# Repeated measurements with a nonlinear model and statistical significance

Let us consider a simple example that explains my problem. Let's say you are doing a simple pendulum experiment, and you measure the position as a function of time, then we have a very good model of this to expect that the position x from some center as function of time will follow a sine function x(t)=Asin(wt+f) and if we have measured 1 time series we can fit this model and obtain estimates of the parameters and their variance using e.g. the method from scipy scipy.optimize.curve_fit. Here we will be assuming that the errors are independent and likely normally distributed (which is fair in this case). Let's say I'm just interested in the w parameter, related to the period. Then what if I repeated this experiment 5 times but I want just 1 estimated value of w and its variance? With the curve fit method I would obtain an estimate of the parameter and it's variance 5 times.. The only correct way I see to do this is to use use the 5 estimates of w and calculate the mean and variance of these 5. But I think this will produce weaker significance than what is possible? I.e. perhaps the estimated variance on each of the runs is smaller than the variance from the 5 repeats of the experiment. How can I use both the assumption of independent and normally distributed errors on each run along with the 5 repeats of the experiment to produce the estimate of w and its variance?