Average standard error I have several measurements from the same population spanning over the course of several years, each year with its own mean and standard error (based on the same replicates at same location each time), and I want to calculate the grand mean and grand standard error. The grand mean seems fairly straight-forward (average the means?) but I am not sure if I can calculate the grand SE averaging annual SE values. For instance:
means <- (35,23,38,40)
se <- (2,4,7,6)

grand_mean <- mean(means)
grand_se <- mean(se) ?

Thanks
 A: Suppose we have observations taken across $m$ different years, $\{ X_{1j} \}_{j=1}^{n_1}, \{ X_{2j} \}_{j=1}^{n_2}, \ldots , \{ X_{mj} \}_{j=1}^{n_m}$ where $\text{Var}(X_{i1}) = \sigma^2_i$ for $1 \leq i \leq m$ and everything is assumed independent. Then we can directly calculate the variance of the overall mean,
$$
\begin{align}
\text{Var} \left ( \frac{\sum_{i=1}^{m} \sum_{j=1}^{n_i} X_{ij}}{ \sum_{i=1}^{m} n_i} \right ) &= \frac{ \sum_{i=1}^{m} \sum_{j=1}^{n_i} \sigma^2_i }{ \left ( \sum_{i=1}^{m} n_i \right )^2 } \\
&= \frac{ \sum_{i=1}^{m} n_i \sigma^2_i }{\left ( \sum_{i=1}^{m} n_i \right )^2}
\end{align}
$$
You could then estimate this quantity by replacing each $\sigma^2_i$ with $\hat{\sigma}^2_i \equiv \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2 / (n_i - 1)$, the sample variance within year $i$.  However, it's not clear what you'd be estimating with the overall average if the means also vary with time.
A: Calculating a "grand" mean and standard error can mean several different things, depending on the exact assumptions on your data and what you want the final "error" to represent. This is a complicated problem in general. For example, if some measurement have much larger errors than others, the "grand mean" should by driven more by the measurements with high certainty, so we don't want to just average the means.
Here is a decent set of notes that goes through this issues in great detail: http://www.nist.gov/itl/sed/training/upload/combine-1.pdf
Discussion of how to combine means and errors starts around page 26.
