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Measurements & Uncertainty Analysis

1 Department of Physics and Astronomy

MeasurementandUncertaintyAnalysisGuide

“Itisbettertoberoughlyrightthanpreciselywrong.”–AlanGreenspan

TableofContentsTHEUNCERTAINTYOFMEASUREMENTS..............................................................................................................2TYPESOFUNCERTAINTY..........................................................................................................................................6ESTIMATINGEXPERIMENTALUNCERTAINTYFORASINGLEMEASUREMENT................................................9ESTIMATINGUNCERTAINTYINREPEATEDMEASUREMENTS...........................................................................9STANDARDDEVIATION..........................................................................................................................................12STANDARDDEVIATIONOFTHEMEAN(STANDARDERROR).........................................................................14WHENTOUSESTANDARDDEVIATIONVSSTANDARDERROR......................................................................14ANOMALOUSDATA.................................................................................................................................................15FRACTIONALUNCERTAINTY.................................................................................................................................15BIASESANDTHEFACTOROFN–1.......................................................................................................................16SIGNIFICANTFIGURES............................................................................................................................................17UNCERTAINTY,SIGNIFICANTFIGURES,ANDROUNDING.................................................................................18PROPAGATIONOFUNCERTAINTY........................................................................................................................23THEUPPER-LOWERBOUNDMETHODOFUNCERTAINTYPROPAGATION...................................................23QUADRATURE..........................................................................................................................................................24COMBININGANDREPORTINGUNCERTAINTIES................................................................................................29CONCLUSION:“WHENDOMEASUREMENTSAGREEWITHEACHOTHER?”...................................................30MAKINGGRAPHS....................................................................................................................................................32USINGEXCELFORDATAANALYSISINPHYSICSLABS......................................................................................34GETTINGSTARTED.................................................................................................................................................34CREATINGANDEDITINGAGRAPH......................................................................................................................35ADDINGERRORBARS............................................................................................................................................36ADDINGATRENDLINE...........................................................................................................................................36DETERMININGTHEUNCERTAINTYINSLOPEANDY-INTERCEPT..................................................................37INTERPRETINGTHERESULTS...............................................................................................................................37FINALSTEP–COPYINGDATAANDGRAPHSINTOAWORDDOCUMENT.....................................................37USINGLINESTINEXCEL......................................................................................................................................38APPENDIX:PROPAGATIONOFUNCERTAINTYBYQUADRATURE...................................................................41


2 University of North Carolina

TheUncertaintyofMeasurementsSomenumericalstatementsareexact:Maryhas3brothers,and2+2=4.However,allmeasurementshavesomedegreeofuncertaintythatmaycomefromavarietyofsources.Theprocessofevaluatingtheuncertaintyassociatedwithameasurementresultisoftencalleduncertaintyanalysisorsometimeserroranalysis.Thecompletestatementofameasuredvalueshouldincludeanestimateofthelevelofconfidenceassociatedwiththevalue.Properlyreportinganexperimentalresultalongwithitsuncertaintyallowsotherpeopletomakejudgmentsaboutthequalityof the experiment, and it facilitates meaningful comparisons with other similarvaluesoratheoreticalprediction.Withoutanuncertaintyestimate,itisimpossibleto answer the basic scientific question: “Does my result agree with a theoreticalprediction or results from other experiments?” This question is fundamental fordecidingifascientifichypothesisisconfirmedorrefuted.Whenmakingameasurement,wegenerallyassume thatsomeexactor truevalueexistsbasedonhowwedefinewhatisbeingmeasured.Whilewemayneverknowthis true value exactly, we attempt to find this ideal quantity to the best of ourability with the time and resources available. As we make measurements bydifferent methods, or even whenmakingmultiple measurements using the samemethod,wemayobtainslightlydifferentresults.Sohowdowereportourfindingsforourbestestimateofthiselusivetruevalue?Themostcommonwaytoshowtherangeofvaluesthatwebelieveincludesthetruevalueis:

measurement=(bestestimate±uncertainty)unitsAsanexample,supposeyouwanttofindthemassofagoldringthatyouwouldliketoselltoafriend.Youdonotwanttojeopardizeyourfriendship,soyouwanttogetanaccuratemassoftheringinordertochargeafairmarketprice.Youestimatethemasstobebetween10and20gramsfromhowheavyitfeelsinyourhand,butthisisnotaverypreciseestimate.Aftersomesearching,youfindanelectronicbalancethat gives amass reading of 17.43 grams.While thismeasurement ismuchmoreprecise than the original estimate, how do you know that it isaccurate, and howconfident are you that this measurement represents the true value of the ring’smass? Since the digital display of the balance is limited to 2 decimal places, youcouldreportthemassasm=17.43±0.01g.Supposeyouusethesameelectronicbalance and obtain several more readings: 17.46 g, 17.42 g, 17.44 g, so that theaveragemass appears to be in the range of 17.44± 0.02 g. By now youmay feelconfidentthatyouknowthemassof thisringtothenearesthundredthofagram,buthowdoyouknowthatthetruevaluedefinitelyliesbetween17.43gand17.45g? Since you want to be honest, you decide to use another balance that gives areadingof17.22g.Thisvalueisclearlybelowtherangeofvaluesfoundonthefirstbalance,andundernormalcircumstances,youmightnotcare,butyouwant tobe



fairtoyourfriend.Sowhatdoyoudonow?Theanswerliesinknowingsomethingabouttheaccuracyofeachinstrument.Tohelpanswerthesequestions,wefirstdefinethetermsaccuracyandprecision:

Accuracyistheclosenessofagreementbetweenameasuredvalueandatrueoracceptedvalue.Measurementerroristheamountofinaccuracy.Precisionisameasureofhowwellaresultcanbedetermined(withoutreferencetoatheoreticalortruevalue).Itisthedegreeofconsistencyandagreementamongindependentmeasurementsofthesamequantity;alsothereliabilityorreproducibilityoftheresult.

Theaccuracyandprecisioncanbepicturedasfollows:

highprecision,lowaccuracy lowprecision,highaccuracy

Figure1.AccuracyvsPrecision



Theuncertainty estimate associatedwith ameasurement should account for boththeaccuracyandprecisionof themeasurement.Precision indicates thequalityofthe measurement, without any guarantee that the measurement is “correct.”Accuracy, on the other hand, assumes that there is an ideal “true” value, andexpresseshow far your answer is from that “correct” answer.These concepts aredirectlyrelatedtorandomandsystematicmeasurementuncertainties(nextsection).Note:Unfortunatelythetermserroranduncertaintyareoftenusedinterchangeablyto describe both imprecision and inaccuracy. This usage is so common that it isimpossibletoavoidentirely.Wheneveryouencountertheseterms,makesureyouunderstandwhethertheyrefertoaccuracyorprecision,orboth.Inthisdocument,wewillemphasizetheterm“uncertainty”butwillusetheterm“error,”asnecessary,toavoidconfusionwithcommonlyfoundexamplesandstandardusageoftheterm.Inordertodeterminetheaccuracyofaparticularmeasurement,wehavetoknowthe ideal, truevalue, sometimes referred to as the “gold standard.” Sometimeswehavea“textbook”measuredvalue,whichiswellknown,andweassumethatthisisour“ideal”value,anduseit toestimatetheaccuracyofourresult.Othertimesweknowatheoreticalvalue,whichiscalculatedfrombasicprinciples,andthisalsomaybetakenasan“ideal”value.Butphysics isanempiricalscience,whichmeansthatthetheorymustbevalidatedbyexperiment,andnottheotherwayaround.Wecanescapethesedifficultiesandretainausefuldefinitionofaccuracybyassumingthat,evenwhenwedonotknowthetruevalue,wecanrelyonthebestavailableacceptedvaluewithwhichtocompareourexperimentalvalue.For thegoldringexample, there isnoacceptedvaluewithwhich tocompare,andbothmeasuredvalueshavethesameprecision,sothereisnoreasontobelieveonemorethantheother.Wecouldlookuptheaccuracyspecificationsforeachbalanceas provided by the manufacturer, but the best way to assess the accuracy of ameasurementistocompareitwithaknownstandard.Forthissituation,itmaybepossible to calibrate the balanceswith a standardmass that is accuratewithin anarrow tolerance and is traceable to a primary mass standard at the NationalInstitute of Standards and Technology (NIST). Calibrating the balances shouldeliminatethediscrepancybetweenthereadingsandprovideamoreaccuratemassmeasurement.Precisionisoftenreportedquantitativelybyusingrelativeorfractionaluncertainty:

Relative Uncertainty = uncertaintymeasured quantity

(1)

Forexample,m=75.5±0.5ghasafractionaluncertaintyof: %7.0600.05.755.0

==gg



Accuracyisoftenreportedquantitativelybyusingrelativeerror:

Relative Error = measured value - expected valueexpected value

(2)

Iftheexpectedvalueformis80.0g,thentherelativeerroris:CriticalNotes:

• Theminus sign indicates that themeasuredvalue is less than theexpectedvalue–unlessexplicitlystated,theterm“relativeerror”doesnotinandofitselfrefertoamagnitude.

• The denominator is neither the measured value nor the average of themeasuredandexpectedvalue–therelativeerrorcanonlybecitedwhenthereisaknownexpectedvalueorgoldstandard.

%6.5056.00.800.805.75

−=−=−



TypesofUncertaintyMeasurement uncertainties may be classified as either random or systematic,depending on how themeasurement was obtained (an instrument could cause arandomuncertaintyinonesituationandasystematicuncertaintyinanother).Random uncertainties are statistical fluctuations (in either direction) in themeasureddata.Theseuncertaintiesmayhavetheirorigininthemeasuringdevice,orinthefundamentalphysicsunderlyingtheexperiment.Therandomuncertaintiesmaybemaskedbytheprecisionoraccuracyofthemeasurementdevice. Randomuncertainties can be evaluated through statistical analysis and can be reduced byaveraging over a large number of observations (see “standard error” later in thisdocument).Systematicuncertaintiesarereproducibleinaccuraciesthatareconsistentlyinthe“samedirection,”andcouldbecausedbyanartifactinthemeasuringinstrument,oraflawintheexperimentaldesign(becauseofthesepossibilities,itisnotuncommonto see the term “systematic error”). Theseuncertaintiesmaybedifficult to detectandcannotbeanalyzedstatistically.Ifasystematicuncertaintyorerrorisidentifiedwhen calibrating against a standard, applying a correction or correction factor tocompensate for the effect can reduce the bias. Unlike random uncertainties,systematicuncertaintiescannotbedetectedorreducedbyincreasingthenumberofobservations.When making careful measurements, the goal is to reduce as many sources ofuncertainty aspossible and tokeep trackof those that cannotbe eliminated. It isusefultoknowthetypesofuncertaintiesthatmayoccur,sothatwemayrecognizethem when they arise. Common sources of uncertainty in physics laboratoryexperimentsinclude:Incomplete definition (may be systematic or random) - One reason that it isimpossible to make exact measurements is that the measurement is not alwaysclearlydefined.Forexample,iftwodifferentpeoplemeasurethelengthofthesamestring, theywouldprobablygetdifferent resultsbecauseeachpersonmaystretchthestringwithadifferenttension.Thebestwaytominimizedefinitionuncertaintyis to carefully consider and specify the conditions that could affect themeasurement.Failuretoaccountforafactor(usuallysystematic)–Themostchallengingpartofdesigning an experiment is trying to control or account for all possible factorsexcept theone independentvariable that isbeinganalyzed.For instance,youmayinadvertently ignore air resistance when measuring free-fall acceleration, or youmayfailtoaccountfortheeffectoftheEarth’smagneticfieldwhenmeasuringthefieldnearasmallmagnet.Thebestwaytoaccountforthesesourcesofuncertaintyistobrainstormwithyourpeersaboutallthefactorsthatcouldpossiblyaffectyour



result.Thisbrainstormshouldbedonebeforebeginningtheexperimentinordertoplan and account for the confounding factors before taking data. Sometimes acorrectioncanbeappliedtoaresultaftertakingdatatoaccountforanuncertaintythatwasnotdetectedearlier.Environmental factors (systematic or random) - Be aware of uncertaintyintroducedbytheimmediateworkingenvironment.Youmayneedtotakeaccountoforprotectyourexperimentfromvibrations,drafts,changesintemperature,andelectronicnoiseorothereffectsfromnearbyapparatus.Instrumentresolution(random)-Allinstrumentshavefiniteprecisionthatlimitsthe ability to resolve small measurement differences. For instance, a meter stickcannotbeusedtodistinguishdistancestoaprecisionmuchbetterthanabouthalfofitssmallestscaledivision(typically0.5mm).Oneof thebestwaystoobtainmoreprecise measurements is to use a null differencemethod instead of measuring aquantitydirectly.Nullorbalancemethodsinvolveusinginstrumentationtomeasurethe difference between two similar quantities, one of which is known veryaccurately and is adjustable. The adjustable reference quantity is varied until thedifference is reduced to zero. The two quantities are then balanced, and themagnitude of the unknown quantity can be found by comparison with ameasurement standard. With this method, problems of source instability areeliminated,andthemeasuringinstrumentcanbeverysensitiveanddoesnotevenneedascale.Thistypeofmeasurementismoresophisticatedandwilltypicallynotbeusedintheintroductoryphysicscourses.Calibration (systematic) –Whenever possible, the calibration of an instrumentshouldbecheckedbeforetakingdata.Ifacalibrationstandardisnotavailable,theaccuracy of the instrument should be checked by comparing with anotherinstrumentthatisatleastasprecise,orbyconsultingthetechnicaldataprovidedbythemanufacturer. Calibrationerrorsareusually linear (measuredasa fractionofthefullscalereading),sothatlargervaluesresultingreaterabsoluteerrors.Zerooffset(systematic)-Whenmakingameasurementwithamicrometercaliper,electronicbalance,orelectricalmeter,alwayscheckthezeroreadingfirst.Re-zerothe instrument if possible, or at leastmeasure and record the zero offset so thatreadings can be corrected later. It is also a good idea to check the zero readingthroughouttheexperiment. Failuretozeroadevicewillresultinaconstantoffsetthatismoresignificantforsmallermeasuredvaluesthanforlargerones.Physicalvariations(random)-Itisalwayswisetoobtainmultiplemeasurementsover the widest range possible. Doing so often reveals variations that mightotherwisegoundetected.Thesevariationsmaycallforcloserexamination,ortheymaybecombinedtofindanaveragevalue.



Parallax (systematic or random) - This error can occur whenever there is somedistance between the measuring scale and the indicator used to obtain ameasurement. If the observer’s eye is not squarely alignedwith the pointer andscale,thereadingmaybetoohighorlow(someanalogmetershavemirrorstohelpwiththisalignment).Instrument drift (systematic) - Most electronic instruments have readings thatdriftovertime.Theamountofdriftisgenerallynotaconcern,butoccasionallythissourceofuncertaintycanbesignificant.Lag time andhysteresis (systematic) - Somemeasuring devices require time toreach equilibrium, and taking ameasurement before the instrument is stablewillresult in a measurement that is too high or low. A common example is takingtemperaturereadingswithathermometerthathasnotreachedthermalequilibriumwithitsenvironment.Asimilareffectishysteresis,whereintheinstrumentreadingslag behind and appear to have a “memory” effect, as data are taken sequentiallymoving up or down through a range of values. Hysteresis is most commonlyassociatedwithmaterialsthatbecomemagnetizedwhenachangingmagneticfieldisapplied.Last but not least, some uncertainties are the result of carelessness, poortechnique, or bias on the part of the experimenter. The experimentermay use ameasuringdeviceincorrectly,ormayusepoortechniqueintakingameasurement,ormayintroduceabiasintomeasurementsbyexpecting(andinadvertentlyforcing)the results toagreewith theexpectedoutcome.Grossuncertaintiesof thisnaturecanbereferredtoasmistakesorblunders,andshouldbeavoidedandcorrectedifdiscovered.Asarule,theseuncertaintiesareexcludedfromanyuncertaintyanalysisdiscussion because it is generally assumed that the experimental result wasobtainedbyfollowingcorrectandwell-intentionedprocedures–thereisnopointtoperforming an experiment and then reporting that it was known to be doneincorrectly. The term human error should be avoided in uncertainty analysisdiscussionsbecauseitistoogeneraltobeuseful.



EstimatingExperimentalUncertaintyforaSingleMeasurementAny measurement will have some uncertainty associated with it, no matter theprecisionofthemeasuringtool.Howisthisuncertaintydeterminedandreported?Theuncertaintyofasinglemeasurementislimitedbytheprecisionandaccuracyofthemeasuringinstrument,alongwithanyotherfactorsthatmightaffecttheabilityoftheexperimentertomakethemeasurement.For example, if you are trying to use ameter stick tomeasure the diameter of atennis ball, the uncertaintymight be±5mm, but if you use a Vernier caliper, theuncertaintycouldbereducedtomaybe±2mm.The limitingfactorwiththemeterstickisparallax,whilethesecondcaseislimitedbyambiguityinthedefinitionofthetennisball’sdiameter(it’sfuzzy!).Inbothofthesecases,theuncertaintyisgreaterthanthesmallestdivisionsmarkedonthemeasuringtool(likely1mmand0.05mmrespectively). Unfortunately, there is no general rule for determining theuncertaintyinallmeasurements.Theexperimenteristheonewhocanbestevaluateandquantifytheuncertaintyofameasurementbasedonallthepossiblefactorsthataffecttheresult.Therefore,thepersonmakingthemeasurementhastheobligationtomakethebestjudgmentpossibleandreporttheuncertaintyinawaythatclearlyexplainswhattheuncertaintyrepresents:Measurement=(measuredvalue±standarduncertainty)(unitofmeasurement)

where “± standard uncertainty” indicates approximately a 68% confidenceinterval(seesectionsonStandardDeviationandReportingUncertainties).

Example:Diameteroftennisball=6.7±0.2cm

EstimatingUncertaintyinRepeatedMeasurementsSupposeyoutimetheperiodofoscillationofapendulumusingadigitalinstrument(that you assume is measuring accurately) and find that T = 0.44 seconds. Thissingle measurement of the period suggests a precision of ±0.005 s, but thisinstrument precision may not give a complete sense of the uncertainty, and youshouldavoidreportingtheuncertainty inthis fashionifpossible. Ifyourepeatthemeasurementseveraltimesandexaminethevariationamongthemeasuredvalues,youcangetabetterideaoftheuncertaintyintheperiod.Forexample,herearetheresultsof5measurements,inseconds:0.46,0.44,0.45,0.44,0.41.Forthissituation,thebestestimateoftheperiodistheaverage,ormean:

N

xxx N+++=

... (mean) Average 21



Whenever possible, repeat ameasurement several times and average the results.Thisaverageisgenerallythebestestimateofthe“true”value(unlessthedatasetisskewedbyoneormoreoutlierswhichshouldbeexaminedtodetermineiftheyarebaddatapointsthatshouldbeomittedfromtheaverageorvalidmeasurementsthatrequire further investigation). Generally, the more repetitions you make of ameasurement,thebetterthisestimatewillbe,butbecarefultoavoidwastingtimetakingmoremeasurementsthanisnecessaryfortheprecisionrequired.Consider, as another example, themeasurement of thewidth of a piece of paperusingameterstick.Beingcarefultokeepthemeterstickparalleltotheedgeofthepaper (to avoid a systematic error which would cause themeasured value to beconsistentlyhigherthanthecorrectvalue),thewidthofthepaperismeasuredatanumberofpointsonthesheet,andthevaluesobtainedareenteredinadatatable.Notethatthelastdigit isonlyaroughestimate,sinceit isdifficulttoreadametersticktothenearesttenthofamillimeter(0.01cm)–weretainthelastdigitfornowtomakeapointlater.

Observation Width(cm)#1 31.33#2 31.15#3 31.26#4 31.02#5 31.20

Table1.FiveMeasurementsoftheWidthofaPieceofPaper

Average = sum of observed widthsnumber of observations

= 155.96 cm5

= 31.19 cm

Thisaverageisthebestavailableestimateofthewidthofthepieceofpaper,butitisnot exact. We would have to average an infinite number of measurements toapproachthetruemeanvalue,andeventhen,wearenotguaranteedthatthemeanvalueisaccuratebecausethereisstill likelysomesystematicuncertaintyfromthemeasuringtool,whichisdifficulttocalibrateperfectlyunlessitisthegoldstandard.Sohowdoweexpresstheuncertaintyinouraveragevalue?Oneway toexpress thevariationamong themeasurements is touse theaveragedeviation. Thisstatistictellsusonaverage(with50%confidence)howmuchtheindividualmeasurementsvaryfromthemean.

The average deviation would seem to be a sufficient measure of uncertainty;however, it is important to understand the distribution of measurements. The

Nxxxxxxd N ||...|||| Deviation, Average 21 −++−+−

=



Central Limit Theorem proves that as the number of independentmeasurementsincreases,andassumingthatthevariationsinthesemeasurementsarerandom(i.e.,there are no systematic uncertainties), the distribution of measurements willapproach the normal distribution,more commonly known as a bell curve. In thiscourse,wewillassumethatourmeasurements,performedinsufficientnumber,willproduceabellcurve(normal)distribution.Inthiscase,thestandarddeviationisthecorrectwaytocharacterizethespreadofthedata.Thestandarddeviationisalwaysslightlygreaterthantheaveragedeviation,andisusedbecauseofitsmathematicalassociationwiththenormaldistribution.



StandardDeviationTocalculatethestandarddeviationforasampleofNmeasurements: 1.SumallthemeasurementsanddividebyNtogettheaverage,ormean. 2.SubtractthisaveragefromeachoftheNmeasurementstoobtainN“deviations.” 3.SquareeachoftheNdeviationsandaddthemtogether. 4.Dividethisresultby(N–1)andtakethesquareroot.Toconvertthisintoaformula,lettheNmeasurementsbecalledx1,x2,…,xN.LettheaverageoftheNvaluesbecalled x .Theneachdeviationisgivenby

xxx ii −=δ ,fori=1,2,...,NThestandarddeviationisthen:

( )( ) ( )11

... 2222

21

−=

−

+++=

∑Nx

Nxxx

s iN δδδδ

Inthemeterstickandpaperexample,theaveragepaperwidth x is31.19cm.Thedeviationsare:

Observation Width(cm) Deviation(cm)#1 31.33 +0.14 =31.33-31.19#2 31.15 -0.04 =31.15-31.19#3 31.26 +0.07 =31.26-31.19#4 31.02 -0.17 =31.02-31.19#5 31.20 +0.01 =31.20-31.19

Table1(completed).FiveMeasurementsoftheWidthofaSheetofPaper

Theaveragedeviationis: d =0.09cmThestandarddeviationis:The significance of the standard deviation is this: if you now make one moremeasurement using the samemeter stick, you can reasonably expect (with about68%confidence)thatthenewmeasurementwillbewithin0.12cmoftheestimatedaverage of 31.19 cm. In fact, it is reasonable to use the standarddeviation as theuncertaintyassociatedwiththissinglenewmeasurement.However,theuncertainty

cm 12.015

)01.0()17.0()07.0()04.0()14.0( 22222

=−

++++=s



oftheaveragevalueisthestandarddeviationofthemean,whichisalwayslessthanthestandarddeviation(seenextsection).Consideranexampleof100measurementsofaquantity, forwhichtheaverageormean value is 10.50 and the standard deviation is s = 1.83. Figure 2 below is ahistogram of the 100 measurements, which shows how often a certain range ofvalueswasmeasured.Forexample,in20ofthemeasurements,thevaluewasintherange9.50to10.50,andmostofthereadingswereclosetothemeanvalueof10.50.Thestandarddeviations forthissetofmeasurementsisroughlyhowfarfromtheaveragevaluemost of the readings fell. Fora largeenoughsample, approximately68%ofthereadingswillbewithinonestandarddeviation(“1-sigma”)ofthemeanvalue,95%ofthereadingswillbeintheinterval x ±2s(“2-sigma”),andnearlyall(99.7%) of the readings will lie within 3 standard deviations (“3-sigma”) of themean.Thesmoothcurvesuperimposedonthehistogramisthenormaldistributionpredictedbytheoryformeasurementsinvolvingrandomerrors.Asmoreandmoremeasurementsaremade,thehistogramwillbetterapproximateabell-shapedcurve,butthestandarddeviationofthedistributionwillremainapproximatelythesame.

sx

→±← 1 sx

→±← 2 sx →±← 3 sx

Figure2.ANormalDistribution(BellCurve)Basedon100Measurements



StandardDeviationoftheMean(StandardError)When reporting the average valueofNmeasurements, theuncertainty associatedwith this average value is the standard deviation of the mean, often called thestandarderror(SE).

Standard Deviation of the Mean, or Standard Error (SE), σ x = sN (3)

Thestandarderrorissmallerthanthestandarddeviationbyafactorof N1 .Thisreflects the fact thatweexpect theuncertaintyof theaveragevaluetogetsmallerwhenweusealargernumberofmeasurements.Inthepreviousexample,wehavedividedthestandarddeviationof0.12by√5toget thestandarderrorof0.05cm.Thefinalresultshouldthenbereportedas“averagepaperwidth=31.19±0.05cm.”

WhentoUseStandardDeviationvsStandardErrorForrepeatedmeasurements,thesignificanceofthestandarddeviationsisthatyoucanreasonablyexpect(withabout68%confidence)thatthenextmeasurementwillbewithinsoftheestimatedaverage.Itisreasonabletousethestandarddeviationas the uncertainty associated with this measurement; however, as moremeasurementsaremade,thevalueofthestandarddeviationmayberefinedbutitwillnotsignificantlydecreaseasthenumberofmeasurementsisincreased.In contrast, if you are confident that the systematic uncertainty in yourmeasurementisverysmall,thenitisreasonabletoassumethatyourfinitesampleofallpossiblemeasurementsisnotbiasedawayfromthe“true”value.Inthiscase,theuncertaintyoftheaveragevaluecanbeexpressedasthestandarddeviationofthemean,whichisalwayslessthanthestandarddeviationbyafactorof√N.

Figure3.StandardDeviationvsStandardError



Ifyouarenotconfidentthatthesystematicuncertaintyinyourmeasurementisverysmall,theuncertaintythatshouldbereportedisthestandardcombineduncertainty(Uc)thatincludesallknownuncertaintyestimates(seesectiononCombiningandReportingUncertaintieslaterinthisdocument).

AnomalousDataThefirststepyoushouldtakeinanalyzingdata(andevenwhiletakingdata)istoexaminethedatasetasawholetolookforpatternsandoutliers.Anomalousdatapointsthatlieoutsidethegeneraltrendofthedatamaysuggestaninterestingphenomenonthatcouldleadtoanewdiscovery,ortheymaysimplybetheresultofamistakeorrandomfluctuations.Inanycase,anoutlierrequirescloserexaminationtodeterminethecauseoftheunexpectedresult.Extremedatashouldneverbe“thrownout”withoutclearjustificationandexplanation,becauseyoumaybediscardingthemostsignificantpartoftheinvestigation!However,ifyoucanclearlyjustifyomittinganinconsistentdatum,thenyoumayexcludetheoutlierfromyouranalysissothattheaveragevalueisnotskewedfromthe“true”mean.Thereareanumberofstatisticalmeasuresthathelpquantifythedecisiontodiscardoutliers,buttheyarebeyondthescopeofthisdocument.Beawareofthepossibilityofanomalousdata,andaddressthetopicasneededinthediscussionincludedwithalabreportorlabnotebook.

FractionalUncertaintyWhenareportedvalueisdeterminedbytakingtheaverageofasetofindependentreadings,thefractionaluncertaintyisgivenbytheratiooftheuncertaintydividedbytheaveragevalue.Forthisexample,

Fractional Uncertainty = uncertaintyaverage

= 0.05 cm31.19 cm

= 0.0016 ≈ 0.2%

Thefractionaluncertaintyisdimensionlessbutisoftenreportedasapercentageorin parts per million (ppm) to emphasize the fractional nature of the value. Ascientistmightalsomakethestatementthatthismeasurement“isgoodtoabout1part in 500” or “precise to about 0.2%”. The fractional uncertainty is importantbecause it is used inpropagating uncertainty in calculations using the result of ameasurement,asdiscussedinthenextsection.



BiasesandtheFactorofN–1Youmayfinditsurprisingthatthebestvalue(average)iscalculatedbynormalizing(dividing)byN,whereasthestandarddeviationiscalculatedbynormalizingtoN–1.ThereasonisbecausenormalizingtoNisknowntounderestimatethecorrectvalueofthewidthofanormaldistribution,unlessNislarge.Thisunderestimateisreferredtoasabiasandistheresultofincompletesampling(thatis,thepopulationofmeasurementsfallsshortoftheentirepopulationofmeasurementsthatcouldbetaken).Ifthenumberofsamplesislessthan10orso,eventheN–1term(knownasBessel’scorrection)canstillinduceabias.Determiningtheexactcorrectiontominimizeoreliminatebiasdependsonthedistributionofthedata,andthereisnosimpleexactequationthatcanbeapplied;however,forsmallsamplesizesthatarequitecommoninintroductoryphysicsclasses,acorrectionofN–1.5maybemoreappropriate.Ifyouuseacorrectionfactorof1.5inyourlabreports,youmustmakethisclearinyouranalysisandcitethisGuideasareference.



SignificantFigures Thenumberofsignificantfiguresinavaluecanbedefinedasallthedigitsbetweenand including the first non-zero digit from the left, through the last digit. Forinstance,0.44hastwosignificant figures,andthenumber66.770has5significantfigures.Zeroesaresignificantexceptwhenusedtolocatethedecimalpoint,asinthenumber 0.00030, which has 2 significant figures. Zeroes may or may not besignificantfornumberslike1200,whereit isnotclearwhethertwo,three,orfoursignificant figuresare indicated.Toavoid thisambiguity, suchnumbers shouldbeexpressed in scientific notation (e.g., 1.20×103 clearly indicates three significantfigures).Whenusingacalculator,thedisplaywilloftenshowmanydigits,onlysomeofwhichare meaningful (significant in a different sense). For example, if you want toestimate theareaof a circularplaying field, youmightpaceoff the radius tobe9metersandusetheformulaA=πr2.Whenyoucomputethisarea,thecalculatorwillreportavalueof254.4690049m2.Itwouldbeextremelymisleadingtoreportthisnumberastheareaofthefield,becauseitwouldsuggestthatyouknowtheareatoanabsurddegreeofprecision–towithinafractionofasquaremillimeter!Sincetheradiusisonlyknowntoonesignificantfigure, it isconsideredbestpracticetoalsoexpressthefinalanswertoonlyonesignificantfigure:Area=3×102m2.Fromthisexample,wecanseethatthenumberofsignificantfiguresreportedforavalue impliesacertaindegreeofprecisionandcansuggestaroughestimateoftherelativeuncertainty: 1significantfiguresuggestsarelativeuncertaintyofabout10%to100% 2significantfiguressuggestarelativeuncertaintyofabout1%to10% 3significantfiguressuggestarelativeuncertaintyofabout0.1%to1%To understand this connection more clearly, consider a value with 2 significantfigures, like 99, which suggests an uncertainty of ±1, or a relative uncertainty of±1/99=±1%(somemightarguethattheimplieduncertaintyin99is±0.5sincetherange of values that would round to 99 are 98.5 to 99.4; however, since theuncertaintyhereisonlyaroughestimate,thereisnotmuchpointarguingaboutthefactor of two.) The smallest 2-significant-figure number, 10, also suggests anuncertaintyof±1,whichinthiscaseisarelativeuncertaintyof±1/10=±10%.Therangesforothernumbersofsignificantfigurescanbereasonedinasimilarmanner.Warning: thisprocedure isopen toawiderangeof interpretation; therefore,oneshould use caution when using significant figures to imply uncertainty, and themethod should only be used if there is no other better way to determineuncertainty. An explicit warning to this effect should accompany the use of thismethod.



Subject to the above warning, significant figures can be used to find a possiblyappropriateprecisionforacalculatedresultforthefourmostbasicmathfunctions.

• Formultiplicationanddivision,thenumberofsignificantfiguresthatarereliablyknowninaproductorquotientisthesameasthesmallestnumberofsignificantfiguresinanyoftheoriginalnumbers.

Example: 6.6 (2significantfigures) ×7328.7 (5significantfigures) 48369.42=48x103 (2significantfigures)

• Foradditionandsubtraction,theresultshouldberoundedofftothelastdecimalplacereportedfortheleastprecisenumber.

Examples: 223.64 5560.5 +54 +0.008 278 5560.5Critical Note: if a calculated number is to be used in further calculations, it ismandatorytokeepguarddigitstoreduceroundingerrorsthatmayaccumulate.Thefinalanswercanthenberoundedaccordingtotheaboveguidelines.Thenumberofguarddigitsrequiredtomaintaintheintegrityofacalculationdependsonthetypeof calculation. For example, the number of guard digits must be larger whenperformingpowerlawcalculationsthanwhenadding.

Uncertainty,SignificantFigures,andRoundingFor the same reason that it is dishonest to report a result with more significantfiguresthanarereliablyknown, theuncertaintyvalueshouldalsonotbereportedwithexcessiveprecision.Forexample,itwouldbeunreasonabletoreportaresultinthefollowingway:

measureddensity=8.93±0.475328g/cm3 WRONG!Theuncertaintyinthemeasurementcannotpossiblybeknownsoprecisely!Inmostexperimentalwork, the confidence in theuncertainty estimate is notmuchbetterthanabout±50%becauseofallthevarioussourcesoferror,noneofwhichcanbeknown exactly. Therefore, uncertainty values should be stated to only onesignificant figure (orperhaps2 significant figures if the firstdigit is a1).Becauseexperimentaluncertaintiesareinherentlyimprecise,theyshouldberoundedtoone,oratmosttwo,significantfigures.



Tohelpgiveasenseoftheamountofconfidencethatcanbeplacedinthestandarddeviation as a measure of uncertainty, the following table indicates the relativeuncertaintyassociatedwith thestandarddeviation forvarioussamplesizes. Notethat inorder foranuncertaintyvaluetobereportedto3significant figures,morethan10,000readingswouldberequiredtojustifythisdegreeofprecision!

N RelativeUncertainty*

SignificantFiguresValid

ImpliedUncertainty

2 71% 1 ±10%to100%3 50% 1 ±10%to100%4 41% 1 ±10%to100%5 35% 1 ±10%to100%10 24% 1 ±10%to100%20 16% 1 ±10%to100%30 13% 1 ±10%to100%50 10% 2 ±1%to10%100 7% 2 ±1%to10%10000 0.7% 3 ±0.1%to1%

Table2.ValidSignificantFiguresinUncertainties

*Therelativeuncertaintyisgivenbytheapproximateformula:Whenanexplicituncertaintyestimateismade,theuncertaintytermindicateshowmany significant figures shouldbe reported in themeasured value (not the otherway around!). For example, the uncertainty in the densitymeasurement above isabout0.5g/cm3,whichsuggeststhatthedigit inthetenthsplaceisuncertain,andshould be the last one reported. The other digits in the hundredths place andbeyondareinsignificant,andshouldnotbereported:

measureddensity=8.9±0.5g/cm3 RIGHT!Anexperimentalvalueshouldberoundedtobeconsistentwiththemagnitudeofitsuncertainty. This generallymeans that the last significant figure in any reportedvalueshouldbeinthesamedecimalplaceastheuncertainty.Inmostinstances,thispracticeofroundinganexperimentalresulttobeconsistentwith the uncertainty estimate gives the samenumber of significant figures as therules discussed earlier for simple propagation of uncertainties for adding,subtracting,multiplying,anddividing.Caution: When conducting an experiment, it is important to keep in mind thatprecisionisexpensive(bothintermsoftimeandmaterialresources).Donotwaste

)1(21−

=Nσ

σσ



yourtimetryingtoobtainapreciseresultwhenonlyaroughestimateisrequired.The cost increases exponentially with the amount of precision required, so thepotentialbenefitofthisprecisionmustbeweighedagainsttheextracost.PracticalTipsforMeasuringUncertaintyI.“Stacking”Assumeyouareaskedtomeasurethemassofatypicalpenny(accordingtotheUSMint,currentlymadepennieshaveanominalmassof2.5grams)withascalewhoseaccuracy is known to be ±0.2 gram. Measuring one penny might yield ameasurementof2.4±0.2grams,andthiswouldbetheonlymeasurementpossibleforthatonepenny.Likewise,anotherpennymightyieldameasurementof2.5±0.2grams.Isthereawaytogetamoreprecisemeasurement?Inthiscase,yes,becauseyouareaskedtofindthemassofatypicalpenny.Bystackingpenniesandmeasuringmorethanoneofthematthesametime,dividingbythenumberofpenniesmeasuredcanprovideamorepreciseanswer.Forexample, assume thatyoumeasure5penniesseparatelywiththeseresults(allwithanaccuracyof±0.2g):2.4,2.4,2.5,2.4,2.6.The relative uncertainty of each measurement is about 8%. Further assume thatwhen youmeasure all five at the same time, the value is 12.3 ± 0.2 g, yielding arelativeuncertaintyofabout2%forthestack.Themeanvalueforatypicalpennyistherefore(retainingguarddigits)2.460g.Butwhatdoweassignastheuncertainty?Onemightarguethattheuncertaintyisstill0.2;however,theuncertaintycanalsobedividedby5,basedontheupper-lowerboundmethod,forwhichthesumoftheindividualmeasurementscanbeplausiblywrittenas:sum=(2.4±0.04)+(2.4±0.04)+(2.5±0.04)+(2.4±0.04)+(2.6±0.04)=12.3±0.2g

This is arguably the same as the stacked value of 12.3 ± 0.2 g. Therefore,we canreasonably divide both the stacked value and its uncertainty byN, and can thusreasonablyassertthevalueforatypicalpennyas2.46±0.04g.II.AlwaysMinimizeYourSigFigsThis “stacking” method can be used for any type of measurement that requirestypicalvalues tobe found fromrepeatedmeasurementsof similarobjectsor timeintervals;however,themethodisnotfoolproof.Supposeinsteadthatthescalehasaquotedaccuracythat ismuchbetterthan itsresolution(e.g.,accuracy=0.2%,andresolution=0.1g).Suchdevicesarenotdesignedtomeasuresmallvalues(the0.2%accuracyforvaluesontheorderof1gissmallerthantheresolution).Inthiscase,thesum,infullprecision,wouldbe:



sum=(2.4±0.005)+(2.4±0.005)+(2.5±0.005)+(2.4±0.005)+(2.6±0.005)=12.300±0.025g

Notethatthe0.025uncertaintyis4timeslessthantheresolution(incontrasttothepreviousexample,wherethe0.2uncertaintyistwicetheresolution).Itwouldthereforebeincorrecttoasserttheansweras2.460±0.005g,becausetheaccuracycitedfarexceedstheresolutionoftheinstrument.Thisresultiseasiertovisualizebylookingatactualdistributions.Supposethatyouhaveadevicethatreportsmeasurementsto4sigfigs.Considerthepreviousexampleofthemeterstickusedtomeasurethewidthofapieceofpaper,where5measurementswereusedtodeterminethatthewidthis31.19±0.12cm(Table1).Table2assertsthatonly1sigfiginthestandarddeviationisjustifiedfor5datapoints.Thus,thisresultshouldproperlybereportedas31.2±0.1cm,consistentwithourbeliefthatitisdifficulttoreadametersticktothenearesttenthofamillimeter.Histogramsofthemeasurementsto4,3,and2sigfigsareshownbelow.Theredandgreenarrowsrepresenttheaverageandstandarddeviation,respectively.



Question:whichchoiceofsigfigsprovidesthebestrepresentationofabellcurvewhoseheightisrepresentedbytheredarrowanditswidthisrepresentedbythegreenarrow?Themostlikelyanswerofcourseisnone:noneofthedistributionslooksmuchlikeabellcurve!Thereasonforthisisthatnotenoughdatapointsexisttocreateabellcurve.Withoutknowingthattheactualdistributionofmeasurementswouldlooklikeabellcurve,wecannotbesurethatthesedatadocreateanormaldistribution,forwhichtheconceptofastandarddeviationmakessense.Giventhislimitation,thecorrectsolutionistotakemanymoredatapoints;however,for4sigfigs,manymanydatapoints(likelythousandsortensofthousands)wouldberequiredtofilleverybininsuchawaythatabellcurvecouldbeapproximated.Likewise,withonly2sigfigs,it’sprobablethateverydatumwillreduceto“31”withnouncertainty.Thecompromiseinthiscaseis3sigfigs:werepresenttheuncertaintywithaminimumnumberofsigfigs.Withoutmoretimetotakedataandusemorepowerfulstatisticaltechniques,wewillinsteadchoosethesmallestnumberofsigfigstocitetheuncertainty.Inreturn,wewillassumethattheaverageandstandarddeviationarethemostreasonableapproachtorepresentingadistributionofdata.Anyassertionofanuncertaintybeyond1sigfigisonlyjustifiedforN=50orabove(Table1).Therefore,forthesecondstackedpennyresult,thesumwouldbeappropriatelyroundedto12.30±0.03g,yieldingafinalvalueforatypicalpennyof2.460±0.006g.Theonlyworkaroundtoabetterresultinthiscaseistostackmanymorepennies,untilthestackeduncertaintyislargeenoughtocompensateforthepoorresolution.



PropagationofUncertaintySupposewewanttodetermineaquantityf,whichdependsonxandmaybeseveralothervariablesy,z,etc.Wewanttoknowtheuncertaintyinfifwemeasurex,y,etc,withuncertaintiesσX,σY, etc.That is,wewant to findouthow theuncertainty inone set of variables (usually the independent variables) propagates to theuncertaintyinanothersetofvariables(usuallythedependentvariables).Therearetwoprimarymethodsofperformingthispropagationprocedure:

• upper-lowerbound• quadrature

Theupper-lowerboundmethodissimplerinconcept,buttendstooverestimatetheuncertainty,whilethequadraturemethodismoresophisticated(andcomplicated)butprovidesabetterstatisticalestimateoftheuncertainty.

TheUpper-LowerBoundMethodofUncertaintyPropagationThismethodusestheuncertaintyrangesofeachvariabletocalculatethemaximumand minimum values of the function. You can also think of this procedure asexaminingthebestandworstcasescenarios.Forexample,supposeyoumeasureanangletobeθ=25°±1°andyouneedtofindf=cosθ,then:

fmax=cos(26°)=0.8988 fmin=cos(24°)=0.9135Then,f=0.906±0.007(where0.007ishalfthedifferencebetweenfmaxandfmin)Notethateventhoughθwasonlymeasuredto2significantfigures,fisknownto3figures.Asanotherimportantexample,considerthedivisionoftwovariables.Acommonexampleisthecalculationofaveragespeed:

vavg =ΔxΔt

Let’ssayanexperimentdonerepeatedlymeasuresdistancetravelledof30±0.5mduring a time of 2 ± 0.1 sec. To find the upper and lower bound of vavg, theuncertaintiesmustbesettocreatethe“worstcasescenario”fortheuncertaintyinvavg:

vavg−max =30 + 0.5 m2 – 0.1 sec

= 16.05 m/s vavg−min =30 – 0.5 m2 + 0.1 sec

= 14.05 m/s



Thebest(expected)valuefortheaveragespeedis30/2=15.00m/s.Theupperboundis1.05m/shigherbutthelowerboundis0.95m/slower(differentfrom1.05).Thisisatypicaloutcomewhenusingtheupper-lowerboundmethod.Theuncertaintyshouldbeexpressedasthemostconservativevalue.Thus: vavg=15.00±1.05m/s PREFERRED!Notethatitisnotcorrecttotakethedifferencebetweentheupperandlowerboundanddividebytwo: vavg=15.05±1.00m/s NOTPREFERRED!Althoughthelastresultsatisfiessymmetrybetweenthebounds,itexplicitlycalculatesanincorrectvalueofthebest-knownexpectedvalueoftheaveragespeed.Manytimes,thedifferencebetweentheso-called“preferred“and“notpreferred”approachesisnotsignificantenoughtobeanissue.Forexample,ifitisappropriatetoroundtheuncertaintyintheabovevaluestoonesigfig,theansweris15±1m/s,regardlessoftheapproach.Nevertheless,youshouldbeawareofthispitfall.The upper-lower bound method is especially useful when the functionalrelationshipisnotclearorisincomplete.Onepracticalapplicationisforecastingtheexpected range in an expense budget. In this case, some expenses may be fixed,whileothersmaybeuncertain,andtherangeoftheseuncertaintermscouldbeusedtopredicttheupperandlowerboundsonthetotalexpense.

QuadratureThe quadrature method yields a standard uncertainty estimate (with a 68%confidence interval) and is especially useful and effective in the case of severalvariables that weight the uncertainty non-uniformly. Themethod is derivedwithseveralexamplesshownbelow.Forasingle-variablefunctionf(x),thedeviationinfcanberelatedtothedeviationinxusingcalculus:

xdxdff δδ ⎟⎠

⎞⎜⎝

⎛=

Takingthesquareandtheaverageyields:

22

2 xdxdff δδ ⎟⎠

⎞⎜⎝

⎛=



Usingthedefinitionofσyields:

xf dxdfσσ =

Examplesforpowerlaws: f=√x f=x2

dfdx

= 12 x

dfdx

= 2x

σ f =σ x

2 xor

σ f

f= 12σ x

x

σ f

f= 2σ x

x

Notethatbyjudiciouslynormalizing,itiseasytoexpresstherelative(fractional)uncertaintyinonevariablewithrespecttotherelative(fractional)uncertaintyinanother.Notealsothattheweightingisdirectlyrelatedtothepowerexponentofthefunction.Nowreconsiderthetrigexamplefromtheupper-lowerboundsection: f=cosθ

θ

θsin−=

ddf

θσθσ sin=f Notethatinthissituation,σθmustbeinradians.Forθ=25°±1°(0.727±0.017)

σf=|sinθ|σθ=(0.423)(π/180)=0.0074Thisisthesameresultasupper-lowerboundmethod.Thefractionaluncertaintyfollowsimmediatelyas:

θσθσ

tan=ff

Thedeeperpowerofthequadraturemethodisevidentinthecasewherefdependsontwoormorevariables;thederivationabovecanberepeatedwithminormodification.Fortwovariables,f(x,y):



yyfx

xff δδδ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠

⎞⎜⎝

⎛∂

∂=

Thepartialderivativexf∂

∂ meansdifferentiatingfwithrespecttoxholdingtheother

variables fixed. Taking the square and the average yields the generalized law ofpropagationofuncertaintybyquadrature:

(δ f )2 = ∂ f∂x

⎛⎝⎜

⎞⎠⎟2

(δ x)2 + ∂ f∂y

⎛⎝⎜

⎞⎠⎟

2

(δ y)2 + 2 ∂ f∂x

⎛⎝⎜

⎞⎠⎟

∂ f∂y

⎛⎝⎜

⎞⎠⎟δ xδ y (4)

Ifthemeasurementsofxandyareuncorrelated,then 0=yxδδ ,andthisreducestoitsmostcommonform:

σ f =∂ f∂x

⎛⎝⎜

⎞⎠⎟2

σ x2 + ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2

σ y2

AdditionandSubtractionExample: f=x±y

∂ f∂x

= 1, ∂ f∂y

= ±1 → σ f = σ x2 +σ y

2

Multiplicationexample: f=xy

∂ f∂x

= y, ∂ f∂y

= x → σ f = y2σ x2 + x2σ y

2

Dividingtheaboveequationbyf=xyyields:

Whenadding(orsubtracting)independentmeasurements,theabsoluteuncertaintyofthesum(ordifference)istherootsumofsquares(RSS)oftheindividualabsoluteuncertainties.Whenaddingcorrelatedmeasurements,theuncertaintyintheresultissimplythesumoftheabsoluteuncertainties,whichisalwaysalargeruncertaintyestimatethanaddinginquadrature(RSS).Addingorsubtractingaconstantdoesnotchangetheabsoluteuncertaintyofthecalculatedvalueaslongastheconstantisanexactvalue.



22

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=

yxfyxf σσσ

Divisionexample: f=x/y

∂ f∂x

= 1y

, ∂ f∂y

= − xy2 → σ f =

1y

⎛⎝⎜

⎞⎠⎟

2

σ x2 + x

y2

⎛⎝⎜

⎞⎠⎟

2

σ y2

Dividingthepreviousequationbyf=x/yyields:

22

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=

yxfyxf σσσ

Note that the relative (fractional) uncertainty in f has the same form formultiplication and division: the relative uncertainty in a product or quotientdependsontherelativeuncertaintyofeachindividualterm.Asanotherexample,considerpropagatingtheuncertaintyinthespeedv=at,wheretheaccelerationisa=9.8±0.1m/s2andthetimeist=1.2±0.1s.

( ) ( ) 3.1%or 031.0029.0010.02.11.0

8.91.0 22

2222

=+=⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛=tavtav σσσ

Notice that the relative uncertainty in t (2.9%) is significantly greater than therelative uncertainty for a (1.0%), and therefore the relative uncertainty in v isessentiallythesameasfort(about3%).Graphically,theRSSislikethePythagoreantheorem:Thetotaluncertaintyisthelengthofthehypotenuseofarighttrianglewithlegsthelengthofeachuncertaintycomponent.

1.0% 3.1%

2.9%

Whenmultiplying(ordividing)independentmeasurements,therelativeuncertaintyoftheproduct(quotient)istheRSSoftheindividualrelativeuncertainties.Whenmultiplyingcorrelatedmeasurements,theuncertaintyintheresultisjustthesumoftherelativeuncertainties,whichisalwaysalargeruncertaintyestimatethanaddinginquadrature(RSS).Multiplyingordividingbyaconstantdoesnotchangetherelativeuncertaintyofthecalculatedvalue.



Timesavingapproximation:“Achainisonlyasstrongasitsweakestlink.”Ifoneof theuncertainty terms ismorethan3 timesgreater thantheother terms,the root-squares formula canbe skipped, and the combineduncertainty is simplythe largest uncertainty. This shortcut can save a lot of time without losing anyaccuracyintheestimateoftheoveralluncertainty.Thequadraturemethodcanbegeneralizedtoallpowerlawsinthefollowingway:

f=xnym

σ f

f= n2 σ x

x⎛⎝⎜

⎞⎠⎟2

+m2 σ y

y⎛⎝⎜

⎞⎠⎟

2

TheproofofthisisshownintheAppendix.

Theuncertaintyestimatefromtheupper-lowerboundmethodisgenerallylargerthanthestandarduncertaintyestimatefoundfromthequadraturemethod,butbothmethodswillgiveareasonableestimateoftheuncertaintyinacalculatedvalue.Note:Onceyouhaveanunderstandingofthequadraturemethod,itisnotrequiredtoperformthepartialderivativeeverytimeyouarepresentedwithapropagationofuncertainty problem in any of the above forms! Instead, simply apply the correctformulafortherelativeuncertainties.



CombiningandReportingUncertaintiesIn1993, the InternationalStandardsOrganization (ISO)published the firstofficialworldwideGuidetotheExpressionofUncertaintyinMeasurement.Beforethistime,uncertainty estimates were evaluated and reported according to differentconventions depending on the context of the measurement or the scientificdiscipline.Hereareafewkeypointsfromthis100-pageguide,whichcanbefoundinmodifiedformontheNISTwebsite(seeReferences).Whenreportingameasurement,themeasuredvalueshouldbereportedalongwithanestimateofthetotalcombinedstandarduncertaintyUcofthevalue.Thetotaluncertainty is found by combining the uncertainty components based on the twotypesofuncertaintyanalysis:TypeAevaluationofstandarduncertainty–methodofevaluationofuncertaintybythestatisticalanalysisofaseriesofobservations.Thismethodprimarilyincludesrandomuncertainties.TypeBevaluationofstandarduncertainty–methodofevaluationofuncertaintybymeansother thanthestatisticalanalysisofseriesofobservations.Thismethodincludes systematic uncertainties and errors and any other factors that theexperimenterbelievesareimportant.The individual uncertainty components ui should be combined using the law ofpropagation of uncertainties, commonly called the “root-sum-of-squares” or “RSS”method.Whenthisisdone,thecombinedstandarduncertaintyshouldbeequivalentto the standard deviation of the result,making this uncertainty value correspondwith a 68% confidence interval. If a wider confidence interval is desired, theuncertaintycanbemultipliedbyacoveragefactor(usuallyk=2or3)toprovideanuncertaintyrangethatisbelievedtoincludethetruevaluewithaconfidenceof95%(fork=2)or99.7%(fork=3).Ifacoveragefactorisused,thereshouldbeaclearexplanationofitsmeaningsothereisnoconfusionforreadersinterpretingthesignificanceoftheuncertaintyvalue.You should be aware that the ± uncertainty notation might be used to indicatedifferentconfidenceintervals,dependingonthescientificdisciplineorcontext.Forexample,apublicopinionpollmayreportthattheresultshaveamarginoferrorof±3%,whichmeansthatreaderscanbe95%confident(not68%confident)thatthereportedresultsareaccuratewithin3percentagepoints.Similarly,amanufacturer’stoleranceratinggenerallyassumesa95%or99%levelofconfidence.



Conclusion:“Whendomeasurementsagreewitheachother?”Wenowhavetheresourcestoanswerthefundamentalscientificquestionthatwasaskedatthebeginningofthiserroranalysisdiscussion:“Doesmyresultagreewithatheoreticalpredictionorresultsfromotherexperiments?”Generally speaking, a measured result agrees with a theoretical prediction if theprediction lies within the range of experimental uncertainty. Similarly, if twomeasured values have standard uncertainty ranges that overlap, then themeasurements are said to be consistent (they agree). If the uncertainty ranges donotoverlap, then themeasurementsaresaid tobediscrepant (theydonotagree).However, you should recognize that these overlap criteria can give two oppositeanswers depending on the evaluation and confidence level of the uncertainty. Itwould be unethical to arbitrarily inflate the uncertainty range just to make ameasurementagreewithanexpectedvalue.Abetterprocedurewouldbetodiscussthe size of the difference between the measured and expected values within thecontextof theuncertainty,andtry todiscover thesourceof thediscrepancy if thedifferenceistrulysignificant.Example:A=1.2±0.4B=1.8±0.4These measurements agree within their uncertainties, despite the fact that thepercentdifferencebetweentheircentralvaluesis40%.Incontrast,iftheuncertaintyishalved(±0.2), thesesamemeasurementsdonotagree sincetheiruncertaintiesdonotoverlap:Furtherinvestigationwouldbeneededtodeterminethecauseforthediscrepancy.Perhapstheuncertaintieswereunderestimated,theremayhavebeenasystematicerror that was not considered, or there may be a true difference between thesevalues.

0 0.5 1 1.5 2 2.5

Measurements and their uncertainties

A

B

0 0.5 1 1.5 2 2.5

Measurements and their uncertainties

A

B



Analternativemethodfordeterminingagreementbetweenvaluesistocalculatethedifferencebetweenthevaluesdividedbytheircombinedstandarduncertainty.Thisratiogivesthenumberofstandarddeviationsseparatingthetwovalues.Ifthisratioislessthan1.0,thenitisreasonabletoconcludethatthevaluesagree.Iftheratioismore than2.0, then it is highlyunlikely (less thanabout5%probability) that thevaluesarethesame.

Examplefromabovewithu=0.4: 1.157.0

|8.12.1|=

− AandBlikelyagree

Examplefromabovewithu=0.2: 1.228.0

|8.12.1|=

− AandBlikelydonotagree



MakingGraphsWhengraphsarerequiredinlaboratoryexercises,youwillbeinstructedto“plotAvs.B”(whereAandBarevariables).Byconvention,A(thedependantvariable)shouldbeplottedalongtheverticalaxis(ordinate),andB(theindependentvariable)shouldbeplottedalongthehorizontalaxis(abscissa).Graphsthatareintendedtoprovidenumericalinformationshouldbedrawnonruledgraphpaper.Useasharppencil(notapen)todrawgraphs,sothatmistakescanbecorrectedeasily.Itisacceptabletouseacomputer(seetheExceltutorialbelow)toproducegraphs.Thefollowinggraphisatypicalexampleinwhichdistancevs.timeisplottedforafreelyfallingobject.Examinethisgraphandnotethefollowingimportantrulesforgraphing:

Figure3.PlotofDistancevsTime

Title.Everygraphshouldhaveatitlethatclearlystateswhichvariablesappearontheplot.Ifthegraphisnotattachedtoanotheridentifyingreport,writeyournameandthedateontheplotforconvenientreference.Axislabels.Eachcoordinateaxisofagraphshouldbelabeledwiththewordorsymbolforthevariableplottedalongthataxisandtheunits(inparentheses)inwhichthevariableisplotted.ChoiceofScale.Scalesshouldbechoseninsuchawaythatdataareeasytoplotandeasytoread.Oncoordinatepaper,every5thand/or10thlineshouldbeselectedas



majordivisionlinesthatrepresentadecimalmultipleof1,2,or5(e.g.,0,l,2,0.05,20,500,etc.)–otherchoices(e.g.,0.3)makeitdifficulttoplotandalsoreaddata.Scalesshouldbemadenofinerthanthesmallestincrementonthemeasuringinstrumentfromwhichdatawereobtained.Forexample,datafromameterstick(whichhaslmmgraduations)shouldbeplottedonascalenofinerthanldivision=lmm,becauseascalefinerthan1div/mmwouldprovidenoadditionalplottingaccuracy,sincethedatafromthemeterstickareonlyaccuratetoabout0.5mm.Frequentlythescalemustbeconsiderablycoarserthanthislimit,inordertofittheentireplotontoasinglesheetofgraphpaper.Intheillustrationabove,scaleshavebeenchosentogivethegrapharoughlysquareboundary;avoidchoicesofscalethatmaketheaxesverydifferentinlength.Notethatitisnotalwaysnecessarytoincludetheorigin(‘zero’)onagraphaxis;inmanycases,onlytheportionofthescalethatcoversthedataneedbeplotted.DataPoints.Enterdatapointsonagraphbyplacingasuitablesymbol(e.g.,asmalldotwithasmallcirclearoundthedot)atthecoordinatesofthepoint.Ifmorethanonesetofdataistobeshownonasinglegraph,useothersymbols(e.g.,∆)todistinguishthedatasets.Ifdrawingbyhand,adraftingtemplateisusefulforthispurpose.Curves.Drawasimplesmoothcurvethroughthedatapoints.Thecurvewillnotnecessarilypassthroughallthepoints,butshouldpassascloseaspossibletoeachpoint,withabouthalfthepointsoneachsideofthecurve;thiscurveisintendedtoguidetheeyealongthedatapointsandtoindicatethetrendofthedata.AFrenchcurveisusefulfordrawingcurvedlinesegments.Donotconnectthedatapointsbystraight-linesegmentsinadot-to-dotfashion.Thiscurvenowindicatestheaveragetrendofthedata,andanypredicted(interpolatedorextrapolated)valuesshouldbereadfromthiscurveratherthanrevertingbacktotheoriginaldatapoints.Straight-lineGraphs.Inmanyoftheexercisesinthiscourse,youwillbeaskedtolinearizeyourexperimentalresults(plotthedatainsuchawaythatthereisalinear,orstraight-linerelationshipbetweengraphedquantities).Inthesesituations,youwillbeaskedtofitastraightlinetothedatapointsandtodeterminetheslopeandy-interceptfromthegraph.Intheexamplegivenabove,itisexpectedthatthefallingobject’sdistancevarieswithtimeaccordingtod=½gt2.Itisdifficulttotellwhetherthedataplottedinthefirstgraphaboveagreeswiththisprediction;however,ifdvs.t2isplotted,astraightlineshouldbeobtainedwithslope=½gandy-intercept=0.



UsingExcelforDataAnalysisinPhysicsLabsStudentshaveanumberofsoftwareoptionsforanalyzinglabdataandgeneratinggraphswiththehelpofacomputer.Itisthestudent’sresponsibilitytoensurethatthecomputationalresultsarecorrectandconsistentwiththerequirementsstatedinthislabmanual.Anysuitablesoftwarecanbeusedtoperformtheseanalysesandgeneratetablesandplotsforlabreportsandassignments;however,sinceMicrosoftExceliswidelyavailableonallCCIlaptopsandinuniversitycomputerlabs,studentsareencouragedtousethisspreadsheetprogram.Inaddition,theremaybeassignmentsduringthesemesterthatspecificallyrequireanExcel(orplatform-equivalent)spreadsheettobesubmitted.

GettingStartedThistutorialwillleadyouthroughthestepstocreateagraphandperformlinearregressionanalysisusinganExcelspreadsheet.Thetechniquespresentedherecanbeusedtoanalyzevirtuallyanysetofdatayouwillencounterinyourphysicsstudio.Tobegin,openExcel.Ablankworksheetshouldappear.EnterthesampledataandcolumnheadingsshownbelowintocellsA1throughD6.Savethefiletoadiskortoyourpersonalfilespaceonthecampusnetwork.

Time(sec) Distance(m) Time± Distance±0.64 1.15 0.1 0.31.1 2.35 0.2 0.51.95 3.35 0.3 0.42.45 4.46 0.4 0.72.85 5.65 0.3 0.5

Notethattheuncertaintiesforthetimeanddistance(denoted±)havebeenincluded.Thesearenotnecessaryforabasicplot,butthestudiolabreportsandassignmentsrequireanuncertaintyanalysis,soyoushouldgetintothehabitofincludingthem.



CreatingandEditingaGraphNote:theseinstructionsmaynotbepreciselycorrectfordifferentversionsofExcelorondifferentplatforms;however,theseinstructionsshouldberoughlycorrectforallmodernversionsofExcelonallplatforms.Otherspreadsheetprogramsshouldhavesimilarfeatures.Wesuggestyouuseon-lineresourcesorconsultwithyourclassmatesoryourTAforspecificquestionsorissues.Youwillbecreatingagraphofthesedatawhosefinishedformlookslikethis:

Followthesestepstoaccomplishthis:1:Useyourmousetoselectallthecellsthatcontainthedatathatyouwanttograph(inthisexample,columnsAandB).Tographthesedata,selectChartonthetoolbar.2:ClickonScatterplotsandchooseXY(Scatter)orMarkedScatterwithnolines.Adefaultplotshouldappearinthespreadsheet,anditshouldbebothmoveableandresizable.3:UsingtheChartLayouttool,experimentwithsettingthetitle,axes,axistitles,gridlinesandlegends.Ataminimum,werequirethattheplotbetitledandthatthex-andy-axesaredescriptivelylabeledwithunits.Westronglysuggestthatallgridlinesandthelegendberemovedforclarity.

y=1.8885x-0.0035R²=0.9767

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

Distance(meters)

Time(sec)

SpeedfromDistancevsTimeData



Mostgraphfeaturescanbemodifiedbydouble-clickingonthefeatureyouwanttochange.Youcanalsoright-clickonafeaturetogetamenu.Trychangingthecoloroftheplotarea,thenumbersontheaxes,andtheappearanceofthedatapoints.Itisrecommendedthatyoualwaysformatthebackgroundareatowhiteusingthe“Automatic”option.

AddingErrorBarsRight-clickonadatumpointandchooseFormatDataSeries...andselectErrorBars.ClickontheappropriateErrorBartab(XorY)andchooseBothunderDisplayandCapfortheEndstyle.Fixedvaluesorpercentagescanbeset,forexample,butifyouhaveseparatecolumnsofuncertaintyvaluesforeachdatum,asshownabove,thenselectCustomtospecifythevalues.Inthesubsequentcustomerrorbarwindow,selectthepositiveerrorvaluefieldandthenclickanddraginthecorrespondingExcelcolumnofuncertainties.RepeatforthenegativevalueandclickOK.Yourcustomerrorbarswillthenbeapplied.Repeatfortheotheraxis.Notethatifyoucreateseparatecolumnsforthepositiveandnegativeerrorbars,theycanbesetindependently.Alsonotethaterrorbarsmaynotbevisibleiftheyaresmallerthanthesizeofthedatumpointontheplot.

AddingaTrendlineTheprimaryreasonforgraphingdataistoexaminethemathematicalrelationshipbetweenthetwovariablesplottedonthex-andy-axes.Toaddatrendlineanddisplayitscorrespondingequation,right-clickonanydatumpointandAddTrendline.Choosethegraphshapethatbestfitsyourdataandisconsistentwithyourtheoreticalprediction(usuallyLinear).Clickonthe"Options"tabandchecktheboxesfor"Displayequationonchart"and“DisplayR-squaredvalueonchart.”AgoodfitisindicatedbyanR2valuecloseto1.Caution:Whensearchingforamathematicalmodelthatexplainsyourdata,itisveryeasytousethetrendlinetooltoproducenonsense.Thistoolshouldbeusedtofindthesimplestmathematicalmodelthatexplainstherelationshipbetweenthetwovariablesyouaregraphing.Lookattheequationandshapeofthetrendlinecritically:

• Doesitmakesenseintermsofthephysicalprincipleyouareinvestigating?• Isthisthebestpossibleexplanationfortherelationshipbetweenthetwo

variables?Usethesimplestequationthatpassesthroughmostoftheerrorbarsonyourgraph.Youmayneedtotryacoupleoftrendlinesbeforeyougetthemostappropriateone.Toclearatrendline,right-clickonitsregressionlineandselectClear.



DeterminingtheUncertaintyinSlopeandY-interceptTheR2valueindicatesthequalityoftheleast-squaresfit,butthisvaluedoesnotgivetheerrorintheslopedirectly.Giventhebestfitliney=mx+b,withndatapoints,thestandarderror(uncertainty)intheslopemcanbedeterminedfromtheR2valuebyusingthefollowingformula:

21)/1( 2

−−

=nRmmσ

Likewise,theuncertaintyinthey-interceptbis:

nx

mb∑

=2

σσ

ThesevaluescanbecomputeddirectlyinExcelorbyusingacalculator.Forthissamplesetofdata,σm=0.1684m/s,andσb=0.333m.NotethatavalueofR2ofexactly1leadstoslopeandinterceptuncertaintiesofzero.CarefullyexamtheExcelR2value–althoughitmaydisplayasexactly1,itlikelyisnotexactly1.Ifyourvalueisindeedexactly1,itindicatesanerrorinhowyouhaveplottedyourdata.Theuncertaintyintheslopeandy-interceptcanalsobefoundbyusingtheLINESTfunctioninExcel.UsingthisfunctionissomewhattediousandisbestunderstoodfromtheHelpfeatureinExcel.

InterpretingtheResultsOncearegressionlinehasbeenfound,theequationmustbeinterpretedintermsofthecontextofthesituationbeinganalyzed.Thissampledatasetcamefromacartmovingalongatrack.Wecanseethatthecartwasmovingatnearlyaconstantspeedsincethedatapointstendtolieinastraightlineanddonotcurveupordown.Thespeedofthecartissimplytheslopeoftheregressionline,anditsuncertaintyisfoundfromtheequationabove:v=1.9±0.2m/s.(Note:Ifwehadplottedagraphoftimeversusdistance,thenthespeedwouldbetheinverseoftheslope:v=1/m)They-interceptgivesustheinitialpositionofthecart:x0=–0.0035±0.33m,whichisessentiallyzero.

FinalStep–CopyingDataandGraphsintoaWordDocumentCopyyourplotsanddatatablefromExceltoWord.Justselectthegraph(orcells)andusetheEditmenuorkeyboardshortcutstocopyandpaste.



UsingLINESTinExcelLINESTisanalternativelinearleastsquarefittingfunctioninExcel.TheresultsoftheLINESTanalysisarevirtuallyidenticaltothelineartrendlineanalysisdescribedabove;however,LINESTprovidesasingle-stepcalculationofboththeslopeandinterceptuncertainties,insteadofthemulti-stepproceduredescribedabove.

• Startwithatablefortimeandvelocity(right).

• TheLINESTfunctionreturnsseveraloutputs;toprepare,selecta2by5arraybelowthedata,asshown.

• Note,velocitywasmistakenlylabeledashavingunitsofm/secinthetabletotheright.

• UndertheInsertmenu,selectionFunction,thenStatistical,andfinallyLINESTasshownbelow(andhitOK).



Selectthey-valuesandx-valuesfromthetableandwrite‘TRUE’forthelasttwooptions(below).Again,pressOK.

TheLINESTfunctionisanarrayfunction;therefore,youmusttellExcelyouaredonewiththearray.Highlighttheformulaintheformulabarasshownbelow.PressCtrl+ShiftsimultaneouslywithEnter(Macusers,pressCommandandEnter)



Theresultisthatthearrayofblankcellsthatwasselectedpreviouslyisnowfilledwithdata(althoughthenewdataisn’tlabeled).Seetheimagebelowforthelabelsthatcanbeaddedafterthefact(e.g.,“Slope,”“R^2Value,”etc):

References:Baird,D.C.Experimentation:AnIntroductiontoMeasurementTheoryandExperimentDesign,3rd.ed.PrenticeHall:EnglewoodCliffs,1995.Bevington,PhillipandRobinson,D.DataReductionandErrorAnalysisforthePhysicalSciences,2nd.ed.McGraw-Hill:NewYork,1991.ISO.GuidetotheExpressionofUncertaintyinMeasurement.InternationalOrganizationforStandardization(ISO)andtheInternationalCommitteeonWeightsandMeasures(CIPM):Switzerland,1993.Lichten,William.DataandErrorAnalysis.,2nd.ed.PrenticeHall:UpperSaddleRiver,NJ,1999.NIST.EssentialsofExpressingMeasurementUncertainty.http://physics.nist.gov/cuu/Uncertainty/Taylor,John.AnIntroductiontoErrorAnalysis,2nd.ed.UniversityScienceBooks:Sausalito,1997.



Appendix:PropagationofUncertaintybyQuadratureConsideraquantityftobecalculatedbymultiplyingtwomeasuredquantitiesxandywhoseuncertaintiesareσxandσy,respectively.Thequestionishowtopropagatethoseuncertaintiestothecalculatedquantityf.Fromthechainruleofcalculus,thechangeinfduetochangesinxandyis:

δ f = ∂ f∂x

δ x + ∂ f∂y

δ y

Squaringandaveragingyields:

δ f( )2 = ∂ f∂x

⎛⎝⎜

⎞⎠⎟2

δ x( )2 + ∂ f∂y

⎛⎝⎜

⎞⎠⎟

2

δ y( )2 + 2 ∂ f∂x

∂ f∂y

δ xδ y

Foruncorrelatedmeasurements,δ xδ y iszero.Considertheaveragesquarechange

inquantitiestobetheuncertaintyineachofx,y,andf;thatis, δ f( )2 =σ f2 ,etc.Then:

σ f =∂ f∂x

⎛⎝⎜

⎞⎠⎟2

σ x2 + ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2

σ y2

To generalize it to arbitrary powers of x and y, consider the function f = xnym;substituting this into the last equation and dividing by f yields the relativeuncertainty:

σ f

f= ∂ f

∂x⎛⎝⎜

⎞⎠⎟2 σ x

2

xnym( )2+ ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2 σ y2

xnym( )2

Thepartialderivativesare ∂ f∂x

= nxn−1ym and ∂ f∂y

= mxnym−1 .Substitutingtheseyields:

σ f

f= nxn−1ym( )2 σ x

2

xnym( )2+ mxnym−1( )2 σ y

2

xnym( )2

Thisexpressionlookscomplicatedbutitsimplifiestosomethingrathersimple:

σ f

f= n2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+m2 σ y

y⎛

⎝⎜⎞

⎠⎟

2



Theresultisthattherelative(fractional)uncertaintyinfistherootofthesquaredsum(RSS)ofindividualuncertaintiesinxandy.Examplesinthebodyofthisdocumentinclude:

f=xy σ f

f= (1)2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+ (1)2σ y

y⎛

⎝⎜⎞

⎠⎟

2

f=x/y σ f

f= (1)2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+ (−1)2σ y

y⎛

⎝⎜⎞

⎠⎟

2

f=xy2 σ f

f= σ x

x⎛⎝⎜

⎞⎠⎟

2

+ 4σ y

y⎛

⎝⎜⎞

⎠⎟

2

Note that the result formultiplication and division is the same (division is just apower lawwith anegative exponent).Alsonote that variables that appearwith ahigherpowerareweightedmoreheavilyinthepropagation.For some functions, especiallynon-linear trig functions, youmayhave toperformthe derivatives to find how the uncertainty propagates; however, for manyfunctions,performingthederivativeseachtime isnotrequired–merelyapply theequationhighlightedinyellow.

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