or ▁estimated ) ▁measurement ▁precision ▁thereby ▁minim izing ▁the ▁Mah al an ob is ▁distance ▁from ▁the ▁points ▁to ▁the ▁line , ▁providing ▁a ▁maximum - lik elihood ▁solution ; ▁the ▁unknown ▁precis ions ▁could ▁be ▁found ▁via ▁analysis ▁of ▁variance . ▁ ▁In ▁short , ▁total ▁least ▁squares ▁does ▁not ▁have ▁the ▁property ▁of ▁units - in var iance — i . e . ▁it ▁is ▁not ▁scale ▁invariant . ▁For ▁a ▁meaning ful ▁model ▁we ▁require ▁this ▁property ▁to ▁hold . ▁A ▁way ▁forward ▁is ▁to ▁real ise ▁that ▁resid uals ▁( dist ances ) ▁measured ▁in ▁different ▁units ▁can ▁be ▁combined ▁if ▁multiplication ▁is ▁used ▁instead ▁of ▁addition . ▁Consider ▁fitting ▁a ▁line : ▁for ▁each ▁data ▁point ▁the ▁product ▁of ▁the ▁vertical ▁and ▁horizontal ▁resid uals ▁equals ▁twice ▁the ▁area ▁of ▁the ▁triangle ▁formed ▁by ▁the ▁resid ual ▁lines ▁and ▁the ▁fitted ▁line . ▁We ▁choose ▁the ▁line ▁which ▁minim izes ▁the ▁sum ▁of ▁these ▁areas . ▁Nobel ▁la ure ate ▁Paul ▁Samuel son ▁proved ▁in ▁ 1 9 4 2 ▁that , ▁in ▁two ▁dimensions , ▁it ▁is ▁the ▁only ▁line ▁express ible ▁sole ly ▁in ▁terms ▁of ▁the ▁r ati os ▁of ▁standard ▁devi ations ▁and ▁the ▁correlation ▁coefficient ▁which ▁( 1 ) ▁fits ▁the ▁correct ▁equation ▁when ▁the ▁observations ▁fall ▁on ▁a ▁straight ▁line , ▁( 2 ) ▁exhib its ▁scale ▁in var iance , ▁and ▁( 3 ) ▁exhib its ▁in var iance ▁under ▁inter change ▁of ▁variables . ▁This ▁solution ▁has ▁been ▁redis cover