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<br />M= <br /> <br />(W2:X + 2:X') <br />N <br /> <br />(8) <br /> <br />[W2: (X-M) 2 + 2: (X' -M)2J <br />(N-l) <br /> <br />(9) <br /> <br />s = <br /> <br />G= <br /> <br />[W2: (X-M)3 + 2: (X' _M)3] N <br />(N - 1) (N - 2) S3 <br /> <br />(10) <br /> <br />in which X' denotes logarithmic magnitudes of historic peaks and high outliers and X denotes logarithmic <br />magnitudes of systematic peaks between the flood base Qo and the historic threshold QH' These formulas <br />are equivalent to those given in Appendix 6 of Bulletin 17B. <br />These formulas remain correct even if there is no historic information (in which case H = N S), no <br />high or low outliers, and no below-gage-base peaks. Thus these formulas are included in PEAKFQ to <br />calculate the Bulletin 17B statistics for all conditions including the unadjusted systematic-record statistics. <br />Letting each observed peak below the historic threshold QH represent an effective number W of <br />"virtual" peaks yields the following formula for the probability plotting position of the m-th ranked <br />observed peak: <br /> <br />m <br />Pm=(H+l) <br /> <br />(11) <br /> <br />where <br /> <br />ill. = cm + 1/2 <br /> <br />(12) <br /> <br />and <br /> <br />cm = m-l/2 <br /> <br />ifm~Z <br /> <br />(Z = NHO + NHP) <br /> <br />= Z+W[(m-Z) -1/2] <br /> <br />ifm>Z <br /> <br />(13) <br /> <br />In this formula, ill. is the historically weighted rank of the m-th largest observed peak and c is the <br />m <br />centroidal position of a conceptual "cell" occupied by the peak. Cells above the historic threshold have unit <br />width; those below have width W. The effective rank ill. always is at the extreme end of a sub-cell of unit <br />width centered at cm . Equation 11 is equivalent to equation 6-8 in Appendix 6 of Bulletin 17B with "a", a <br />constant characteristic of a given plotting position formula equal to O. <br /> <br />PEAKFQ <br /> <br />8 <br /> <br />DRAFT -1/30/98 <br />