ations ▁of ▁the ▁probability ▁distribution ▁of ▁the ▁errors ▁is ▁the ▁same ▁number ▁ σ ▁on ▁different ▁we igh ings ; ▁errors ▁on ▁different ▁we igh ings ▁are ▁independent . ▁ ▁Den ote ▁the ▁true ▁weights ▁by ▁ ▁We ▁consider ▁two ▁different ▁experiments : ▁▁ ▁We igh ▁each ▁object ▁in ▁one ▁pan , ▁with ▁the ▁other ▁pan ▁empty . ▁ ▁Let ▁X i ▁be ▁the ▁measured ▁weight ▁of ▁the ▁object , ▁for ▁i ▁= ▁ 1 , ▁... , ▁ 8 . ▁ ▁Do ▁the ▁eight ▁we igh ings ▁according ▁to ▁the ▁following ▁schedule ▁and ▁let ▁Y i ▁be ▁the ▁measured ▁difference ▁for ▁i ▁= ▁ 1 , ▁... , ▁ 8 : ▁▁▁▁▁ ▁Then ▁the ▁estimated ▁value ▁of ▁the ▁weight ▁ θ 1 ▁is ▁▁▁▁ ▁Similar ▁estimates ▁can ▁be ▁found ▁for ▁the ▁weights ▁of ▁the ▁other ▁items . ▁For ▁example ▁▁▁▁ ▁The ▁question ▁of ▁design ▁of ▁experiments ▁is : ▁which ▁experiment ▁is ▁better ? ▁ ▁The ▁variance ▁of ▁the ▁estimate ▁X 1 ▁of ▁ θ 1 ▁is ▁ σ 2 ▁if ▁we ▁use ▁the ▁first ▁experiment . ▁ ▁But ▁if ▁we ▁use ▁the ▁second ▁experiment , ▁the ▁variance ▁of ▁the ▁estimate ▁given ▁above ▁is ▁ σ 2 / 8 . ▁ ▁Thus ▁the ▁second ▁experiment ▁gives ▁us ▁ 8 ▁times ▁as ▁much ▁precision ▁for ▁the ▁estimate ▁of ▁a ▁single ▁item , ▁and ▁estimates ▁all ▁items ▁simultaneously , ▁with ▁the ▁same ▁precision . ▁What ▁the ▁second ▁experiment ▁ach ieves ▁with ▁eight ▁would ▁require ▁ 6 4 ▁we igh ings ▁if ▁the ▁items ▁are ▁we