# Combining measurements of time periods with unequal number of periods

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Suppose you want to estimate the mean period and its uncertainty with N measurements of the time for a number of periods per measurement ni.

We assume that each time measurement has the same uncertainty .

We estimate the mean period by minimizing the of the measurements with respect to the mean period,

by setting the first derivative to zero:

We solve for the mean period:

To find the uncertainty on we use the second derivative of the :

which gives us

We can estimate from the standard deviation of the measured times with respect to the estimated mean period times the number of periods measured in measurement :

These formulae for variable reduce to well-known results for two simplified cases of constant :

1) All (measure 1 period at a time):

2) All (measure a constant number periods at a time):