# Error propagation for the mean of time-series

From what I have read, when measure repeatedly the same quantity X N times and the measurements follow a normal distribution the uncertainty of the mean is $$σ_{mean} = \frac{σ}{\sqrt{N}}$$ where σ is the standard deviation of the measurements.

Now let's assume that we don't measure the same thing all the time, but we have got a set of different measurements without autocorrelation of a parameter P that changes with time for a time period t. The question is how do we calculate the uncertainty propagated to mean value of the period t from the uncertainties of the measurements.

1. If each measurement has the same uncertainty u (random error) and the dataset follows a normal distribution would that be correct to use $$u_{mean} = \frac{u}{\sqrt{N}}$$ ?
2. If each measurement has a different uncertainty $$u_{i}$$ would that be correct to use the same formula like this: $$u_{mean} = \frac{(1/Ν)\sqrt{\sum{u_i^2}}}{\sqrt{N}}$$ ?
• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Mar 31, 2022 at 15:04