]

Significant figures are a way of dealing with "relative error". That is, error as a fraction of the actual value.

If the actual value is smaller, the relative error is larger.

But what is the connection of significant figures to the precision? By precision I mean the number of decimal points to which a measured value is known. For ex, a precision of 0.1 or 0.01.

 

[DrGreg]

For a particular unit that I chose and is Volts, only two leading zeros.

And what if you chose millivolts? Or kilovolts?

 

[RaduAndrei]

And what if you chose millivolts? Or kilovolts?

8.84 in mV would be 8850 and 0.01 in mV would be 10.

So 3 significant figures vs 1 significant figure. The 0 at the end should not count.

That's the thing. I do not get the connection between significant figures and the precision. An instrument with a certain precision (two decimal points for a particular unit) gives of two numbers with different number of significant figures. Or maybe there is a connection for numbers larger than 1 and another connection for numbers between 0 and 1.

 

[jbriggs444]

But what is the connection of significant figures to the precision? By precision I mean the number of decimal points to which a measured value is known. For ex, a precision of 0.1 or 0.01.

log(value/error).

 

[RaduAndrei]

log(value/error).

Value being the true value or measured value? Error being the relative error or absolute error? And log(value/error) representing what?

 

[jbriggs444]

Value being the true value or measured value?

As long as the absolute error is small compared to the the true value (or to the measured value) then it does not matter which.

Error being the relative error or absolute error?

The absolute error. For instance, if the accuracy of the measuring device is +/- .001 then the error is .001.

And log(value/error) representing what?

If you know what the value is and what the error is, what do you think the log(value/error) is?

 

[RaduAndrei]

8.84 in mV would be 8850 and 0.01 in mV would be 10.

So 3 significant figures vs 1 significant figure. The 0 at the end should not count.

That's the thing. I do not get the connection between significant figures and the precision. An instrument with a certain precision (two decimal points for a particular unit) gives of two numbers with different number of significant figures. Or maybe there is a connection for numbers larger than 1 and another connection for numbers between 0 and 1.

Maybe the connection is this.
Let's say we have an instrument which has a precision of 2 decimal points for a particular unit V.
Then the number of significant figures can be: 3 for numbers of the form 4.55, 2 for numbers of the form 0.85, 1 for numbers of the form 0.03. That's it if the numbers are below 10. And if the the numbers can be whatever, then the number of significant figures can be from 1 to n, whatever n is.

So a significant figure is a digit that adds to the precision, but I thought there was a connection between the number of decimal points and number of significant figures. Anyway.

 

[jbriggs444]

That's it if the numbers are below 10. And if the the numbers can be whatever, then the number of significant figures can be from 1 to n, whatever n is.

What does this mean? What numbers and what n are supposed to be whatever?

 

[RaduAndrei]

What does this mean? What numbers and what n are supposed to be whatever?

The numbers being the measured values. If the measured value if 88.85 then I have 4 sign figures.
For 4544.34 I have 6 sign figures.

The precision is always at two decimal points, but the number of signif figures can be from 1 to n depending on what the measured value is.

 

[RaduAndrei]

As long as the absolute error is small compared to the the true value (or to the measured value) then it does not matter which.

The absolute error. For instance, if the accuracy of the measuring device is +/- .001 then the error is .001.

If you know what the value is and what the error is, what do you think the log(value/error) is?

The number of significant figures? The integer part at least plus 1.

 

[jbriggs444]

The numbers being the measured values. If the measured value if 88.85 then I have 4 sign figures.
For 4544.34 I have 6 sign figures.

The precision is always at two decimal points, but the number of signif figures can be from 1 to n depending on what the measured value is.

Again, what is "n" supposed to denote?

 

[RaduAndrei]

Again, what is "n" supposed to denote?

The number of significant figures.

 

[jbriggs444]

The number of significant figures.

So it is your position that the number of significant figures can range from one to the number of significant figures.

 

[RaduAndrei]

So it is your position that the number of significant figures can range from one to the number of significant figures.

Yes. n can vary from 1 to whatever.

I understand that it is a mistake to say that x can vary from 1 to x. But you got my point.

 

[jbriggs444]

Yes. n can vary from 1 to whatever.

If n is the number of significant figures then the number of significant figures is n. But that statement tells no one anything about what the number of significant figures is.

 

[RaduAndrei]

If n is the number of significant figures then the number of significant figures is n. But that statement tells no one anything about what the number of significant figures is.

I do not get where you are going with this.

 

[jbriggs444]

I do not get where you are going with this.

Re-read what you have written about n and see if there is any sense to be made of it.

 

[Isaac0427]

I think I can translate--tell me if I'm wrong:
Teachers say that the number of sig figs represents the precision of the instrument, however the OP has found a situation in which that does not make sense: you measure the mass of something to be 4.329 kg (4 sig figs). The same device measures something far more massive and comes up with 563214903.271 kg (12 sig figs). The teacher's definition of precision seems to fail; a device measures two different masses with the same precision, but gets answers with drastically different amounts of sig figs. Both times the device measures to the nearest gram, but the second object is about 8 orders of magnitude larger than the first object, so the result has 8 more sig figs.

I believe the answer to this is that sig figs don't really have any thing to do with precision, it's all in the number of known decimals. Is that correct?

 

[Isaac0427]

Also, look at post number 2 again. @jbriggs444 explains what sig figs actually do; show relative error.

 

[Svein]

Anecdote: Way back when, in my first year at the University, we had a crash course in measurements and error estimates. The favorite story of the lecturer dealt with a new world record in javelin throw. The American Bud Held had thrown the javelin 263 feet 10 inches - and a journalist had converted that to metric measures (this was before the inch was standardized to 25.4mm) and come up with a result of 80.41754m.

His point was: A grain of sand is about 1mm in diameter or 0.001m. The mark left by a javelin when it lands (usually in a grass field) is at least 4inches long. Thus, 263 feet 10 inches indicates a measurement precision of about one inch - which is reasonable. Translated to metric it indicates a precision of about 2.5cm. So - the journalist invented a precision that was not present in the original measurement.

----------

I measure some quantity with an instrument that it is precise to two decimal points.
So maybe I get 8.84 V. Then I do some changes in my parameters and get 0.01 V.

A digital instrument has a resolution and a precision. The usual digital multimeter has a precision of 3½ digits or about 1/1000 - which tells us something about the analog part of the meter. The resolution is the "step size" of the digital part of the meter. 1/1000 is the equivalent of 10 bits - and an analog-to-digital converter with 12 bit resolution is basic technology. Thus you can get a measurement with lots of figures, not all of which have any significance.

Anecdote: When digital thermometers first came out, they had a tendency to show the temperature with at least two decimals (as in 17.65°C). That represented the resolution of the digital part of the thermometer. Just for fun I checked it against a calibrated mercury thermometer and got a calibrated reading of about 1°C higher. The precision of the mercury thermometer was 0.25°C, so the precision of the digital thermometer was about 1°C. The decimals given by the digital thermometer were useless. Decide for yourself whether or not they were significant figures.

 

[RaduAndrei]

I think I can translate--tell me if I'm wrong:
Teachers say that the number of sig figs represents the precision of the instrument, however the OP has found a situation in which that does not make sense: you measure the mass of something to be 4.329 kg (4 sig figs). The same device measures something far more massive and comes up with 563214903.271 kg (12 sig figs). The teacher's definition of precision seems to fail; a device measures two different masses with the same precision, but gets answers with drastically different amounts of sig figs. Both times the device measures to the nearest gram, but the second object is about 8 orders of magnitude larger than the first object, so the result has 8 more sig figs.

I believe the answer to this is that sig figs don't really have any thing to do with precision, it's all in the number of known decimals. Is that correct?

That is right. So the number of sig figures do not represent the precision, they just add to the precision.

 

[RaduAndrei]

Re-read what you have written about n and see if there is any sense to be made of it.

There is sense. I just did not say it formally. It is a simple idea. No need to be pedantic.

 

[Svein]

That is right. So the number of sig figures do not represent the precision, they just add to the precision.

No, they do not add to the precision.

If the larger mass is measured with 12 significant figures - a precision of one part in 10¹², I would like to see the measuring device!

 

[RaduAndrei]

No, they do not add to the precision.

If the larger mass is measured with 12 significant figures - a precision of one part in 10¹², I would like to see the measuring device!

So. In a book it says:
"The uncertainty is just an estimate and it should not be stated with too much precision. For example, instead of saying g = 9.82 +/- 0.02385 m/s2, we should say g = 9.82 +/- 0.02 m/s2 because you cannot know the uncertainty to four significant figures. As a rule, experimental uncertainties should almost always be rounded to one significant figure"

So observe that the author talks about precision and significant figures as they are related and I too believe that. Many authors seem to play with this word precision while talking about significant figures.

What I am asking here is what is the connection between them two more precisely. I know that the precision refers to the repeatability of a measurement. For example, if I measure 4.54, 4.56, 4.55, etc, then I can say that I have a precision of 0.01. And the number of significant figures is 3. If the precision were to be 0.001 then the number of significant figures would have been 4.

It seems to me that the significant figures add to the precision of a measurement. Or I can say that the precision represents the number of significant figures in the decimal places. Exception can be like 0.005, 0.004, etc There is only one significant figure here while the precision is 0.001. But if the precision increases then the number of sign figures also increases. So I can safely say that the significant figures add to the precision of a measurement and just that.

Am I wrong?

 

[jbriggs444]

Significant figures do not add to precision. They express the precision in a different way.

 

[RaduAndrei]

Significant figures do not add to precision. They express the precision in a different way.

I am not saying 'add' in the way that I actually add the number of significant figures with the precision. I'm saying it in a figuratively way.
The more significant figures you have in the decimal places, the more precision you have.

Or is this wrong?

 

[Svein]

I am not saying 'add' in the way that I actually add the number of significant figures with the precision. I'm saying it in a figuratively way.
The more significant figures you have in the decimal places, the more precision you have.

Or is this wrong?

If the person doing the measurements knows what he/she is doing, the number of significant figures given represents the precision of the measurement. If not (see my earlier anecdote about introducing a lot of figures when converting from one standard to another), the number of "significant" figures are neither significant nor bear any relation to the measurement precision.

 

[jbriggs444]

I am not saying 'add' in the way that I actually add the number of significant figures with the precision. I'm saying it in a figuratively way.
The more significant figures you have in the decimal places, the more precision you have.

Or is this wrong?

That idea is correct. A better word choice might be "related".

 

[RaduAndrei]

Significant figures do not add to precision. They express the precision in a different way.

Ok. I finally understood it. (with the help of the book introduction to error analysis by Taylor)

A measured value with n significant figures means an uncertainty of one unit in the nth significant figure. Sometimes it means a bigger uncertainty, sometimes it means a smaller uncertainty, depending on the situation. But we adopt a middle of the road definition that it is one unit.

An uncertainty of one unit in the nth significant figure means some variable precision depending on the number (here precision is defined as uncertainty/measured value). So for 10, the precision is 10%, while for 99 the precision is 1%. We can say that the precision is roughly 50%.

In general for n sign figures, the precision can vary from 10^(-n+3)% to 10^(-n+2)%.

Thus, there is an approximate correspondence between the number of significant figures (as we defined them) and the precision (as we defined it) given by:
roughly precision [%] = 10^(-n+3)/2 %

PS: Also we adopted the convention that all trailing zeros are significant.