Uncertainties Statistics
5. Proportion of quarks and leptons
When studying a small number of events, the statistical uncertainty can be very large. The statistical uncertainty in N_i observations is squareroot(N_i). The relative uncertainty is squareroot(N_i)/N_i that is 1/squareroot(N_i). The relative uncertainty decreases with increasing number of events one observes. That is why it is interesting to study a large number of events. Use this relation to determine the uncertainty of the determined quantities. Regard the total number of events as fixed, that is having no uncertainty.

6. In the case that the total number of events is fixed and there are only two possible outcomes (like either lepton event or quark event), then the uncertainty is estimated somewhat differently. If:

• The total number of events is N,
• the number of events of type i is N_i ,
• the proportion of type i is p_i = N_i/N,

then the uncertainty (standard deviation) in number of events N_i becomes squareroot(N_i(1-p_i)).

7. Ratio R = N_3jets/N_{2 jets}
Both N_3jets and N_2jets have uncertainties. These have to be combined to get the uncertainty of the ratio. The relative uncertainty in the jet ratio is the square root of 1/N_3jets plus 1/N_{2 jets}, or more mathematically: squareroot(R)/R = squareroot( 1/N_3jets + 1/N_{2 jets} ).