Analyzing sets of data without the original values One of my colleagues is collecting data from an experiment, and analyzing the average value and standard deviation of each sample set individually. My plan is to do some extra analysis on this same data, but I may not get access to his original data other than his statistics.
So say, if he takes eight measurements ten times from the same population, then calculates the average and standard deviation for each of the eight samples. If I only get the averages and standard deviations, is there a way to calculate the standard deviation of the eighty data points considered as a single sample?
 A: Iff you know what $n_j$ is for each $j$ subgroup of $N$ subgroups you can by working backwards to calculate $RSS= \sum_{k=1}^N (y_k - f(x_k))^2$ for the whole population piecewise, and then calculate the group standard deviation. 
That is, in detail, as follows. For the whole population, $RSS$ is the sum of the $j$ individual $RSS$ values.
$RSS= \sum_{j=1}^{N}\sum_{i=1}^{n_{j}} (y_{i,j} - f(x_{i,j}))^2=\sum_{j=1}^{N}RSS_j$.
Note that calculated standard deviations are 
$\text{SD}_j=\sqrt{\frac{\sum_{i=1}^{n_{j}} (y_{i,j} - f(x_{i,j}))^2}{n_j-1}}=\sqrt{\frac{RSS_j}{n_j-1}}$. Thus, $RSS_j=(n_j-1)SD_j^2$,
then total 
$RSS=\sum_{j=1}^{N}(n_j-1)SD_j^2$. 
Since the sum total number of samples is $n_{all}=\sum_{j=1}^{N}n_j$, our group SD is finally
$\text{SD}_{all}=\sqrt{\frac{RSS}{n_{all}-1}}=\sqrt{\frac{\sum_{j=1}^{N}[(n_j-1)SD_j^2]}{n_{all}-1}}$
Note also, that the calculated standard deviation generally underestimates population standard deviation when $n<10$ due to small number bias, and only gets to about 1% error when $n$ is approximately $100$. Thus, the last thing you want to do is use some averaging technique other than that suggested above.
