Instrument accuracy and comparison measurements

[fog37]

TL;DR Summary

Instrument accuracy and comparison measurements

Hello,

All measurement instruments have a finite accuracy. For example, a thermometer may provide a temperature measurement that is +-2F from the "actual" and true temp value. For example, if the reported measurement is 70F, the actual temperature may be between 72F and 68F.

That said, let's assume we have 2 different materials and measure their temperatures $T_1$ and $T_2$. Both $T_1$ and $T_2$ are affected by the finite accuracy of +-2F but their difference $|T_1 - T_2|$ is not. Is that correct? Even if $T_1$ and $T_2$ are not the actual temperatures, their difference is constant (assuming the same environment conditions). If we repeated, under the same conditions, both measurements 1 hr later, $T_1$ and $T_2$ may assume different values but their difference would be the same...

However, for the same and single material, the finite accuracy imply that multiple measurements would produce different temperature values even under the same conditions...

 

[Baluncore]

Summary:: Instrument accuracy and comparison measurements

That said, let's assume we have 2 different materials and measure their temperatures T1 and T2. Both T1 and T2 are affected by the finite accuracy of +-2F but their difference |T1−T2| is not.

It will depend on how many different thermometers you use to make the measurements.

 

[Vanadium 50]

It also depends on if your 2F is really 2F or if it's really 3%, or something else.

 

[Dullard]

Not strictly correct. Your reasoning assumes that the error is a constant offset - that may be true, but cannot be assumed from a simple +/- 2% accuracy claim.

 

[fog37]

Hello, thanks everyone.

Yes, I am assuming that the same thermometer is being used to measure both samples. The rated thermal accuracy is:

Thermal Accuracy

±2°C or ±2%

So when I measure the temperature of each sample, each temperature value will be off. But does this matter if all I care about is the relative temperature difference? Same reasoning goes if there are more than 2 samples and compare their temperatures to determine the hottest and the coolest.

Would any temperature difference less than 2C between the samples not be meaningful?

 

[jrmichler]

Measurement error can be random, offset, or both. Search terms precision vs accuracy bring up a number of sites that discuss this. The difference is summarized in the following graphic:

Averaging a number of readings reduces the effects of random error (precision in the above graphic), but does not reduce the effect of offset error (accuracy in the above graphic).

 

[fog37]

Thanks jrmichler.

I see. So would you agree that, regardless of what the offset error may be, comparing the temperatures of two different samples is meaningful even if the temperature difference is a value lower than the stated instrument accuracy, in my example 2C? For example, the two samples differ in temperature by 1.3C. Would that difference be scientifically make sense or would it be nonsense since it is lower than the accuracy of 2C?

 

[jrmichler]

Measurement error is normally both offset and random. If you are comparing differences using the same instrument, the offset error should drop out, while averaging multiple readings will reduce the random error.

So yes, the comparison of two samples should be valid.

You test this by taking multiple temperature readings of the same sample, then plotting the temperatures vs time. A smooth curve has low random error.

 

[hutchphd]

I agree with these responses but this brings up one of the most useful (IMHO) results in all of physics: the Central Limit Theorem.
Loosely it says that if there are enough variable inputs that affect the value of the measurement (temp, phase of moon, time of day, passing truck, shaky hand, line voltage, humidity, etc etc etc) then the result will be Gaussian distributed. In my real world experience this result is both remarkably true and fabulously useful. Then you can characterize the system and anticipate results.

 

[fog37]

Measurement error is normally both offset and random. If you are comparing differences using the same instrument, the offset error should drop out, while averaging multiple readings will reduce the random error.

So yes, the comparison of two samples should be valid.

You test this by taking multiple temperature readings of the same sample, then plotting the temperatures vs time. A smooth curve has low random error.

So, somehow, that stated offset error value due to the limited accuracy is an interval and I thought that the offset would vary from measurement to measurement with offset error values within the stated +-2C or +-2% of the measurement. But the conclusion is that for comparison measurements, one done after the other, the offset error washes out and the comparison difference is valid...

 

[hutchphd]

But it behooves the investigator to vigorously investigate such an offset error. This is unlikely to be a designed feature in a system and so the prudent course of action is to calibrate excessively until one understands the reason.

 

[alan123hk]

All the above replies are very worthy of reference.

But I think for users, if the specification of the thermometer says that the accuracy is +/-2C or 2% under certain environmental conditions, and if the measured temperature is 50C, the real temperature should be in range of 48C to 52C or 49C to 51C, so of course we have to use the worse range (48C to 52C) for the measurement result, which can reasonably expect the manufacturer to guarantee.

On the other hand, before obtaining further information from the manufacturer, it is not certain to use one thermometer for multiple tests, or use multiple thermometers for multiple tests, and then on average can improve accuracy. I think this approach is safer because it reduces the possibility of disappointment caused by overly optimistic expectations.

There is one thing that should be more certain. Referring to the example mentioned above, if two thermometers of the same brand and the same model are used to measure the temperature of the same thing under the same environmental conditions, the temperature difference obtained is more than 4C, then people will indeed lose confidence in this thermometer.

 

[fog37]

All the above replies are very worthy of reference.

But I think for users, if the specification of the thermometer says that the accuracy is +/-2C or 2% under certain environmental conditions, and if the measured temperature is 50C, the real temperature should be in range of 48C to 52C or 49C to 51C, so of course we have to use the worse range (48C to 52C) for the measurement result, which can reasonably expect the manufacturer to guarantee.

On the other hand, before obtaining further information from the manufacturer, it is not certain to use one thermometer for multiple tests, or use multiple thermometers for multiple tests, and then on average can improve accuracy. I think this approach is safer because it reduces the possibility of disappointment caused by overly optimistic expectations.

There is one thing that should be more certain. Referring to the example mentioned above, if two thermometers of the same brand and the same model are used to measure the temperature of the same thing under the same environmental conditions, the temperature difference obtained is more than 4C, then people will indeed lose confidence in this thermometer.

Thanks alan123hk.

I guess I am still hang up on the fact that, for comparison measurements between 2 different samples with the same thermometer we can take the finite accuracy of +-2C can be take to be a constant offset making the comparison measurement valid.

Example: temperature $T_1= 60C$ of sample 1 is measured at time $t_1$. Temperature $T_2=58C$ of sample 2 is measured at time $t_2$ later. The temp difference $(60-58)=2C$ makes the first sample 5C higher. Is that meaningful even if the the accuracy is 2C?

In theory, the first sample could be lower or higher than 60C and the 2nd sample be lower or higher than 58C making the the comparison difference very different from 2C...

 

[alan123hk]

@fog37 I hope I don't misunderstand what you mean.

Manufacturers usually only provide the simplest and most straightforward specifications, such as +/-2C, they do not need to provide the error in measuring the temperature difference, and I have not seen product specifications detailing its the random error, offset error and multiplier error.

Of course, if you know that the thermometer has only a constant offset error, then apart from the displayed resolution limit, the temperature difference it measures basically has no other errors.

Under normal circumstances, the user only knows that the measurement error is +/-2C, then the error of the measured temperature difference should be [(+2)-(-2)] to [(-2)-(+2)], which means +/-4C.

https://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html

 

[fog37]

@fog37 I hope I don't misunderstand what you mean.

Manufacturers usually only provide the simplest and most straightforward specifications, such as +/-2C, they do not need to provide the error in measuring the temperature difference, and I have not seen product specifications detailing its the random error, offset error and multiplier error.

Of course, if you know that the thermometer has only a constant offset error, then apart from the displayed resolution limit, the temperature difference it measures basically has no other errors.

Under normal circumstances, the user only knows that the measurement error is +/-2C, then the error of the measured temperature difference should be [(+2)-(-2)] to [(-2)-(+2)], which means +/-4C.

https://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html

Thanks again. No, you didn't misunderstand I believe.

In essence, when comparing 2 samples, the larger is their temp difference, relative to the stated accuracy, the better.
In the above example, the temp difference is 2C (sample 1 is hotter than sample 2) with an accuracy of +/-4C which is a lot, i.e. there is a lot of variability in the temp difference. All we can say is that one sample is hotter than the other but by how much does not seem very meaningful if the temp difference is smaller than the accuracy. There is a big difference in saying that the two samples differ by 6C, 4C or -2C...

Same goes if we compare 3 or 4 samples temperaturewise with the same thermometer with the +/-4C accuracy...

 

[fog37]

Considering the rule that the uncertainty of a difference is the sum of the uncertainties:

A = T1 +/-2C
B = T2 +/-2C

A - B = (T1-T2)+/-4C

 

[hutchphd]

The importance of the potential error in a measurement depends upon the circumstances. Whether a difference is "meaningful" is also similarly situation dependent. And "absolute" error bars without some specification of confidence are a fiction. If you ask the uninformed what is an acceptable rate of failure the answer is invariably "zero"
There are conventional methods for characterizing error that are useful for general commerce, particularly where life and limb depend upon risk assessments and lawyers are involved. It behooves scientists to know these standards.
But to really characterize the errors in a system is seldom a "one size fits all" exercise. Attempts to do so lack focus and utility.

Considering the rule that the uncertainty of a difference is the sum of the uncertainties:

A = T1 +/-2C
B = T2 +/-2C

Even uniform "box" uncertainties for a larger numbers of measurements will get more and more peaked to the average (you work it out). Suppose $T_1=T_2$ in your example.

 

[Tom.G]

There is a special case applicable to dial thermometers not present in thermometers based on expansion of a fluid in a capillary tube or in electronic thermometers..

The indicator needle for these dials is often driven by a mechanical rack-and-pinion gearset. Any friction, dirt, dried lubricant in the gears leads to hysteresis, or deadband, in their readings.

Know you equipment before betting your reputation on it!

Cheers,
Tom