Uncertainty of Mean - Test rig that measures water temperatures



We have a test rig that measures water temperatures.

I have created the Uncertainty budget attached.(we calibrate the probes our selves with a UKAS calibrated calibrator).

We measure the water temperature over a period of 300 seconds taking a reading every second and averaging at the end.

Am I correct in thinking that I can state a measurement uncertainty for the average result as Uncertainty/SQRT(300). As this is more precision than accuracy how does this relate to the uncertainty in terms of the accuracy. ?

Bearing in mind all results are corrected for the calibration.

Thank you in advance for any help.


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The question that pops up to me regards "Drift".

Are you using this to create an accuracy tolerance for your measurements, i.e. you can expect the accuracy of your reading to be within 0.68°C over one year? If so, I don't feel the number is valid: it is a mere snapshot of two readings taken a year apart. To create a tolerance you will need to collect more samples so that you can see the distribution over time and determine a tolerance from that data. You might up with a normal distribution, it may turn out to be rectangular.

On the other hand, I assume that your probes have a published accuracy tolerance and drift rate from the manufacturer. It might be better to use that data.

You also did not account for repeatability and reproducibility. Chances are these will not be significant contributors, but they should be accounted for.


The measurement is taken for every pH buffer that we manufacture so they are not measurements taken years apart but simultaneously. Both probes are calibrated simultaneously immediately before measurement.

Looking data coming from 26 sets of data, I see that the pH measured by the two sensors are slightly different but never the less the differences are significantly different on 25/26 occasions.

With regard to R&R - that comes next. I am first of all estimating the uncertainty theoretically by looking at existing measurement results and the published data from our supplier of primary standards. This I use to calculate my uncertainty budget. I make stability, linearity and R&R tests to check that the value that I have estimated is being achieved in practice.

In any case, I have been reading around this. It seems there are two ways to deal with a bimodal distribution. The best way would be to resolve the issues that are causing the difference in measurement between the two probes. I am not confident this is achievable as we are talking about bringing the difference between the means down from a few thousandths to a few 100000ths of a pH unit. In practical terms this is not measureable (remember, primary standards have a combined uncertainty of 0.003 pH.

So the other way is to treat the bimodal distribution as a uniform distribution, this delivers a somewhat pessimistic evaluation but I do not see any other reasonable way.


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It truly depends on usage. If you are curing concrete cylinders for example, you have a broader application than if you are doing aerospace.

Based on description only, my reply will gear more towards concrete. So, if much more precision needed, lest us know and I will revise.

Given my assumption, yes, a mean average is likely OK. However, compare it also to a median average to see how close you may be. If doing concrete cylinders, after such comparison, include an undefined number, sometimes called "slop" into the equation. I do not like the term, but it applies to the difference between methods of average. However, make sure the slop is small, JUST enough to cover distance between Mean and Median averages, and not one bit more.

However, if you are looking for almost anything more accurate, forget the previous two paragraphs.

Hope this helps.
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