# Systematic+Random Uncertainty for velocity measurement

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.

• What this question calls "random uncertainties" are generally called "statistical uncertainties" rather than "random" undercertainties. Commented May 22 at 5:18

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.