- squ ared ▁: ▁▁▁ ▁where ▁S ▁is ▁the ▁minimum ▁value ▁of ▁the ▁( weight ed ) ▁objective ▁function : ▁ ▁The ▁denomin ator , ▁, ▁is ▁the ▁number ▁of ▁degrees ▁of ▁freedom ; ▁see ▁effective ▁degrees ▁of ▁freedom ▁for ▁general izations ▁for ▁the ▁case ▁of ▁correl ated ▁observations . ▁ ▁In ▁all ▁cases , ▁the ▁variance ▁of ▁the ▁parameter ▁estimate ▁ ▁is ▁given ▁by ▁ ▁and ▁the ▁cov ar iance ▁between ▁the ▁parameter ▁estimates ▁ ▁and ▁ ▁is ▁given ▁by ▁. ▁The ▁standard ▁deviation ▁is ▁the ▁square ▁root ▁of ▁variance , ▁, ▁and ▁the ▁correlation ▁coefficient ▁is ▁given ▁by ▁. ▁These ▁error ▁estimates ▁reflect ▁only ▁random ▁errors ▁in ▁the ▁measurements . ▁The ▁true ▁uncertainty ▁in ▁the ▁parameters ▁is ▁larger ▁due ▁to ▁the ▁presence ▁of ▁system atic ▁errors , ▁which , ▁by ▁definition , ▁cannot ▁be ▁quant ified . ▁Note ▁that ▁even ▁though ▁the ▁observations ▁may ▁be ▁un cor related , ▁the ▁parameters ▁are ▁typically ▁correl ated . ▁ ▁Parameter ▁confidence ▁limits ▁ ▁It ▁is ▁often ▁assumed , ▁for ▁want ▁of ▁any ▁concrete ▁evidence ▁but ▁often ▁appe aling ▁to ▁the ▁central ▁limit ▁theorem — see ▁Normal ▁distribution # Occ urrence — that ▁the ▁error ▁on ▁each ▁observation ▁belongs ▁to ▁a ▁normal ▁distribution ▁with ▁a ▁mean ▁of ▁zero ▁and ▁standard ▁deviation ▁. ▁Under ▁that ▁assumption ▁the ▁following ▁prob abilities ▁can ▁be ▁derived ▁for ▁a ▁single ▁scalar ▁parameter ▁estimate ▁in ▁terms ▁of ▁its ▁estimated ▁standard ▁error ▁ ▁( g iven ▁here ): ▁ 6 8 % ▁that ▁the ▁interval ▁ ▁en comp ass es ▁the