Effect of random binning on significance

with one bin (ie: counting experiment) the significance of the discrepancy is given by

\[ S_0 = \frac{D - B }{\sqrt{D}} \] with k random bins of equal size, (meaning there is no "shape" or trend of the D or B)

Then the significance is given by,

\[ S_k = \sqrt{ \sum_k \left( \frac{D_k - B_k }{\sqrt{D_k}} \right)^2 } = \sqrt{ \left(\sum_k \frac{ \frac{D}{k} - \frac{B}{k} }{\sqrt{ \frac{D}{k} }} \right)^2 } = \sqrt{ \sum_k \left( \frac{1}{\sqrt{k}} \frac{D - B }{\sqrt{D}} \right)^2} = \sqrt{ \sum_k \left( \frac{1}{\sqrt{k}} S_0 \right)^2} \] \[ = \sqrt{ \sum_k \frac{1}{k} S_0^2 } = S_0 \]

D3 is number of 3b data B = $∑ w$D3, with \(w\sim\) B / D3

(Note that the analysis is typically setup such that w << 1, ie more D3 than expected D4 background)

Now with one bin the significance of the discrepancy is given by

\[ S_0 = \frac{D - B }{\sqrt{D + \sum w^2}} = \frac{D - B }{\sqrt{D + \sum (\frac{B}{D3})^2}} = \frac{D - B }{\sqrt{D + (\frac{B}{D3})^2 D3} } = \frac{D - B }{\sqrt{D + (\frac{B}{D3}) B} } = = \frac{D - B }{\sqrt{D + w B} } \] (Again, note that the 2nd term in the denominator is usually arranged to be small bc w<<1)

Now with k bins:

\[ S_k = \sqrt{ \sum_k \left( \frac{\frac{D}{k} - \frac{B}{k} }{\sqrt{\frac{D}{k} + (\frac{B}{D3})^2 \frac{D3}{k}} } \right)^2 } = \sqrt{ \sum_k \left(\frac{1}{\sqrt{k}} \frac{D - B }{\sqrt{D + w B}} \right)^2 } = S_0 \]

Stats Data-Driven SR

202504181245