# How does one combine independent repeatability and accuracy uncertainties on the same quantity in a reported uncertainty value?

I have a NIST calibrated photodiode with reported $\pm$2% uncertainty in the calibration. I've interpreted this as the "accuracy uncertainty", though I apologize if my terminology is misleading.

If I measure the same quantity multiple times using this photodiode, I find a standard deviation over $N$ measurements of $\pm$0.5%. I've called this the "repeatability uncertainty".

How can I reconcile these uncertainties in a measurement? What is the standard method of reporting an uncertainty value? Specifically, when the same measured quantity has both an uncertainty in calibration (which may not systematically lie to one side or the other of the mean) and an uncertainty arising to repeatability, how can the uncertainties be utilized to predict the overall uncertainty?

Here are two scenarios:

Scenario I

I use the same measurement device to measure a relative value, e.g. the light intensity in front of and behind an optic. I propose that by not relying on the calibrated value that I can essentially ignore the accuracy uncertainty and report uncertainty as \begin{equation} \sigma_{measurement} = \sqrt{\sigma_{input}^2+\sigma_{output}^2} = \sqrt{2}\sigma_{input} \end{equation}

Scenario II

I characterize a second device (sample) to calibrate its absolute performance against the calibrated reference. Each measurement (calibrated reference and sample) has the same repeatability uncertainty, but the reference to the calibrated values requires dividing once by the values with reported accuracy uncertainty. I am uncertain of how to proceed in determining the overall uncertainty in the measured value of the calibrated reference in this case where \begin{equation} \mathrm{Value\enspace of\enspace interest} = \frac{I_{\mathrm{sample}}}{I_{\mathrm{{reference}}} * f_{\mathrm{calibration, reference}}} \end{equation}

Is it as simple as \begin{equation} \sigma_{measurement} = \sqrt{\sigma_{\mathrm{sample}}^2+\sigma_{\mathrm{reference}}^2+\sigma_{\mathrm{calibration\enspace factor}}^2} \enspace \enspace \mathrm{?} \end{equation}

## 1 Answer

It sounds to me as if you mean the difference between a systematic error and a random error.

In this case the systematic error is 2% i.e. when you report an absolute value for the light intensity that value is subject to a 2% error. All your measurements will have the same systematic error. The random errors are the differences between individual measurements and will be different for all the measurements.

I'm not sure I can claim any great knowledge of current concventions in publishing, but I generally see systematic and random errors reported seperately in papers i.e. you'd report a 0.5% random error and 2% systematic error.

You certainly can't combine them as you've done in your last equation because the systematic error is constant for any one diiode and not randomly distributed.

• Thank you @john for pointing out my misconception with regard to randomly distributed and systematic or constant error. I've mostly encountered systematic error as referring to a constant offset in a consistent direction. Can I interpret your answer as saying instead that it is more generally any constant error independent of random distribution, regardless of whether it's unidirectionally offset from the mean? – CFlowers Dec 19 '16 at 7:37
• @CFlowers: Yes, I think that's a pretty good description. – John Rennie Dec 19 '16 at 8:53