Variances of many series of random variables

#1
Hello all,

I have to use some statistics in my bachelor thesis, but I had little contact before this. So I am a litte confused on how to proceed.

I have about 100.000 series, lets call them Yn, of weather temperature measurements. Each containing about 50 values (X1, .. X50) and each is extracted for different times of the year or different geographical locations. So the expected average value mu differs for every series.

I am interested in the average standard deviation, and the average trend c. My first attempt was to just sum up all Yn and to a linear fit for the trend.

Then but then calculating sigma^2 was not straight forward.

So I also calculated a linear trend for each individual Yn, to then calculate sigma_n^2 = SUM(Xi- (mu(Yn) + c(Yn)*i))^2.
and then calculate sigma^2 = 1/N * SUM(sigma_n^2).

I hope it is clear what I was trying to do.
I would be glad for help.