Another uncertainty/error question

Ok let me give this a try.

The reading you did with two different instruments demonstrated the resolution and tolerance of these to instruments. You measured the exact same parameter with similar instruments and ended up with an overlapping uncertainty region. As you implied this is what one would expect, the cheaper lower quality instrument produced a larger uncertainty region +/- 0.017 V than the higher quality instrument of +/- 0.004 V. Also the entire high quality instrument's uncertainty reading lies inside the uncertainty region of the less precise instrument.

Now if you don't include your tolerance value in your recorded data you should record your data so that 1/2 of your least significant digit is larger than the actual tolerance of your reading. So to maintain all of the accuracy of your precise reading you should record 1.525 +/- 0.004 V. But you can record this as 1.53 V for the implied +/- 0.005 tolerance is larger than your measurement tolerance. This does mean that your recorded data is less accurate than your instrument, but one is permitted to record less accurate data. Your other meter though has a tolerance of +/- 0.017 so the next largest half digit above this is +/- 0.05. This means that if you don't wish to include any tolerance value you should record this as 1.5 V. But you are entitled to choose 1.52 +/- 0.02 V for 0.02>0.017.

Now for the statistical analysis, one must remember that the analysis program does not know the origin of the numbers entered. Your fractional number entries maybe in reality integer numbers times an absolutely known fraction, the software does not know. So your identical set of 28 readings does have a zero standard deviation. But each of your readings has a tolerance of +/- 0.017 V. That precision limit does not disappear.

Measurements, tolerances and statistical analysis are a confusing subjects. We have all misused them in our work. By understanding this though and by being creative with circuit topology you can apparently exceed the resolution limitations of your instrumentation by including dithering. But that's another even longer story.

_{I wish I could cite my source written by D.C. Baird on measurement and precision but that old text is at work.}

OK, I got part of it. A GA for that and thanks. I wasn't picking up on that idea of moving up to the next half digit point.

Who and what is D.C. Baird? I don't know about that. All I have is John Taylor and that's about useless for stuff like this.

I got to check on this dithering. I've used dithering in gas flow, but in electrical work? Thanks again for something new to check out.

D. C. Baird was the author of "Experimentation: An Introduction to Measurement Theory and Experiment Design" This concise little 200 page text covered literally everything one needs to know about measurements. Well at least its always contained my needs.

Yep. Found it on ABE. Maybe that'll make me a little smarter? Well, there's always hope.

Thanks for all your help. I know I'm asking some dumb questions, but I always thought I understood some of this stuff and maybe I was just too dumb to know I didn't.

Anyway, if I take 29 readings and they're all the same (or close), and I run a standard deviation (or can I use standard error - not sure), that tells me the distribution of the repeatability of that meter. Is that right?

Yep, in your statistical analysis you've found the repeatability of that particular instrument being used by you on that object.

But this is anything but a dumb question. In contrast I would call this a very astute question. Many people never question measurement errors, neither their origin or meaning. Because of this many work for a long time not knowing how accurately they do know a measurement and how accurately they know an indirect measurement.

Because many people don't know that there really is a legitimized formal methodology to making uncertainty measurements and calculations, one should always refer to a reference when using this confusing subject. I have often cited this text book at work to bolster my data and to be certain that I'm using the correct terms.

This leads me to your question about standard error or deviation. I had to review the Wikipedia site to see if my memory was correct on the subtle difference between these two numbers, it wasn't. The difference in these two terms hinges on knowing the population size that you are taking samples from. How that applies to an electronics voltage measurement that doesn't have an obvious population size, I haven't thought through yet. But since standard error does not require knowledge of the population size, using that term will be accurate.

There are two different entities

Standard Error and Standard Error of Measurement (SEM)

The standard deviation (or call it the variability) may point to either depending on the way you are conduction your experiment.

However that is likely to complicate the issue just a bit more than we may need here. And we will be entering into the depth of measurement reliability and so on.

And yes, here you are only calculating the repeatability of the instrument.

You are mixing up two different things. One is the design of the meter, each meter is expected to be within the stated range, +/- 1%, but each reading of the same measure by the same meter will not vary.

Your zero error is an experimental result that has nothing to do with the above.

Hmmmm, this is one of those great Qs...real simple, until you start thinking about it.
I think one part comes into the 'use and abuse of statistic' category.
There is self evidently no point in doing a mean and standard deviation etc on a bunch of identical readings so it's just a red herring...the problem is explaining why.
Intuitively it's just because there is nothing to average so you are not gaining anything, effectively the stats are a way of consolidating a bunch of information into one figure.
The stats show the standard deviation of you measurements but only relative to their resolution in the first instance.
If you repeat the stats just using the 1.5 figure (pretending you meter only read to 1dp) you get a mean of 1.5 and the error of zero...relative to the resolution and accuracy you plugged in.

If you had some figure 1.5, 1.5, 1.4 etc then the Q is how many places of decimals can you legitimately use in your answer... this links in with sample size etc...
Argghhh brevquot error must nap...<flump thud>

Del
(Here's one I prepared earlier....)

Yeah, the trouble is that I'm not the world's sharpest cookie when it comes to stats, so I tend to believe folks who seem to be smarter. Then later, I start to wonder if I was following a wild goose. That's why I asked on here. I'm a little chagrinned to have you guys see how little I really know, but I'm grateful for the chance to finally get some of this squared away.

In a measurement system many sources of error exist. The five main sources are

Stability : Producing same value over time when same measurement is being done, eg your voltmeter reads 1.527 today as well as the next week.

Bias - The difference between the actual value and the known true value (1.525-1.517 = 0.008V)

Linearity - The linear relation between the actual and the read value, eg 1.525 reads 1.517 , 15 reads 15.005 then 10 should not read 9.80

Repeatability - same operator, same instrument, (ie the process and the instrument is same) how the readings vary.

Reproduceability - Different operator, same documented process, variations in readings.

Here in your case you have studied the repeatability only (and upto certain extent bias)

You can have a look at this from Chat-3 at least and get some Idea about measurement uncertainties. GRR is a part of this. or have a look at NIST or another

Hey, thanks for that Chat-3 link. I glanced through and saw, for the second time this weekend, the recommendation to set up an error model BEFORE measuring. I never knew to do that. Durn! I may be even dumber than my family thinks.

I am familiar with a little of this, but I'm trying to question everything I think I know so I can figure out what I really know and what I just think I know. I appreciate all your help and patience.

The NIST stuff is great, but I'm pretty much a kinesthetic learner. I have to actually take a meter in my hands and measure something and then say to myself, "Self, what do you know about what you just did?"

OK, I'm starting to get some of this sorted out. Turns out I've been getting by through overestimating my uncertainty over the years. I was never wrong, but sometimes I wasn't right either. I figure I actually knew about 10% of what I thought I did. Thanks to RedFred and Fyz and others who have helped so far.

I think I'm close to clear on the difference between error and uncertainty. Probability Distribution Functions seem OK. Type A uncertainties seem OK for N > 25. I think I've got the rules for propagation.

But, I'm not clear on how to calculate calibration uncertainty. I think I know how to calibrate, but how do I calculate that?

By the way, Fyz, I'm pretty PO'd that you didn't mentional the National Physical Labs. What a treasure! Were you keeping that all to yourself?