Forskjell mellom versjoner av «Combining measurements of time periods with unequal number of periods»
Fra mn/fys/epf
Linje 33: | Linje 33: | ||
<math>\sigma ={\sigma_t\over \sqrt{n}}</math> | <math>\sigma ={\sigma_t\over \sqrt{n}}</math> | ||
− | 2) All <math>n_i=m</math> (measure a constant number m periods at a time): <br> | + | 2) All <math>n_i=m</math> (measure a constant number <math>m</math> periods at a time): <br> |
<math>\bar{T}={1\over m}{\sum^N_{i=1}t_i\over N}</math><br> | <math>\bar{T}={1\over m}{\sum^N_{i=1}t_i\over N}</math><br> | ||
<math>\sigma ={1\over m}{\sigma_t\over\sqrt{N}}</math> | <math>\sigma ={1\over m}{\sigma_t\over\sqrt{N}}</math> |
Nåværende revisjon fra 4. feb. 2013 kl. 22:50
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 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
2) All