282215597
5/28/2016
1/1
Standard Deviation of F({f})
Assume that there is a set {f1,…,fN}I of values and a set of rules for using this set of values to produce a value F({f} I). If F
can be differentiated with respect to fi, the difference between F for the {f} I and that for the {f}J is given by
F  f I   F  f J   
N
i 1
F
 fi , I  fi , J 
fi
(1.1)
The difference squared is
 FI
 FJ  
2
F  F 


i 1, j 1 f i   0 f j 
 j 0
i
N

 f
i,I
 fi , J   f j , I  f j , J 

(1.2)
The standard deviation in F is the limit that MI and MJ → infinity in

2
F ,M I ,M J
M I ,M J
1 N
1
F F
 

2 i 1 M I  M J  1 I 1 fi f j
j 1
 f
i,I

 fi , J   f j , I  f j , J  (1.3)
J 1
J I
If fi and fj are statistically independent, the terms with i ≠ j average to zero so that
 F2 , M


M I ,M J

 F 
2 
1

 
f

f



 i,I i, J 
i 1  f i   2 M I  M J  1 I 1
J 1


J I


2
N
I
,M J
(1.4)
The term in parenthesis in the limit that MI and MJ become infinite is the standard deviation of the values of fi
M I ,M J
2
1
 
fi , I  fi , J 


2M I  M J  1 I 1
2
i
(1.5)
J 1
J I
Finally the standard deviation in F is
2
 F  2
  
 i
i 1  f i 
N
2
F
(1.6)
The relationship of  to the standard deviation in fi for a Gaussian distribution is derived in Deviations.docx. In the case of Poisson
data with more than just a few counts i2=fi. The critical step is the assumption in going from (1.3) to (1.4) that the terms for i≠j sum
to zero.