Measurement Science: True Magnitude, Error & Accuracy

[Logical Dog]

Hello.

So one of the axioms of measurement science is that every measurement we make is an approximation and can never reflect the true magnitude of a physical phenomena.

We would then go on to teach that there is a certain "true" magnitude which is something that experts or majority would agree represents that particular physical phenomena "best"

Then one would define absolute error as being:
$$\left | Measured magnitude - Expected/"true" \right |$$

How is this expected/ "true" value arrived at? I can understand theoretically by calculations one might get some ideal number but in real life due to instruments not working properly or precision limitations, are there any other examples? Also is it correct to say accuracy is the reproducibility of the measured quantity through repeated measurements? how does one measure accuracy? through statistics I imagine.

Are there any examples where people do not "agree" on the "true" value of a measure? eg some constant in physics?

 

[anorlunda]

Are there any examples where people do not "agree" on the "true" value of a measure? eg some constant in physics?

Your question sounds very philosophical having to do with psychology rather than physics.

The only constant that even remotely sounds like what you ask for is Plank's Constant. Look it up on Wikipedia.

But for a simple example, think of a wheel rotating on its axis. We can measure speed and direction of a spot on the perimeter and the clock time and add up those measurements to estimate where the spot is after a whole revolution. But we can make much better estimates by simply noting that it returns to the starting point after one or 1000 or 1000000 revolutions. Use that as a constraint on the speed & direction measurements and we can estimate the errors in those measurements. There is no magic, no mathematical, no physics trick there; just common sense.

 

[Logical Dog]

Your question sounds very philosophical having to do with psychology rather than physics.

The only constant that even remotely sounds like what you ask for is Plank's Constant. Look it up on Wikipedia.

But for a simple example, think of a wheel rotating on its axis. We can measure speed and direction of a spot on the perimeter and the clock time and add up those measurements to estimate where the spot is after a whole revolution. But we can make much better estimates by simply noting that it returns to the starting point after one or 1000 or 1000000 revolutions. Use that as a constraint on the speed & direction measurements and we can estimate the errors in those measurements. There is no magic, no mathematical, no physics trick there; just common sense.

Oh sorry, I don't mean to invoke philosophy here. I just wanted to know a little bit as this is one of my favourite topics though I am ill educated on it.

 

[kith]

I like your post and I don't see it as mainly philosophical.

The true value is usually estimated. This may sound vague but there are theorems in statistics how the best estimate looks like under very general assumptions (see Maximum likelihood estimation for example).

I recommend the book "An Introduction to Error Analysis" by Taylor. In your profile, I've read that you are still in high school but I think mathematically inclined high schoolers should be able to read it.

After you have read a couple of chapters in the book, you will be able to separate the philosophical bits ("is there really a true value?") from the actual science.

 

[Logical Dog]

I like your post and I don't see it as mainly philosophical.

The true value is usually estimated. This may sound vague but there are theorems in statistics how the best estimate looks like under very general assumptions (see Maximum likelihood estimation for example).

I recommend the book "An Introduction to Error Analysis" by Taylor. In your profile, I've read that you are still in high school but I think mathematically inclined high schoolers should be able to read it.

After you have read a couple of chapters in the book, you will be able to separate the philosophical bits ("is there really a true value?") from the actual science.

No haha I finished high school (a levels) in 2015 *well the results came then* haha. Thats pretty much the last qualification I "finished"..So I can't put my background as bachelors until I finish it. that's how I understand at least.

 

[kith]

Ah, ok. Have you considered updating your profile? ;-)

 

[f95toli]

Also is it correct to say accuracy is the reproducibility of the measured quantity through repeated measurements? how does one measure accuracy? through statistics I imagine.

No, that would be precision. If you repeat the same measurement many times and get nearly the same value every time your measurement is precise. If you measure something and the results is close to the true value the measurement is accurate.
It is not at all uncommon for measurements to be precise but inaccurate; this typically happens because of systematic errors (e.g. measuring using a piece of kit that has not been properly calibrated). You can also make accurate but imprecise measurements; by repeating that same measurement many times and taking the average your final measurement will then hopefully be accurate (this is not always true, there are lots of caveats in this case).

This is often illustrated using a dartboard
http://blog.minitab.com/blog/real-w...nt/accuracy-vs-precision-whats-the-difference

 

[Andy Resnick]

Also is it correct to say accuracy is the reproducibility of the measured quantity through repeated measurements? how does one measure accuracy? through statistics I imagine.

Are there any examples where people do not "agree" on the "true" value of a measure?

as f95toli states, there's a difference between precision and accuracy- precision is more easily quantified through statistics (many measurements).

There are lots of situations where accuracy is important and there is no "truth" value available: most measurements rely on the sensor to be highly repeatable but not to have 'absolute' accuracy. These sensors have to be calibrated prior to use (and so many have internal calibration mechanisms), and better measurement protocols use "two-point calibrations", where each calibration occurs near each end of the useful measurement range. pH meters often require 3-point calibrations because the electrode response is nonlinear.

Some sensors (I use a calcium indicator dye as an example) are not really repeatable- the response depends on a lot of variables that I can't really control. Then I have to 2-point calibrate the response in situ every time I use the dye to make a measurement.

 

[Logical Dog]

as f95toli states, there's a difference between precision and accuracy- precision is more easily quantified through statistics (many measurements).

There are lots of situations where accuracy is important and there is no "truth" value available: most measurements rely on the sensor to be highly repeatable but not to have 'absolute' accuracy. These sensors have to be calibrated prior to use (and so many have internal calibration mechanisms), and better measurement protocols use "two-point calibrations", where each calibration occurs near each end of the useful measurement range. pH meters often require 3-point calibrations because the electrode response is nonlinear.

Some sensors (I use a calcium indicator dye as an example) are not really repeatable- the response depends on a lot of variables that I can't really control. Then I have to 2-point calibrate the response in situ every time I use the dye to make a measurement.

Wow!

 

[Khashishi]

We typically are comparing some measured value to some theoretical value. The "true" measurement is whichever the scientist thinks is closer to the correct value. It really doesn't matter much if the calculated error is off by a little bit, since we're talking about an error on an error. It's just not important.

 

[Andy Resnick]

We typically are comparing some measured value to some theoretical value. The "true" measurement is whichever the scientist thinks is closer to the correct value. It really doesn't matter much if the calculated error is off by a little bit, since we're talking about an error on an error. It's just not important.

This is patently false and unscientific.

 

[sophiecentaur]

The notion of a 'true' value is just philosophical because all that Science can do is to come up with a value which is within a set of boundaries*. Those boundaries get closer together as the repeatability and accuracy of measurement improves. You can 'believe' or not that there is a true value in there somewhere. It makes no difference to the result of your measurements and we all should present experimental results with appropriate error limits.
* In the same way that good Science does not present any absolute truth (or even pretend that there is one).

 

[Andy Resnick]

The notion of a 'true' value is just philosophical because all that Science can do is to come up with a value which is within a set of boundaries.

Yes and no- there are certain physical quantities defined to be a specific numerical value: the triple point of water, the speed of light in vacuum, things like that. There are also physical artifacts that possess, by definition, a standard (hopefully) fixed value: the standard Ohm and Kilogram*, for example.

*The case of the standard mass is interesting for a variety of reasons, and the development of the Watt balance (which would fix Planck's constant to a specific value) is of interest.

 

[sophiecentaur]

In which case, I would call them Arbitrary quantities. (It's TRUE that someone made them up.) The uncertainty then gets off loaded onto the other, dependent variables in a formula.

 

[anorlunda]

I thought of one example that does seem to fit the OP's question about a "true value" as opposed to empirical evidence -- Newton's Law of Gravitation.

$F=G\frac{m_1m_2}{R^2}$

Since we have no quantum gravity theory yet, gravitation must be considered empirical. All observations so far seem to support 2 as the "true value" of the exponent of R. However, there are those who argue for MOND (https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics) in which gravity varies as some function other than $R^2$. MOND does not seem consistent with obervational data. Nevertheless, it is an example of a "true value" that most (but not all) people believe in that is not [yet] derived from theory but is justified by empirical evidence with finite precision and accuracy.

 

[sophiecentaur]

I thought of one example that does seem to fit the OP's question about a "true value" as opposed to empirical evidence -- Newton's Law of Gravitation.

$F=G\frac{m_1m_2}{R^2}$

Since we have no quantum gravity theory yet, gravitation must be considered empirical. All observations so far seem to support 2 as the "true value" of the exponent of R. However, there are those who argue for MOND (https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics) in which gravity varies as some function other than $R^2$. MOND does not seem consistent with obervational data. Nevertheless, it is an example of a "true value" that most (but not all) people believe in that is not [yet] derived from theory but is justified by empirical evidence with finite precision and accuracy.

That law makes massive assumptions about the nature of space. It assumes Cartesian space, in which the area of a sphere is 4πr², whatever the value of r happens to be. Cosmologists do not assume that any more, afaiaa.

 

[Andy Resnick]

In which case, I would call them Arbitrary quantities. (It's TRUE that someone made them up.) The uncertainty then gets off loaded onto the other, dependent variables in a formula.

Yes, in that '1 meter' is an arbitrary length. That's one reason why the Pioneer plaque and Voyager record reference the hyperfine transition of neutral hydrogen as a 'secret decoder ring'/rosetta stone to correctly translate the message contents. There's no uncertainty with that quantity- only measurement error.

 

[sophiecentaur]

There's no uncertainty with that quantity- only measurement error.

I would say that anything where QM is involved (i.e. Everything) involves uncertainty - not just measurement error. The Heisenberg Uncertainty Principle does more than just describe problems in measuring things.

 

[Andy Resnick]

I would say that anything where QM is involved (i.e. Everything) involves uncertainty - not just measurement error. The Heisenberg Uncertainty Principle does more than just describe problems in measuring things.

Not exactly- the uncertainty principle covers the limits to simultaneous measurements of conjugate quantities, e.g. position and momentum. It is possible to measure a single quantity with arbitrary precision.

 

[sophiecentaur]

Not exactly- the uncertainty principle covers the limits to simultaneous measurements of conjugate quantities, e.g. position and momentum. It is possible to measure a single quantity with arbitrary precision.

We're being a bit Angels on a Pinhead here, I think. Of course, we can say, for certain, that we can talk in terms of Integer Quantum Numbers but they only refer to atoms in isolation. As soon as another atom is brought significantly close, a new set of quantum levels are introduced and we soon get into a spread of possible quantum levels, even in a monatomic gas. So doesn't that imply that there has to be a bandwidth associated with atomic transitions? Your notion of 'certainty' in these matters has to rely on an ideal case which involves an arbitrarily limited number of variables. It's ok within the model but we are aware that the model itself is limited.

 

[Andy Resnick]

We're being a bit Angels on a Pinhead here, I think. Of course, we can say, for certain, that we can talk in terms of Integer Quantum Numbers but they only refer to atoms in isolation. As soon as another atom is brought significantly close, a new set of quantum levels are introduced and we soon get into a spread of possible quantum levels, even in a monatomic gas. So doesn't that imply that there has to be a bandwidth associated with atomic transitions? Your notion of 'certainty' in these matters has to rely on an ideal case which involves an arbitrarily limited number of variables. It's ok within the model but we are aware that the model itself is limited.

I think we are getting off track- my point is that there are certain physical quantities that have a specific and absolute value.

 

[sophiecentaur]

I think we are getting off track- my point is that there are certain physical quantities that have a specific and absolute value.

We'll have to disagree there, I think - unless you add the words "... under certain conditions" to that statement. Their values and the way they are specified hang on a particular model and are not necessarily universal. The upper and lower limits, such as 0K and c certainly apply in mainstream Science but our Science could easily have limits to its applicability - history is full of examples where a paradigm shift takes us outside a well held set of views.

 

[Andy Resnick]

We'll have to disagree there, I think - unless you add the words "... under certain conditions" to that statement. Their values and the way they are specified hang on a particular model and are not necessarily universal. The upper and lower limits, such as 0K and c certainly apply in mainstream Science but our Science could easily have limits to its applicability - history is full of examples where a paradigm shift takes us outside a well held set of views.

I'm somewhat willing to go along with this, since I agree that we don't know everything. Even so, it's important to note that there is currently no evidence that c_0 is not a fixed, universal, constant.

 

[WWGD]

Just wanted to recommend John Mandel's (Former from The National Bureau of Standards ) " The Statistical Analysis of Experimental Data". Chapter 6 deals with this specific topic/area.

 

[WWGD]

No, that would be precision. If you repeat the same measurement many times and get nearly the same value every time your measurement is precise. If you measure something and the results is close to the true value the measurement is accurate.
It is not at all uncommon for measurements to be precise but inaccurate; this typically happens because of systematic errors (e.g. measuring using a piece of kit that has not been properly calibrated). You can also make accurate but imprecise measurements; by repeating that same measurement many times and taking the average your final measurement will then hopefully be accurate (this is not always true, there are lots of caveats in this case).

This is often illustrated using a dartboard
http://blog.minitab.com/blog/real-w...nt/accuracy-vs-precision-whats-the-difference

Is it fair to say that , however precise, if the average of the measured values does not approach the (accepted) true value, then your measurements are innacurate and this is a sign of systemic errors? EDIT: I am assuming that random errors would cancel each other out. I am trying to understand this too in terms of the needed condition on errors in(multi) linear regression: Errors are required to have mean 0, be uncorrelated with each other as well as with the independent variables and I.I.D (Independent and Identically-Distributed; basically meaning all having equal variance). Having mean 0 means measuremements are accurate. I guess being uncorrelated with each other and/or the independent variables means errors are random rather than systematic. Sorry for the rambling, I am relatively new to this ( At least in a more precise/deeper sense).

 

[f95toli]

Is it fair to say that , however precise, if the average of the measured values does not approach the (accepted) true value, then your measurements are innacurate and this is a sign of systemic errors?

No, it can also just mean that there are inherent limitation in the way you are trying to measure something,
if you e.g. are trying to measure a quantity that is smaller than the 1/f noise level of your signal it won't matter how long you measure since 1/f noise does not average out; this is a VERY common situation
If you e.g are trying to measure a spectra by averaging for a long time you might still just end up getting a very precise measurement of the 1/f noise level.