Error on distribution of values with uncertainties

I'm currently analysing a measurement which results in a high number of values, each having a slightly different uncertainty.

These values are following a gaussian distribution, so I wrote a python program, which fits a gaussian distribution over a histogram of my measured data. The program returns both the expected value of my measurement and its uncertainty.

The program however does not consider the uncertainty of the measurement.

My question is how do i combine these two?

Currently I am considering the uncertainty on the gaussian fit as the statistical error.

Also I assume, that the uncertainties of the measurement produce a systematic error.

To calculate this error I took a wild guess and calculated it as i would calculate the error of a weighted mean:

systematic error = sqrt( sum( 1/(uncertainty of each measured value)^2)).

Then I added these two, to get my overall error.

Is this anywhere near correct and if not, what is the correct way to do this?