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I come from an Engineering background, and I am familiar with some basics of error treatment. However, discussing with a friend over some data he had to analyze, we couldn't quite figure out what to do. We searched the internet and could find no answer (maybe we didn't know exactly how to look for it), and I have searched this forum for some related problem but couldn't find anything either (sorry if some related question has already been asked, some pointers to the answer should suffice in this case).

My friend is trying to characterize a transfer coefficient ($k$) for a chemical substance. The formula is given by

$$\ln \left(\frac{C_s - C}{C_{max} - C}\right) = k(t - t_0)$$

where $C_s$ and $C_{max}$ are considered known. Through a series of experiments, the pairs $(t,C)$ have been measured to characterize the transfer, and $k$ has been estimated as the mean of the angular coefficient of the different straight lines. However, considering the error, there seems to be two different approaches: the values of $t$ and $C$ are measured with a certain precision (which we could call $\Delta t$ and $\Delta C$), which can be propagated to the error in $k$ using the "standard" error propagation formulas. However, since several experiments have been performed, there is also a way to measure the standard deviation ($\sigma_k$) and standard error of the mean ($\sigma_\bar{k}$). Are these two errors (propagated measurement error and standard error of the mean) complementary ? Should we consider both of them, or are the measurement errors implicitly included in the standard error? If both are to be considered, how should we do it?

I'm sorry if this is somewhat of a basic question, but we couldn't seem to grasp it. Thanks in advance.

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Standard error = Standard Deviation /$\sqrt{N}$, where N is the number of measurements in the dataset used to derive standard deviation.

If you are measuring one value multiple times, use standard error. If you are measuring one quantity in multiple cases, then use standard deviation.

In other words, if you could make a perfect measurement, and you would always get the same number, then use standard error.

As k represents some physical property, with a true value, it seems standard error is most appropriate.

Referece: http://lben.epfl.ch/files/content/sites/lben/files/users/179705/Error%20Propagation_2012.pdf

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