UNI-T UT181A question: what is confidence interval used in the datasheet?

Question

So the question is: https://www.batronix.com/pdf/uni-t/UT181A-Manual-English.pdf

The manual has

So if you are measuring perfect 50 Hz source you have resolution 0.001 Hz the values are in interval from 50 Hz - 0.2% 50 Hz - 0.008 Hz = 49.982 Hz are okay. In practice I can get 49.989 Hz, so it is in spec., in fact it appears it has a constant mistake -0.022 (as the source was 50.001 Hz, also see this https://youtu.be/PjNXbKlr3MI?t=1792) in all modes, so 500 is measured as 499.89 Hz, 5 kHz as 4.9989 kHz, 50 kHz as 49.89 kHz and 500 kHz as 499.89 kHz.

The sigma accuracy does not apply for frequency apparently, as Gnuradio developer clarified. So then the question is for voltage.

I am pretty sure it is 100%. Or at least 99.73%. Unless something really bad happens and the SOC inside the multimeter gives up I cannot see how do you get such low confidense of 1 sigma which is mere ~68%.

Also, how do they measure that interval? Do they decrease the interval until you get needed sigma? What happens if sigma always stays high?

Also what does it mean for military equipment to have more than one sigma? Do they have two datasheet entries, on for 68.27%, 1 sigma and another for 95%?

Answer 1

What is the confidence interval given in the datasheet

The datasheet did not specify a confidence interval! It specifies an accuracy, and that's a different thing.

In this case, what that means is probably that given (a hypothetical) noise-free input, how accurately this device measures the frequency. It measures the frequency accurately with an error less than what's specified in the parentheses, for example a 500 kHz tone would be detected as something between 500 kHz · (1-0.01%) - 5 Hz to 500 kHz · (1+0.01%) + 5Hz, so 499.945 kHz to 500.055 kHz. The actual confidence interval would also need to incorporate precision, which would be a function depening on the noise your signal is exposed to, and not specified in the datasheet.

Also what does it mean for military equipment to have more than one sigma?

It makes no sense to say "for military equipment to have more than one sigma". It makes sense to specify some quality for three sigmas of system noise.

I think you have a misconception on what "sigma" is: \$\sigma\$ is the standard deviation of your measurement, or of a variable derived from that measurement.

So, when I say "this is an error bound that holds true for \$3 \sigma\$", I actually need to specify the nature of the probability distribution that describes the system noise. Now, as you'll find out, under normally-distributed noise, which happens to be a common model for noise happening through thermal effects, the error of a frequency estimator is not normally distributed. In fact, the error will depend on the estimator used, and typically it is quite complicated to derive a probability density function for. So, knowing that something holds true for three sigma tells you very little – unless you know the probability density of the error function. (For a more in-depth discussion of this, I'd have to refer you to the literature on the noise probabilities of FM, and error probabilities of FSK communinicatons. But you'd really need a bit of stochastics to work on that; if you don't mind, I think that would leave the scope of your question.)

And that's exactly where your 68% and 99.7% come from: for a normally distributed variable, around 68% of realizations fall within \$sigma\$ of the mean, and around 99.7% fall within \$3\sigma\$. For other distributions, that is not the case! And that's why it's wrong to assume everyone assumes 68% of measurements fall into \$\sigma\$: we don't assume that! We check our assumptions before we apply the math: only if the measurement noise is normal, then we might say that.

Now, a lot of things behave approximately normal, and then it's fine for everyday usage to assume that 68% of measurements do fall into \$\sigma\$ of the true value (let's ignore biases for a moment). But at \$3\sigma\$, you need something to be very precisely normally distributed for that 99.7% to hold true. And that's not always the case, which is why some precision equipment (I assure you, being military equipment has nothing to do with good equipment, in general), will specify more than just the standard deviation.

Also, how do they measure that interval?

Measure a reference, measure it often with different devices, than do statistics and derive the standard deviation.

Answer 2

Here's an answer:

TLDR Answer:

It depends on your application. I will give you my two answers below, which I parse into the following:

• The General answer: It doesn't matter for most
• Specific answer: It does matter for the few, usually critical application. In that specific answer, I provide a potential method to obtain your own confidence interval data to your liking.

First, we'll start with a prelude. Then, with that as context, you can read the general answer (for general technicians and non-critical applications) then specific answers (for more serious applications).

Prelude

First, just to interpret the question as: "How confident is UNI-T, statistically speaking, in its measurements variation to be due to noise across multiple samples for the UT181 device?"

If the question is interpreted corerctly, it's a very interesting question to be answered and can be answered in various ways depending on the application and purpose.

General Answer

Generally speaking, if the product quality is high (from a reputable brand), most electronics technicians don't care (and, the non-engineer electricians generally don't get into such deep electronics considerations as its almost irrelevantly philosophical for them as their goal is to simply get the job done in as little steps as possible without worrying about such abstractions.)

So, in that case, generally speaking, I don't think it matters here as it would in other scientific fields as, due to the nature of electricity, the measurements are made so frequently that, if the measurement value is constant over say a few seconds, the mean sample measurement approaches the value of all possible measurements (sigma, i.e., the population measurement) so the confidence interval can be quite tight for a series of very low-variance readings.

Specific Answer

So, then why does this question matter and is this a good question? Answer follows:

Yes, I do think the question is valid to ask. After considering the general answer above, on the other hand, when high precision and high accuracy DOES matter (depending on your application--say NASA stuff or next-gen military tech), this topic becomes incredibly relevant. Furthermore, what you are asking does matter and is important when you are trying to assess the quality of a product (to which company employees may disagree with and respond negatively here).

That said, in this specific situation, you would focus on the sample rate of your device, and possibly use a fancy benchtop oscilloscope/signal generator to send various known very precise signals to your device. Then see how long it takes, on average, in time (seconds) for you to see a variation from the specifically set signal on your measurement device under test. You can then compute the number of samples using the sampling rate. Then, you can develop a confidence of measurement. If, while sampling, you don't see variation (your measurement device reads 50.00 Hz and matches your signal generator), you start to increase the fineness of the changes to your supply test signal (voltage or current or whatever, in your case, the test frequency of 50 Hz).

Caveat

• You need to know statistics and basic electronics to pull this off and also need the devices. You probably do since you are talking about confidence intervals.
• One issue is the electrical signal produced by the device may vary at some level of precision from the stated value of that signal (i.e., it says 50.000000000 Hz but it's really 49.99999995 Hz. So, your measurement is only good as your signal generator/oscilloscope.

Hope this helps.

Answer 3

As you are a good lover for UNI-T equipment (which I personally find too a really price efficient alternative), we cannot left Fluke's team work without any merit.

They give an explanation of what Accuracy Percentage and Digits mean for them (Fluke.com, Why digital multimeter accuracy and precision matter):

Accuracy may also include a specified amount of digits (counts) added to the basic accuracy rating. For example, an accuracy of \$\pm(2\%+2)\$ means that a reading of \$100.0 V\$ on the multimeter can be from \$97.8 V\$ to \$102.2 V\$. Use of a digital multimeter with higher accuracy allows for a great number of applications.

As you can read from your equipment datasheet:

• Selected Range: \$0-60 Hz\$

• Resolution: \$0.001 Hz\$

• Accuracy: \$\pm 0.02%\$ +8 Digits (% Reading + Digits)

Hence, for a \$50.000 Hz\$ signal, with normally 5 significant digits for the UN181A, we should expect a \$\pm50\cdot0.0002 Hz=\pm0.01 Hz\$ error from the % Reading.

The typical format should be 50.000, so the digits accuracy should be taken from this representation of significative digits.

\$\pm1\$ digit means if you have a 50.000 reading, you could expect a reading between 49.999 and 50.001. Thus, 8 digits error means an \$\pm 0.008\$ Hz error when reading 50.000.

Then, you should expect a total error of: $$ \pm(0.01 + 0.008) Hz = \pm0.018 Hz. $$

And that is!. We cannot infer anything else outside from this range. Electrical gaussian noise, temperature drifts, sensor drifts, internal compensations, etc. are all in this range and every other further extrapolation, without any specific advice from UNI-T is a side conclusion.

As you could realize, every sigma interpretation would assume a distribution, for which, in this case, do not have information to conclude. Typically we shouldn't too, since the drifts would be more significative.