re: Statistics or sadistics?

[trammell@fn872.fnal.gov (John Trammell)]

Hello all:

Ludwik Kowalski <KOWALSKIL@alpha.montclair.edu> wrote:

> A long-T radiactive sample was "counted" for one munute. The outcome
> was A=145 -->st.dev.=12. Then the background was measured for one minute
> yielding B=41 --> st.dev=6.4.
>
> 1) The estimated net A-B is 104. What error bar (st.dev.) should be
>   used for this result. I know it would be 18.4 if the result were
>   obtained from a multiplication, or from a division of A and B. But
>   what is the expected standard deviation of a differnce, or sum, in
>   terms of known sig_A and sig_B?
>
> 2) Subsequently the background, measured for 60 minutes, were 2110.
>   Thus B=35 --> st.dev.=sqrt(2110)/60=0.76. This is much better that
>   41 +/-6.4 . But the sample is no longer available for long counting.
>   Can we really benefit from the long counting of the background? My
>   answer is NO. We might know the long-term mean value of B very well
>   but we have no way of knowing what fraction of the net count of
>   (145 +/-12) was actually due to the background. A very long
>   counting of B does not help us to reduce the error bar around the
>   value of 104. Do you agree?

OK, I'll have a go at it.  Use 'd' for full derivative,
'D' for partial.

    S = A - B

         DS          DS
   dS = ---- * dA + ---- * dB
         DA          DB

      = dA - dB

So, assuming gaussian statistics:

   |dS| = sqrt((dA)^2 + (dB)^2)

This means the answers are:

1. what is the expected standard deviation of a difference,
   in terms of known sig_A and sig_B?

     sig_tot = sqrt(sig_A^2 + sig_B^2)

2. Can we really benefit from the long counting of the
   background?  My answer is NO.

     I'd say "sort of".  The uncertainty in your measurement
     of A dominates; your improvement of your B measurement
     decreases your error from 13.6 (the error with 1-minute
     background estimate) to 12.0, about 10% improvement.

Cheers,
John Trammell