Uses and abuses of logarithmic axes - GraphPad Software
Uses and abuses of logarithmic axes - GraphPad Software
Uses and abuses of logarithmic axes - GraphPad Software
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
difference<strong>of</strong>90,000).Whatisconsistentistheratio.Eachaxistickrepresentsavaluetenfoldhigherthantheprevioustick.Thereddotsplotadatasetwithequallyspacedvalues.EachdotrepresentsavaluewithaYvalue500higherthanthedotbelow.Thedotsareequallyspacedonthegraphontheleft,butfarfromequallyspacedonthegraphontheright.Topreventoverlap,thepointsarejitteredtotheright<strong>and</strong>leftsotheydon'toverlap.Thehorizontalposition<strong>of</strong>thereddotshasnoothermeaning.ThebluedotsrepresentadatasetwhereeachvaluerepresentsaYvalue1.5timeshigherthantheonebelow.Onthegraphontheleft,thelowervaluesarealmostsuperimposed,makingitveryhardtoseethedistribution<strong>of</strong>values(evenwithhorizontaljittering).Onthegraphontherightwitha<strong>logarithmic</strong>axis,thepointsappearequallyspaced.Interpolating between log ticksWhatvalueishalfwaybetweenthetickfor10<strong>and</strong>theonefor100?Yourfirstguessmightbetheaverage<strong>of</strong>thosetwovalues,55.Butthatiswrong.Valuesarenotequallyspacedona<strong>logarithmic</strong>axis.Thelogarithm<strong>of</strong>10is1.0,<strong>and</strong>thelogarithm<strong>of</strong>100is2.0,sothelogarithm<strong>of</strong>themidpointis1.5.Whatvaluehasalogarithm<strong>of</strong>1.5?Theansweris10 1.5 ,whichis31.62.Sothevaluehalfwaybetween10<strong>and</strong>100ona<strong>logarithmic</strong>axisis31.62.Similarly,thevaluehalfwaybetween100<strong>and</strong>1000ona<strong>logarithmic</strong>axisis316.2.Why “<strong>logarithmic</strong>”?Intheexampleabove,theticksat1,10,100,1000areequallyspacedonthegraph.Thelogarithms<strong>of</strong>1,10,100<strong>and</strong>1000are0,1,2,3,whichareequallyspacedvalues.Sincevaluesthatareequallyspacedonthegraphhavelogarithmsthatareequallyspacednumerically,thiskind<strong>of</strong>axisiscalleda“<strong>logarithmic</strong>axis”.LingoThetermsemilogisusedtorefertoagraphwhereoneaxisis<strong>logarithmic</strong><strong>and</strong>theotherisn’t.Whenboth<strong>axes</strong>are<strong>logarithmic</strong>,thegraphiscalledaloglog plot.Other BasesAllthelogarithmsshownabovearecalledbase10logarithms,becausethecomputationstake10tosomepower.Thesearealsocalledcommonlogarithms.Mathematiciansprefernaturallogarithms,usingbasee(2.7183...).Buttheydon’tseemverynaturaltoscientists,<strong>and</strong>arerarelyusedasanaxisscale.Abase2logarithmisthenumber<strong>of</strong>doublingsittakestoreachavalue,<strong>and</strong>thisiswidelyusedbycellbiologists<strong>and</strong>immunologists.Ifyoustartwith1<strong>and</strong>doubleitfourtimes(2,4,8,<strong>and</strong>16),theresultis16,sothelogbase2<strong>of</strong>16is4. 2
Change language