I have a measurement method for which I want to study the measurement error by an error budget. Therefore, I listed all possible errors (error sources) (lets say $x_1, x_2, x_3,\ldots$). For each error $x_i$, I have derived an analytical expression from which I can calculate the resulting measurement error $\epsilon_i$. To calculate the total measurement error epsilon, I specified an interval for the possible values of each error value $x_i$ and performed an Monte-Carlo simulation calculating the measurement error $\epsilon_i$ 100.000 times for each error $x_i$. This means that I have 100.000 error values $\epsilon_i$ for each $x_i$. From this data I can calculate the standard deviation $\sigma_i$ of each measurement error $\epsilon_i$. However, I am interested in the total measurement error epsilon, not in the independent measurement errors $\epsilon_i$. How can I combine each measurement error $\epsilon_i$ to come to the total measurement error $\epsilon$? Suppose the errors $x_i$ are independent.