Measurement Error

It is generally accepted that any scale; linear, dial, or other, can be read to the nearest one half of the smallest division of the scale. In your example; divisions of one half millimeter (0.5 mm) can be read accurately to the nearest one quarter millimeter (0.25). The accuracy of the measuring device is usually specified by standard as a percent full scale, such as in ASME hydrostatic testing "the accuracy of the pressure gauge shall be 0.5% minimum of full scale." Therefore; the accuracy of the measuring device is not generally included in the reported measurement.

Back to your measurement; if you observe that the edge length is closer to 111.5 mm than to half way between, then you report that measurement as 111.5 mm. If your observation is that the measurement is closer to half way to the next increment then you should report that as 111.75mm. If it is closer to the next division on the scale than the half way point, then it would be recorded as 112.0 mm even if it is actually less than 112.0 mm.

I haven't read all the discussions on the challenge question, so I don't know if this helps solve your problem, but I am sure there will be lots of disagreement on this thread.

Yes, thank you. I wrote that incorrectly didn't I? I'll copy it here and fix my boneheaded mistake.

This is a take-off from the discussions in the Weight Lifter Challenge blog, but I'd like to start a discussion devoted to a practical way to understand measurement error. I think I generally understand the subject, but I can't explain it clearly and sometimes I don't know which way to use it. So, let me propose an example, and please tell me if I'm seeing this wrong or too simple or too complicated.

Suppose I want to measure the length of a piece of metal along an edge. I have a machinist's scale which is marked in cm and ruled to 0.5 mm, so each cm has 20 divisions. The scale is made by a well-respected company and I have checked the overall accuracy and found it to be good within at least 0.02 mm.

So, I think the scale has no systematic error of any significance since I know enough to avoid parallax errors. So, the error of measurement will come from the resolution of the scale and my ability to distinguish between divisions.

I measure the edge and find it to lie between 111.5 mm and 112.0 mm. In fact, I think I can tell that the edge length is nearer the middle of the division than either line, so I write down 111.7 mm. Now, I understand the 0.7 mm to be uncertain, so I could leave it that way, or I could make an estimate of my uncertainty and write 111.7 mm +/- 0.2 mm. I could have written this as 111.8 mm and I could have estimated my uncertainty slightly differently. But I chose 111.7 mm +/- 0.2 mm. I think I cannot write 111.75 mm since I don't have the ability to resolve 0.05 mm.

I understand this to be a worse case error, and I think it is limited by the resolution of the scale. Even if I get a loupe, I don't think I can do much better.

So far, how am I doing? Have I made any stupid assumptions? Have I made any (more) mistakes?

I'll clarify my mistake. If the required accuracy is 0.25 mm, then you can use the scale you have. Remember you can measure accurately to the nearest one half division. It would be incorrect to record the measurement to the nearest tenth because your accuracy is 0.25 mm. You can only take measurements of 0.25, and 0.5 beyond the last whole mm. If your divisions were in 0.2 mm, the it would be acceptable to record measurements of 0.1, 0.3, 0.5...etc.

If you round the measurement to the next tenth beyond 0.25 or 0.75, and the measurement is used in a calculation later, then you will begin what is known as propagation of error. Measure and record only to the nearest one half division of the scale on your measuring device. If you require more accuracy in you measurement, use a scale with smaller divisions.

All of the above has nothing to with the overall accuracy of your scale. It is only about the accuracy of your measurement and how you record it.

Thanks. Let me explain that I'm not being difficult, but I have to explain this to someone who is, and I'm trying to make sure I can justify everything I normally do (and in the process of asking you guys, I hope to fill in some of the gaps in what I know), so please don't mind if I question closely.

So, since I have no appreciable systematic error, my error is presumably a random one, which could be improved by multiple trials and statistics. But, here I have only one measurement, so I'm treating my best estimate of the reading as though it were the mean and I'm treating my best estimate of the uncertainty as though it were the standard error. Is that right?

My second question is in these odd number cases. I see that, if I have a scale marked in 1 mm increments, that my uncertainty might, by rule, be +/- 0.5 mm. But, in this case, I deliberately chose 0.5 mm divisions because I don't know how to justify using either +/- 0.25 mm or +/- 0.3 mm for the uncertainty.

I'm not asking about a guard digit that I would need for subsequent calculations; I'm trying to treat this as a single measurement that will never go into any calculation. I guess I could handle the idea of +/- 1/4 mm (groan), but I'm suspicious of saying +/- 0.25 mm. My suspicion may just be a residue of my drafting days when half of 0.25" was either 0.12" or 0.13" depending on which rule I was following for that job.

if I have a scale marked in 1 mm increments, that my uncertainty might, by rule, be ±0.5 mm

Not necessarily. See the next post the ILE and uncertainty are two different things.

ILE depends on the spacing between the markings (0.5mm) and then the error in the markings (0.02mm)

Let me just enlarge the view assume that the markings are in 10cm (least count), the errors in marking is 1mm

Then reasonably say you will be able to read (or interpolate) 2 cm ? (visually/ with whatever instrument you have)

So the possible error in the estimate will be this 2 cm along with the 1 mm for the marking error.

In other example, take a spring balance, marked in Kgs but the gap between markings is say 2 cm. Accuracy of markings is 0.05 Kg, so with all reasonableness you should be able to measure even 0.25kg on the scale, that will be compounded with the other uncertainty 0.05Kg in marking itself.

The measurement terminology

a) Least count - the smallest division on the scale - that is 0.50 here.

b) Instrument Limit of Error- That is the value that can be read by interpolating the gap between the minimum division. Of course it depends on the

- spacing between division

- The measurerer's comprehension/ eyesight (missing the exact term)

- usage of additional instruments

etc.

For an example, for an electronic load cell of 3000 Kg the Least count we have asked is 1 Kg

The Error (repeatability, accuracy etc all included) is 0.24 Kg - and no it is not an error it is peermitted in class A instruments (there was a controversy on exact this point)

This is digital, then how do you check the accuracy and repeatability < LC ?

Simple - use of additional instrument - use a volt meter with required accuracy to read the signal directly before it is fed to ADC circuit.

The ILE is always equal or less than the LC now for your case it can be as low as 0.02 (you may always use a microscope you know)

So it all depends.

End of the day you have to go to statistics and do GR&R

Do most people really do GR & R for a length measurement? That seems like a lot of effort and time? Is there a sort of "quick and dirty" GR & R?