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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.