Suppose you have some function $f(x)$ which represents the mean value of an experiment related to $x$. Then $\epsilon = \sigma(x)/\sqrt{n}$ is the error associated with each point $x$. Assuming that each experiment considers the same number of samples $n$ and $\sigma(x)$ is the standard deviation of each experiment, what is the process behind accounting for the errors $\epsilon$ when modelling the function $f(x)$?
For example, suppose we have reason to believe $f(x)$ is an exponential function and we wish to perform a non-linear regression on $f$. Do we simply assign weights as $w = 1/\sigma(x)$? Does this differ if $f$ was expected to follow a power law trend?
edit: The means $f(x)$ and standard deviation are defined in the usual way. For instance suppose we do finitely many experiments $x_1, x_2, ..., x_k$. For each $x_k$ we obtain a series of observations $X_1^{(k)}, ..., X_n^{(k)}$ which means $f(x_k) = \frac{1}{n} \sum_{i=1}^n X_i^{(k)}$ and $\sigma(x_k) = \frac{1}{n} \sum_{i=1}^n (X_i^{(k)} - f(x_k))^2$. The $x_k$'s themselves could represent anything; a parameter, time, space etc. Finally, I chose $\epsilon$ above because $f(x)$ represents the mean of an experiment, so presumably the error should be the standard error of the mean.