This is an old question but the answer accepted is not actually correct or complete.
The user wants to calculate the standard deviation over 12-month data where the mean and standard deviation is already calculated over each month.
Assuming that the number of samples in each month is the same, then it is possible to calculate the sample mean and variance over the year from each month's data.
For simplicity assume that we have two sets of data:
$X=\{x_1,....x_N\}$
$Y=\{y_1,....,y_N\}$
with known values of sample mean and sample variance, $\mu_x$, $\mu_y$,$\sigma^2_x$,$\sigma^2_y$.
Now we want to calculate the same estimates for
$Z=\{x_1,....,x_N, y_1,...,y_N\}$.
Consider that $\mu_x$,$\sigma^2_x$ are calculated as:
$\mu_x = \frac{\sum^N_{i=1} x_i}{N}$
$\sigma^2_x = \frac{\sum^N_{i=1} x^2_i}{N}-\mu^2_x$
To estimate mean and variance over the total set we need to calculate:
$\mu_z = \frac{\sum^N_{i=1} x_i +\sum^N_{i=1} y_i }{2N}= (\mu_x+\mu_y)/2$
which is given in the accepted answer.
For variance however the story is different:
$\sigma^2_z = \frac{\sum^N_{i=1} x^2_i +\sum^N_{i=1} y^2_i }{2N}-\mu^2_z$
$\sigma^2_z = \frac{1 }{2}(\frac{\sum^N_{i=1} x^2_i}{N}-\mu^2_x + \frac{\sum^N_{i=1} y^2_i}{N}-\mu^2_y )+\frac{1 }{2}(\mu^2_x+\mu^2_y) -(\frac{\mu_x+\mu_y}{2})^2$
$\sigma^2_z = \frac{1 }{2}(\sigma^2_x+\sigma^2_y )+(\frac{\mu_x-\mu_y}{2})^2$
So if you have the variance over each subset and you want the variance over the whole set then you can average the variances of each subset if they all have the same mean. Otherwise, you need to add the variance of mean of each subset.
Let's say that over the first half of the year we produce exactly 1000 MWh per day and in the seconds half, we produce 2000 MWh per day. Then the mean and variance of energy production in first and seconds half are 1000 and 2000 for mean and variance is 0 for both halves. Now there are two different things that we may be interested in:
1-We want to calculate the variance of energy production over the whole year: then by averaging the two variance we arrive at zero, which is not correct since the energy per day over the the whole year is not constant. In this case we need to add the variance of all the means from each subset.
Mathematically in this case the random variable of interest is energy production per day. We have sample statistics over subsets and we want to calculate the sample statistics over a longer time.
2-We want to calculate the variance of energy production per year: In other words we are interested in how much energy production changes from one year to another year. In this case averaging the variance leads to the correct answer which is 0, since in each year we are producing exactly 1500 MHW on average.
Mathematically in this case the random variable of interest is average of energy production per day where the averaging is done over the whole year.