High School Least / Smallest / Minimum Detectable Difference

[Roger Dodger]

In scientific experiment, we often have a physical property that can change but have no detectable impact on the measurement.

For example, suppose I have a mass of (say) 30 grams attached to a string passing over a pulley. I can add up to another 2 grams and the system doesn't budge.

In our lab manual, we call this additional 2 grams the least detectable difference and write the mass as 30 +- 2 grams. However, I cannot find any literature that uses this term. In fact, it doesn't appear to be discussed much at all.

What is the proper name for this effect? Is it the least detectable difference, minimum detectable difference, or smallest detectable difference. Or, is there a better term to use, such as (say) sensitivity uncertainty or sensitivity error?

 

[berkeman]

For example, suppose I have a mass of (say) 30 grams attached to a string passing over a pulley. I can add up to another 2 grams and the system doesn't budge.

Why, because of bearing friction? Your example would appear to be under-defined, so it's hard to know exactly what you are asking about. Certainly the FBD changes, right?

There is a predicted error in many measurements, and ways to describe that and to deal with it in data analysis. Can you give a different example, or refine your first one please? Thanks.

 

[Roger Dodger]

Yes, because of bearing friction a little mass can be added or subtracted without affecting the apparatus. Why? Because friction can take on any value (up to a maximum) to keep the system from accelerating.

It may be that other uncertainties, such as the standard deviation, already take this into account.

 

[berkeman]

In scientific experiment, we often have a physical property that can change but have no detectable impact on the measurement.

For example, suppose I have a mass of (say) 30 grams attached to a string passing over a pulley. I can add up to another 2 grams and the system doesn't budge.

Yes, because of bearing friction a little mass can be added or subtracted without affecting the apparatus. Why? Because friction can take on any value (up to a maximum) to keep the system from accelerating

I don't think that's a valid example. You can measure the friction force by adding a 4g weight and measuring a(t) of the weight. That tells you the force (or torque) from friction. If you want to try to claim that your margin of error is +/-2g, you would have to make that only for masses that are less than the threshold to get the pulley moving.

I don't think this example helps us to try to answer your question. Can you come up with a different one? Are they all related to the level of noise in the measurement itself?

 

[Roger Dodger]

The force table, where you have three pulleys hanging by strings, with each string tied to a small loop centered over the table. Theoretically, there is one combination of masses and angles that should work. But you can change the angle slightly with no effect on the position of the loop. .

 

[berkeman]

In scientific experiment, we often have a physical property that can change but have no detectable impact on the measurement.

The force table, where you have three pulleys hanging by strings, with each string tied to a small loop centered over the table. Theoretically, there is one combination of masses and angles that should work. But you can change the angle slightly with no effect on the position of the loop. .

Not if you can make bearing friction arbitrarily small...

 

[Roger Dodger]

But I can't. I'm stuck with the pulleys we have in the cabinet.

 

[Roger Dodger]

I was thinking about this a bit more. Could this sensitivity error simply be a crude alternative to performing a large number of trials and finding the standard deviation? For example, suppose the nature of an experiment doesn't allow for a large number of trials, or even multiple trials. Could this least detectable difference serve as a crude estimate of the standard deviation? If so, perhaps this is why I haven't seen much mention of this error.

 

[berkeman]

i'd suggest a couple of lines of reading...

• Compare and contrast stiction with thermal noise and experimental error
• Investigate dithering in noise reduction techniques in DSP hardware and algorithms...

 

[Dale]

What is the proper name for this effect? Is it the least detectable difference, minimum detectable difference, or smallest detectable difference. Or, is there a better term to use, such as (say) sensitivity uncertainty or sensitivity error?

I think that the definitive reference for reporting and handling error and uncertainty is given by the NIST publication below:

https://www.nist.gov/sites/default/files/documents/2017/05/09/tn1297s.pdf

 

[Roger Dodger]

Thanks. I'm not sure they address this issue, but I'll plow through it and see.

 

[Roger Dodger]

i'd suggest a couple of lines of reading...

• Compare and contrast stiction with thermal noise and experimental error
• Investigate dithering in noise reduction techniques in DSP hardware and algorithms...

I'm fully aware of the causes of stiction, but I'm not sure what you are asking me to do.

 

[Stephen Tashi]

There is the concept of "just noticeable difference" as used in perception testing.