Combining measurements of time periods with unequal number of periods
Suppose you want to estimate the mean period 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 theof 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 onwe use the second derivative of the :
which gives us
We can estimatefrom 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 variablereduce to well-known results for two simplified cases of constant :