Uncertainty in Measurement: Significant Figures | Protocol

101 has two captive zeros and 6 significant figures. Leading zeros are zeros to the left of the first nonzero digit. Leading digits are never significant; they ... Loginprocessing... Skiptocontent Trialendsin RequestFullAccess TellYourColleagueAboutJove Chapter1:Introduction:MatterandMeasurement 1.11:UncertaintyinMeasurement:SignificantFigures TABLEOFCONTENTS X Chapter1:Introduction:MatterandMeasurement 301.1:ScientificLawsandTheories301.2:TheScientificMethod301.3:ClassifyingMatterbyState301.4:ClassifyingMatterbyComposition301.5:PhysicalandChemicalPropertiesofMatter301.6:WhatisEnergy?301.7:Measurement:StandardUnits301.8:Measurement:DerivedUnits301.9:UncertaintyinMeasurement:AccuracyandPrecision301.10:UncertaintyinMeasurement:ReadingInstruments301.11:UncertaintyinMeasurement:SignificantFigures301.12:DimensionalAnalysis Chapter2:AtomsandElements 302.1:TheAtomicTheoryofMatter302.2:SubatomicParticles302.3:Elements:ChemicalSymbolsandIsotopes302.4:IonsandIonicCharges302.5:ThePeriodicTable302.6:AtomicMass302.7:MolarMass Chapter3:Molecules,Compounds,andChemicalEquations 303.1:MoleculesandCompounds303.2:ChemicalFormulas303.3:MolecularModels303.4:ClassificationofElementsandCompounds303.5:IonicCompounds:FormulasandNomenclature303.6:MolecularCompounds:FormulasandNomenclature303.7:OrganicCompounds303.8:FormulaMassandMoleConceptsofCompounds303.9:ExperimentalDeterminationofChemicalFormula303.10:ChemicalEquations Chapter4:ChemicalQuantitiesandAqueousReactions 304.1:ReactionStoichiometry304.2:LimitingReactant304.3:ReactionYield304.4:GeneralPropertiesofSolutions304.5:SolutionConcentrationandDilution304.6:ElectrolyteandNonelectrolyteSolutions304.7:SolubilityofIonicCompounds304.8:ChemicalReactionsinAqueousSolutions304.9:PrecipitationReactions304.10:Oxidation-ReductionReactions304.11:OxidationNumbers304.12:Acids,BasesandNeutralizationReactions304.13:SynthesisandDecompositionReactions Chapter5:Gases 305.1:PressureandMeasuringPressure305.2:GasLaws305.3:ApplicationsoftheIdealGasLaw:MolarMass,Density,andVolume305.4:MixtureofGases-Dalton'sLawofPartialPressures305.5:ChemicalStoichiometryandGases305.6:KineticMolecularTheory:BasicPostulates305.7:KineticMolecularTheoryandtheGasLaws305.8:MolecularVelocitiesandKineticEnergy305.9:EffusionandDiffusion305.10:RealGases-DeviationfromtheIdealGasLaw Chapter6:Thermochemistry 306.1:EnergyBasics306.2:FirstLawofThermodynamics306.3:InternalEnergy306.4:QuantifyingHeat306.5:QuantifyingWork306.6:Enthalpy306.7:ThermochemicalEquations306.8:ConstantPressureCalorimetry306.9:ConstantVolumeCalorimetry306.10:Hess'sLaw306.11:StandardEnthalpyofFormation306.12:EnthalpiesofReaction Chapter7:ElectronicStructureofAtoms 307.1:TheWaveNatureofLight307.2:TheElectromagneticSpectrum307.3:InterferenceandDiffraction307.4:PhotoelectricEffect307.5:TheBohrModel307.6:EmissionSpectra307.7:ThedeBroglieWavelength307.8:TheUncertaintyPrinciple307.9:TheQuantum–MechanicalModelofanAtom307.10:QuantumNumbers307.11:AtomicOrbitals307.12:ThePauliExclusionPrinciple307.13:TheEnergiesofAtomicOrbitals307.14:TheAufbauPrincipleandHund'sRule307.15:ElectronConfigurationofMultielectronAtoms Chapter8:PeriodicPropertiesoftheElements 308.1:PeriodicClassificationoftheElements308.2:AtomicRadiiandEffectiveNuclearCharge308.3:IonicRadii308.4:IonizationEnergy308.5:ElectronAffinity308.6:AlkaliMetals308.7:Halogens308.8:NobleGases Chapter9:ChemicalBonding:BasicConcepts 309.1:TypesofChemicalBonds309.2:LewisSymbolsandtheOctetRule309.3:IonicBondingandElectronTransfer309.4:TheBorn-HaberCycle309.5:TrendsinLatticeEnergy:IonSizeandCharge309.6:CovalentBondingandLewisStructures309.7:Electronegativity309.8:BondPolarity,DipoleMoment,andPercentIonicCharacter309.9:LewisStructuresofMolecularCompoundsandPolyatomicIons309.10:Resonance309.11:FormalCharges309.12:ExceptionstotheOctetRule309.13:BondEnergiesandBondLengths309.14:BondinginMetals Chapter10:ChemicalBonding:MolecularGeometryandBondingTheories 3010.1:VSEPRTheoryandtheBasicShapes3010.2:VSEPRTheoryandtheEffectofLonePairs3010.3:PredictingMolecularGeometry3010.4:MolecularShapeandPolarity3010.5:ValenceBondTheory3010.6:HybridizationofAtomicOrbitalsI3010.7:HybridizationofAtomicOrbitalsII3010.8:MolecularOrbitalTheoryI3010.9:MolecularOrbitalTheoryII Chapter11:Liquids,Solids,andIntermolecularForces 3011.1:MolecularComparisonofGases,Liquids,andSolids3011.2:IntermolecularvsIntramolecularForces3011.3:IntermolecularForces3011.4:ComparingIntermolecularForces:MeltingPoint,BoilingPoint,andMiscibility3011.5:SurfaceTension,CapillaryAction,andViscosity3011.6:PhaseTransitions3011.7:PhaseTransitions:VaporizationandCondensation3011.8:VaporPressure3011.9:Clausius-ClapeyronEquation3011.10:PhaseTransitions:MeltingandFreezing3011.11:PhaseTransitions:SublimationandDeposition3011.12:HeatingandCoolingCurves3011.13:PhaseDiagrams3011.14:StructuresofSolids3011.15:LatticeCenteringandCoordinationNumber3011.16:MolecularandIonic Solids3011.17:IonicCrystalStructures3011.18:MetallicSolids3011.19:BandTheory3011.20:NetworkCovalentSolids3011.21:X-rayCrystallography Chapter12:SolutionsandColloids 3012.1:SolutionFormation3012.2:IntermolecularForcesinSolutions3012.3:EnthalpyofSolution3012.4:AqueousSolutionsandHeatsofHydration3012.5:SolutionEquilibriumandSaturation3012.6:PhysicalPropertiesAffectingSolubility3012.7:ExpressingSolutionConcentration3012.8:VaporPressureLowering3012.9:IdealSolutions3012.10:FreezingPointDepressionandBoilingPointElevation3012.11:OsmosisandOsmoticPressureofSolutions3012.12:Electrolytes:van'tHoffFactor3012.13:Colloids Chapter13:ChemicalKinetics 3013.1:ReactionRate3013.2:MeasuringReactionRates3013.3:ConcentrationandRateLaw3013.4:DeterminingOrderofReaction3013.5:TheIntegratedRateLaw:TheDependenceofConcentrationonTime3013.6:Half-lifeofaReaction3013.7:TemperatureDependenceonReactionRate3013.8:ArrheniusPlots3013.9:ReactionMechanisms3013.10:Rate-DeterminingSteps3013.11:Catalysis3013.12:Enzymes Chapter14:ChemicalEquilibrium 3014.1:DynamicEquilibrium3014.2:TheEquilibriumConstant3014.3:EquilibriaforGaseousReactionsandHeterogeneousReactions3014.4:CalculatingtheEquilibriumConstant3014.5:ReactionQuotient3014.6:CalculatingEquilibriumConcentrations3014.7:TheSmallxAssumption3014.8:LeChatelier'sPrinciple:ChangingConcentration3014.9:LeChatelier'sPrinciple:ChangingVolume(Pressure)3014.10:LeChatelier'sPrinciple:ChangingTemperature Chapter15:AcidsandBases 3015.1:Bronsted-LowryAcidsandBases3015.2:Acid/BaseStrengthsandDissociationConstants3015.3:Water:ABronsted-LowryAcidandBase3015.4:pHScale3015.5:RelativeStrengthsofConjugateAcid-BasePairs3015.6:StrongAcidandBaseSolutions3015.7:WeakAcidSolutions3015.8:WeakBaseSolutions3015.9:MixturesofAcids3015.10:IonsasAcidsandBases3015.11:DeterminingthepHofSaltSolutions3015.12:PolyproticAcids3015.13:AcidStrengthandMolecularStructure3015.14:LewisAcidsandBases Chapter16:Acid-baseandSolubilityEquilibria 3016.1:CommonIonEffect3016.2:Buffers3016.3:Henderson-HasselbalchEquation3016.4:CalculatingpHChangesinaBufferSolution3016.5:BufferEffectiveness3016.6:Acid-BaseTitrationCurves3016.7:TitrationCalculations:StrongAcid-StrongBase3016.8:TitrationCalculations:WeakAcid-WeakBase3016.9:Indicators3016.10:TitrationofaPolyproticAcid3016.11:SolubilityEquilibria3016.12:FactorsAffectingSolubility3016.13:FormationofComplexIons3016.14:PrecipitationofIons3016.15:QualitativeAnalysis Chapter17:Thermodynamics 3017.1:Spontaneity3017.2:Entropy3017.3:SecondLawofThermodynamics3017.4:ThirdLawofThermodynamics3017.5:StandardEntropyChangeforaReaction3017.6:GibbsFreeEnergy3017.7:EffectsofTemperatureonFreeEnergy3017.8:CalculatingStandardFreeEnergyChanges3017.9:FreeEnergyChangesforNonstandardStates3017.10:FreeEnergyandEquilibrium Chapter18:Electrochemistry 3018.1:BalancingRedoxEquations3018.2:ElectromotiveForce3018.3:Voltaic/GalvanicCells3018.4:StandardElectrodePotentials3018.5:CellPotentialandFreeEnergy3018.6:TheNernstEquation3018.7:ConcentrationCells3018.8:BatteriesandFuelCells3018.9:Corrosion3018.10:Electrolysis Chapter19:RadioactivityandNuclearChemistry 3019.1:RadioactivityandNuclearEquations3019.2:TypesofRadioactivity3019.3:NuclearStability3019.4:NuclearBindingEnergy3019.5:RadioactiveDecayandRadiometricDating3019.6:NuclearFission3019.7:NuclearPower3019.8:NuclearFusion3019.9:NuclearTransmutation3019.10:BiologicalEffectsofRadiation Chapter20:TransitionMetalsandCoordinationComplexes 3020.1:PropertiesofTransitionMetals3020.2:CoordinationCompoundsandNomenclature3020.3:Metal-LigandBonds3020.4:CoordinationNumberandGeometry3020.5:StructuralIsomerism3020.6:Stereoisomerism3020.7:ValenceBondTheory3020.8:CrystalFieldTheory-OctahedralComplexes3020.9:CrystalFieldTheory-TetrahedralandSquarePlanarComplexes3020.10:ColorsandMagnetism Chapter21:Biochemistry 3021.1:FunctionalGroups3021.2:Polymers3021.3:ChemistryoftheCell3021.4:StructureofLipids3021.5:ChemistryofCarbohydrates3021.6:Aminoacids3021.7:PeptideBonds3021.8:ProteinandProteinStructure3021.9:NucleicAcids3021.10:DNABasePairing3021.11:DNAReplication3021.12:FromDNAtoProtein FullTableofContents Thisisasampleclip.Signinorstartyourfreetrial. JoVECoreChemistry AsubscriptiontoJoVEisrequiredtoviewthiscontent.Youwillonlybeabletoseethefirst20seconds. Education UncertaintyinMeasurement:SignificantFigures PreviousVideo NextVideo TheJoVEvideoplayeriscompatiblewithHTML5andAdobeFlash.OlderbrowsersthatdonotsupportHTML5andtheH.264videocodecwillstilluseaFlash-basedvideoplayer.WerecommenddownloadingthenewestversionofFlashhere,butwesupportallversions10andabove. Ifthatdoesn'thelp,pleaseletusknow. Unabletoloadvideo.PleasecheckyourInternetconnectionandreloadthispage.Iftheproblemcontinues,pleaseletusknowandwe'lltrytohelp. Anunexpectederroroccurred.PleasecheckyourInternetconnectionandreloadthispage.Iftheproblemcontinues,pleaseletusknowandwe'lltrytohelp.   PreviousVideo 1.10:UncertaintyinMeasurement:ReadingInstruments NextVideo 1.12:DimensionalAnalysis Embed Share AddtoFavorites ADDTOPLAYLIST English中文DeutschFrançaisPortuguêsTürkçeРусскийEspañolNederlands한국어العربيةעִבְרִיתItaliano日本語 TRANSCRIPT Allofthenumbersinascientificmeasurementarecertain,exceptforthelastdigit.Thecertaintyofmeasurementdependsontwofactors:thenumberofdigitsinthemeasurementandtheprecisionoftheinstrumentused.  Inameasuredquantity,allofthedigits,includingthelastuncertaindigit,arecalledsignificantfigures andcanbedeterminedusingspecificrules. Anynon-zerodigitsandallcaptivezeros, whichliebetweentwonon-zerodigits, aresignificant.Forexample,28hastwosignificantfigures,while26.25hasfour,and208hasthree. Leadingzerosareneversignificant,astheyjustlocatethedecimalpoint.Forexample,0.00208hasthreesignificantfigures.Suchquantitiescanbeexpressedusingexponentialnotations.Thus,0.00208canbewrittenas2.08×10−3. Trailingzerosareonlysignificantindecimalformattednumbers.2200hastwotrailingzerosandtwosignificantfigures,whereas2200.0and2200.1bothhave5significantfigures.  Forquantitieswithoutdecimalpoints,thesignificanceoftrailingzerosbecomesambiguous.Thus,2200canbewrittenas2.2×103withtwosignificantfiguresor2.20×103withthreesignificantfigures. Significantfigureshelpachievecertaintyinmathematicaloperations,too.Inadditionorsubtraction,theresultshouldberoundedofftohavethesamenumberofdecimalplacesasthemeasurementwiththefewestdecimalplaces.  Roundingdownshouldbeperformedwhenthelastdigitisbelow5,androundingupcarriedoutwhenitis5orabove.Otherroundingmethodsaresometimesusedwhenthelastdigitis5. Forinstance,thesumof2.052and1.2isroundedoffas3.3. However,whilemultiplyingordividing,theresultshouldberoundedtohavethesamenumberofsignificantfiguresasthemeasurementwiththefewestsignificantfigures.Thus,theproductof2.052and1.2isroundedoffas2.5. Scientistsoftenrepeatexperimentstoachieveprecisionintheirmeasurements.Standarddeviationisthestatisticalexpressionofsuchprecisionandmeasuresthedispersionfromtheexpectedvalue.Ifprecisionishigh,thestandarddeviationissmall,andviceversa.  Forexample,twogroupsmeasuredthethicknessofabookincentimeters.Theyfoundthesameaverage: 10.6cm.However,themeasurementsbythefirstgrouparemoreprecise,andthushavealowerstandarddeviation.Thesecondgrouphasmorespreadoutmeasurementsandahigherstandarddeviation. 1.11:UncertaintyinMeasurement:SignificantFigures Allthedigitsinameasurement,includingtheuncertainlastdigit,arecalledsignificantfiguresorsignificantdigits.Notethatzeromaybeameasuredvalue;forexample,ifascalethatshowsweighttothenearestpoundreads“140,”thenthe1(hundreds),4(tens),and0(ones)areallsignificant(measured)values. Ameasurementresultisproperlyreportedwhenitssignificantdigitsaccuratelyrepresentthecertaintyofthemeasurementprocess.Belowareasetofrulestodeterminethenumberofsignificantfiguresinameasurement: Allnonzerodigitsaresignificant.Startingwiththefirstnonzerodigitontheleft,countthisdigitandallremainingdigitstotheright.Thisisthenumberofsignificantfiguresinthemeasurement.Forexample,843hasthreesignificantdigits,843.12has5significantdigits.  Captivezeros,whicharezerosbetweentwononzerodigits,aresignificant.Forexample,808.101hastwocaptivezerosand6significantfigures. Leadingzerosarezerostotheleftofthefirstnonzerodigit.Leadingdigitsareneversignificant;theymerelyrepresentthepositionofthedecimalpoint.Forexample,theleadingzerosin0.008081arenotsignificant.Thisnumbercanbeexpressedusingexponentialnotationas8.081×10−3,thenthenumber8.081containsallthesignificantfigures,and10−3locatesthedecimalpoint. Thesignificanceoftrailingzeros,whicharezerosattheendofanumber,dependsontheirposition.Trailingzerosbefore(butafteranonzerodigit)andafterthedecimalpointaresignificant.However,fornumbersthatdonothavedecimalpoints,trailingzerosmayormaynotbesignificant.Thisambiguitycanberesolvedwiththeuseofexponentialnotation.Forexample,themeasurement1300canbewrittenas1.3×103(twosignificantfigures),1.30×103(threesignificantfigures,ifthetensplacewasmeasured),or1.300×103(foursignificantfigures,iftheonesplacewasalsomeasured). SignificantFiguresinCalculations Uncertaintyinmeasurementscanbeavoidedbyreportingtheresultsofcalculationwiththecorrectnumberofsignificantfigures.Thiscanbedeterminedbythefollowingtherulesforroundingnumbers: Whenaddingorsubtractingnumbers,roundtheresulttothesamenumberofdecimalplacesasthenumberwiththeleastnumberofdecimalplaces. Whenmultiplyingordividingnumbers,roundtheresulttothesamenumberofdigitsasthenumberwiththeleastnumberofsignificantfigures. Ifthedigittobedropped(theoneimmediatelytotherightofthedigittoberetained)islessthan5,“rounddown”andleavetheretaineddigitunchanged. Ifthedigittobedropped(theoneimmediatelytotherightofthedigittoberetained)is5orgreater,“roundup”andincreasetheretaineddigitby1.Alternativeroundingmethodsmayalsobeusedifthedroppeddigitis5.Theretaineddigitisroundedupordown,whicheveryieldsanevenvalue. Animportantnoteisthatroundingofsignificantfiguresshouldpreferablybedoneattheendofamultistepcalculationtoavoidtheaccumulationoferrorsateachstepduetorounding.Thus,significantfiguresandrounding,facilitatecorrectrepresentationofthecertaintyofthemeasuredvaluesreported. ThistextisadaptedfromOpenstax,Chemistry2e,Section1.5:MeasurementUncertainty,Accuracy,andPrecision. Pleaseenteryourinstitutionalemailtocheckifyouhaveaccesstothiscontent Submit hasaccessto Pleasecreateanaccounttogetaccess CreateAccount Back LogintoaccessJoVE EMAIL PASSWORD ForgotPassword? 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