Systematic+Random Uncertainty for velocity measurement

Question

I have a question regarding systematic and random uncertainties. I have to measure the mean value of a velocity measurement in flow field at a point. I've recorded say 1000 samples in time, which I indicate with u_i (i=1,2,...,N)., where N=1000. I have to establish the uncertainty on the mean value U of the velocity. To do that, I use the sample mean relationship

$$U=\frac{1}{N} \sum_i^N (u_i)$$

Now each measurement in time u_i is also affected by a certain calibration uncertainty. I guess this is a systematic uncertainty, which I indicate with du_SYS. In addition there is a random uncertainty, because the value of u_i fluctutates. How can I establish the total uncertainty in U (systematic + random)?

Do I have to compute the standard deviation ($\sigma$) of the samples, and consider this as a random uncertainty? I guess this value decreaeses with N, according to the relationship

$$du_{RND}=\frac{1.95 * \sigma}{\sqrt N}$$ (confidence level 95%)

However, I think that the systematic uncertainty du_sys does not decrease with N, so it is the same in u_i as in U. What do you think about this? I'm a little bit confused and would like to have a clarification.

I guess in the end I have to use the formula

$$dU_{TOTAL}=({du_{SYS}}^2 + {du_{RND}}^2)$$

but I don't know if the systematic uncertainty also depends on N or not...

I would also take this opportunity to ask a question: does the formula $$du_{RND}=\frac{1.95 * \sigma}{\sqrt N}$$ apply only for the random part of the uncertainty in the velocity measurement?

Thank you, E.

Answer 1

To begin with ask yourself two questions:

• Has the device been re-calibrated in the middle of data taking?
• Is the calibration known to drift over time similar to the length of the data taking?

If both of the answers are "no" then you can reasonably assume that the calibration effect is the same on each and every data point. So the mean is off by that same calibration error no matter how many data points you take and the systematic uncertainty is unaffected by the number of data points.

This is sometimes referred to as the "systematic floor" and with the $1/\sqrt{N}$ behavior of the error on the mean it leads to very strong diminishing returns on running an experiment for much longer than it takes to get the random error down to about the same size as the systematic error.