How do I perform error propagation of a polynomial function? I'm trying to perform error propagation for some photometry code I'm writing, but I'm having some trouble with it!
I have a value $x$ that I draw from a distribution with standard deviation $\sigma_x$, $x = \mu_x \pm \sigma_x$. I feed $x$ into a polynomial function,
$f(x) = \sum_i c_i x^i$
How do I find the standard deviation $\sigma_{f(x)}$? I understand how to perform error propagation for the sum of two variables, but for some reason polynomial error propagation is tripping me up.

There are (at least) three different approaches to compute the variance of $f(x)$:

*

*Gauss' law of error propagation, which is in this case $(\Delta f)^2\approx \left(\left.\frac{\partial f}{\partial x}\right|_{x=\mu_x}\right)^2 (\Delta x)^2$

*The Jackknife Variance: $\sigma_{JK}^2=\frac{n-1}{n}\sum_{i=1}^n (f_{(i)} - f_{(.)})^2$, where $f_{(i)}^2$ is the value of $f$ computed with sample $i$ omitted, and $f_{(.)}$ is the mean of these leave-one-out values.

*A Bootstrap estimate of the variance, which is a Monte Carlo method and thus yields a random value, i.e. two researchers starting with the same data will obtain (somewhat) different results.

Edit: Method 1 assumes that $f(x)\approx f(\mu_x)+f'(\mu_x)(x-\mu_x)$ in the region $\pm 2\sigma_x$ around $x=\mu_x$, which you can check graphically. Note that if this does not hold and the function $f$ is not odd around $x=\mu_x$, the approximation (which you presumably use) $\mu_f\approx f(\mu_x)$ does not hold either.
Methods 2&3 are only applicable, if $\mu_x$ and $\sigma_x$ are empirical values and you still have the data $x_1,\ldots,x_n$. If these are theoretical values and all you have is $\mu_x$ and $\sigma_x$, but you are pretty sure that $x$ is normally distributed, then you can follow the suggestion in the comment by @whuber and calculate $\mu_f$ and $\sigma_f^2$ numerically from their exact definitions:
$$\mu_f=\int_{-\infty}^{\infty} f(x)\,\varphi_{\mu_x,\sigma_x}(x)\,dx \quad\mbox{ and }\quad\sigma_f^2 =\int_{-\infty}^{\infty} (f(x)-\mu_f)^2\,\varphi_{\mu_x,\sigma_x}(x)\,dx$$
where $\varphi$ is the probability density of the normal distribution. Apart from a proper numerical integration, you can also use a Monte Carlo approach: simulate normally distrbuted $x_i$ and compute the mean value and variance of $f(x_i)$. This is very slow to converge compared to proper numerical methods, but it has the advatnage that you also obtain an estimate of the distribution of $f$ and can compute confidence intervals therefrom. The confidence intervals $\mu_f\pm 2\sigma_f$ will have poor coverage probability, if the distribution of $f$ deviates considerably from normality.
