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Measurements & Uncertainty Analysis

1 Department of Physics and Astronomy

Measurement and Uncertainty Analysis Guide

“Itisbettertoberoughlyrightthanpreciselywrong.”–AlanGreenspan

Table of Contents THEUNCERTAINTYOFMEASUREMENTS...................................................................................................................2RELATIVE(FRACTIONAL)UNCERTAINTY..................................................................................................................4RELATIVEERROR.............................................................................................................................................................5TYPESOFUNCERTAINTY................................................................................................................................................6ESTIMATINGEXPERIMENTALUNCERTAINTYFORASINGLEMEASUREMENT..................................................9ESTIMATINGUNCERTAINTYINREPEATEDMEASUREMENTS...............................................................................9STANDARDDEVIATION................................................................................................................................................12STANDARDDEVIATIONOFTHEMEAN(STANDARDERROR).............................................................................14WHENTOUSESTANDARDDEVIATIONVSSTANDARDERROR..........................................................................14ANOMALOUSDATA.......................................................................................................................................................15BIASESANDTHEFACTOROFN–1.............................................................................................................................15SIGNIFICANTFIGURES..................................................................................................................................................16UNCERTAINTY,SIGNIFICANTFIGURES,ANDROUNDING.....................................................................................17PROPAGATIONOFUNCERTAINTY..............................................................................................................................20THEUPPER-LOWERBOUNDMETHODOFUNCERTAINTYPROPAGATION.....................................................20QUADRATURE.................................................................................................................................................................21COMBININGANDREPORTINGUNCERTAINTIES.....................................................................................................26CONCLUSION:“WHENDOMEASUREMENTSAGREEWITHEACHOTHER?”......................................................27MAKINGGRAPHS...........................................................................................................................................................29USINGEXCELFORDATAANALYSISINPHYSICSLABS..........................................................................................31GETTINGSTARTED........................................................................................................................................................31CREATINGANDEDITINGAGRAPH............................................................................................................................32ADDINGERRORBARS...................................................................................................................................................33ADDINGATRENDLINE..................................................................................................................................................33DETERMININGTHEUNCERTAINTYINSLOPEANDY-INTERCEPT.....................................................................34INTERPRETINGTHERESULTS.....................................................................................................................................34FINALSTEP–COPYINGDATAANDGRAPHSINTOAWORDDOCUMENT........................................................34USINGLINESTINEXCEL............................................................................................................................................35APPENDIXI:PROPAGATIONOFUNCERTAINTYBYQUADRATURE....................................................................39APPENDIXII.PRACTICALTIPSANDEXAMPLESFORMEASURINGANDCITINGUNCERTAINTY................41

lastupdatedJuly25,2019

ifyourcopydatesfromtheSP19semester,itisnotthelatestversion!


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 whenmaking multiple 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 is accurate, 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, youcouldreport themassasm=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 feelconfident thatyouknowthemassof thisringto thenearesthundredthofagram,buthowdoyouknowthatthetruevaluedefinitelyliesbetween17.43gand17.45g? Since you want to be honest, you decide to use another balance that gives areadingof17.22g.Thisvalueisclearlybelowtherangeofvaluesfoundonthefirstbalance,andundernormalcircumstances,youmightnotcare,butyouwanttobe



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 faryouranswer is from that “correct” answer.These conceptsaredirectlyrelatedtorandomandsystematicmeasurementuncertainties(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 toas the “gold standard.” Sometimeswehavea“textbook”measuredvalue,whichiswellknown,andweassumethatthisisour“ideal”value,anduse it toestimatetheaccuracyofourresult.Othertimesweknowatheoreticalvalue,whichiscalculatedfrombasicprinciples,andthisalsomaybetakenasan“ideal”value.Butphysicsisanempiricalscience,whichmeansthatthetheorymustbevalidatedbyexperiment,andnottheotherwayaround.Wecanescapethesedifficultiesandretainausefuldefinitionofaccuracybyassumingthat,evenwhenwedonotknowthetruevalue,wecanrelyonthebestavailableacceptedvaluewithwhichtocompareourexperimentalvalue.Forthegoldringexample, there isnoacceptedvaluewithwhichtocompare,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.

Relative(Fractional)UncertaintyPrecisionisoftenreportedquantitativelybyusingrelativeorfractionaluncertainty,givenbytheratiooftheuncertaintydividedbytheaveragevalue:

(1)Relative Uncertainty = uncertaintymeasured quantity



The relativeuncertainty isdimensionlessbut isoftenreportedasapercentageor inparts permillion (ppm) to emphasize the fractional nature of the value. A scientistmightalsomakethestatementthatthismeasurement“isgoodtoabout1partin500”or“precisetoabout0.2%”.Therelativeuncertaintyisimportantbecauseitisusedinpropagating uncertainty in calculations using the result of a measurement, asdiscussedinalatersection.Forexample,m=75.5±0.5ghasarelativeuncertaintyof

RelativeErrorAccuracyisoftenreportedquantitativelybyusingrelativeerror:

(2)

Iftheexpectedvalueformis80.0g,thentherelativeerroris:CriticalNotes:

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

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

• Arelativeerrorof100%means that theupperboundcouldbeasmuchastwicethevalue.Arelativeerrorof200%meansthattheupperboundcouldbeasmuchastriplethevalue,andsoon.Donotmisinterpreta200%errorastwicethevalue.

%7.0600.05.755.0

==gg

Relative Error = measured value - expected valueexpected value

%6.5056.00.800.805.75

-=-=-



TypesofUncertaintyMeasurement uncertainties may be classified as either random or systematic,depending on how the measurement was obtained (an instrument could cause arandomuncertaintyinonesituationandasystematicuncertaintyinanother).Random uncertainties are statistical fluctuations (in either direction) in themeasureddata.Theseuncertaintiesmayhavetheirorigininthemeasuringdevice,orinthefundamentalphysicsunderlyingtheexperiment.Therandomuncertaintiesmaybemaskedbytheprecisionoraccuracyof themeasurementdevice. 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”).Theseuncertaintiesmay be difficult 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 ofuncertaintyaspossibleand tokeep trackof those that cannotbeeliminated. 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 ameasurementbefore 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 experimenter may use ameasuringdeviceincorrectly,ormayusepoortechniqueintakingameasurement,ormayintroduceabiasintomeasurementsbyexpecting(andinadvertentlyforcing)the results toagreewith theexpectedoutcome.Grossuncertaintiesof thisnaturecanbereferredtoasmistakesorblunders,andshouldbeavoidedandcorrected ifdiscovered.Asarule,theseuncertaintiesareexcludedfromanyuncertaintyanalysisdiscussion because it is generally assumed that the experimental result wasobtainedbyfollowingcorrectandwell-intentionedprocedures–thereisnopointtoperforminganexperimentandthenreportingthatitwasknowntobeincorrect.Thetermhumanerrorshouldbeavoidedinuncertaintyanalysisdiscussionsbecauseitistoogeneraltobeuseful.



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 limiting factorwiththemeterstickisparallax,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 in thisfashion ifpossible. Ifyourepeat themeasurementseveraltimesandexaminethevariationamongthemeasuredvalues,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 errorwhichwould cause themeasured value to beconsistentlyhigherthanthecorrectvalue),thewidthofthepaperismeasuredatanumberofpointsonthesheet,andthevaluesobtainedareenteredinadatatable.Notethatthelastdigitisonlyaroughestimate,sinceitisdifficulttoreadametersticktothenearesttenthofamillimeter(0.01cm)–weretainthelastdigitfornowtomakeapointlater.

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

Table1.FiveMeasurementsoftheWidthofaPieceofPaper

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 is a sufficient measure of uncertainty; however, it isimportant to understand the distribution of measurements. The Central Limit

Average = sum of observed widthsnumber of observations

= 155.96 cm5

= 31.19 cm

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

=



Theoremprovesthat as thenumberof independentmeasurements increases, andassumingthat thevariations in thesemeasurementsarerandom(i.e., therearenosystematic uncertainties), the distribution of measurements will approach thenormaldistribution,morecommonlyknownasabellcurve. In thiscourse,wewillassumethatourmeasurements,performedinsufficientnumber,willproduceabellcurve(normal)distribution.Inthiscase,thestandarddeviationisanalternatewayto characterize the spread of the data. The standard deviation is always slightlygreater than the average deviation, and is used because of its mathematicalassociationwiththenormaldistribution.



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

,fori=1,2,...,NThestandarddeviationisthen:

Inthemeterstickandpaperexample,theaveragepaperwidth 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: =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

x

xxx ii -=d

( )( ) ( )11

... 2222

21

-=

-+++

= åNx

Nxxx

s iN dddd

x

d

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 ±2s(“2-sigma”),andnearlyall(99.7%) of the readings will lie within 3 standard deviations (“3-sigma”) of themean.Thesmoothcurvesuperimposedonthehistogramisthenormaldistributionpredictedbytheoryformeasurementsinvolvingrandomerrors.Asmoreandmoremeasurementsaremade,thehistogramwillbetterapproximateabell-shapedcurve,butthestandarddeviationofthedistributionwillremainapproximatelythesame.

Figure2.ANormalDistribution(BellCurve)Basedon100Measurements

x

sx

®±¬ 1 sx®±¬ 2 sx®±¬ 3 sx



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

(3)

Thestandarderrorissmallerthanthestandarddeviationbyafactorof .Thisreflects the fact thatweexpect theuncertaintyof theaveragevaluetogetsmallerwhen we use a larger number of measurements. In the previous example, wedividedthestandarddeviationof0.12byÖ5toget thestandarderrorof0.05cm.Thefinalresultshouldthenbereportedas“averagepaperwidth=31.19±0.05cm.”Forthisexample,therelativeuncertaintywouldbe(0.05/31.19),or≈0.2%.

WhentoUseStandardDeviationvsStandardErrorForrepeatedmeasurements,thesignificanceofthestandarddeviationsisthatyoucanreasonablyexpect(withabout68%confidence)thatthenextmeasurementwillbe within s of the estimated average. It may be reasonable to use the standarddeviationas theuncertaintyassociatedwith thismeasurement;however, asmoremeasurementsaremade, thevalueof thestandarddeviationwillberefinedbut itwill not significantly decrease as the number of measurements is increased;therefore, it will not be the best estimate of the uncertainty of a set ofmeasurements. In contrast, if youare confident that the systematicuncertainty inyourmeasurement is very small, then it is reasonable to assume that your finitesample of all possiblemeasurements is not biased away from the “true” value. Inthis case, the uncertainty of the average value can be expressed as the standarddeviationofthemean,whichisalwayslessthanthestandarddeviationbyafactorofÖN.

Figure3.StandardDeviationvsStandardError

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

N1



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.

BiasesandtheFactorofN–1Youmayfinditsurprisingthatthebestvalue(average)iscalculatedbynormalizing(dividing)byN,whereasthestandarddeviationiscalculatedbynormalizingtoN–1.ThereasonisbecausenormalizingtoNisknowntounderestimatethecorrectvalueofthewidthofanormaldistribution,unlessNislarge.Thisunderestimateisreferredtoasabiasandistheresultofincompletesampling(thatis,thepopulationofmeasurements,orsample,fallsshortoftheentirepopulationofmeasurementsthatcouldbetaken),alsoknownas“samplingerror.”Ifthenumberofsamplesislessthanorabout10,eventheN–1term(knownasBessel’scorrection)isknowntoinduceabias.Determiningtheexactcorrectiontominimizeoreliminatebiasdependsonthedistributionofthedata,andthereisnosimpleexactequationthatcanbeapplied;however,forsmallsamplesizesthatarequitecommoninintroductoryphysicsclasses,acorrectionofN–1.5maybemoreappropriate.Ifyouuseacorrectionfactorof1.5inyourlabreports,youmustmakethisclearinyouranalysisandcitethisGuideasareference.



SignificantFigures Thenumberofsignificantfigures(sigfigs)inavaluecanbedefinedasallthedigitsbetweenand includingthe firstnon-zerodigit fromthe left, throughthe lastdigit.Forinstance,0.44hastwosigfigs,andthenumber66.770has5sigfigs.Zeroesaresignificantexceptwhenusedtolocatethedecimalpoint,asinthenumber0.00030,whichhas2 sig figs.Zeroesmayormaynotbe significant fornumbers like1200,whereitisnotclearwhethertwo,three,orfoursigfigsareindicated.Toavoidthisambiguity,suchnumbersshouldbeexpressed inscientificnotation(e.g.,1.20×103indicates3sigfigs).Whenusingacalculator,thedisplaywilloftenshowmanydigits,onlysomeofwhichare meaningful (significant in a different sense). For example, if you want toestimate theareaof a circularplaying field, youmightpaceoff the radius tobe9metersandusetheformulaA=pr2.Whenyoucomputethisarea,thecalculatorwillreportavalueof254.4690049m2.Itwouldbeextremelymisleadingtoreportthisnumberastheareaofthefield,becauseitwouldsuggestthatyouknowtheareatoanabsurddegreeofprecision–towithinafractionofasquaremillimeter!Sincetheradiusisonlyknowntoonesignificantfigure,itisconsideredbestpracticetoalsoexpressthefinalanswertoonlyonesignificantfigure:Area=3´102m2.Fromthisexample,wecanseethatthenumberofsignificantfiguresreportedforavalueimpliesacertaindegreeofprecisionandcansuggestaroughestimateoftherelativeuncertainty: 1significantfiguresuggestsarelativeuncertaintyofabout10%to100% 2significantfiguressuggestarelativeuncertaintyofabout1%to10% 3significantfiguressuggestarelativeuncertaintyofabout0.1%to1%To understand this connection more clearly, consider a value with 2 significantfigures, like99,whichsuggestsanuncertaintyof±1,ora fractionaluncertaintyof±1/99=±1%(onemightarguethattheimplieduncertaintyin99is±0.5sincetherange of values that would round to 99 are 98.5 to 99.4; however, since theuncertainty here is only a rough estimate, there is not much point debating thefactor of two.) The smallest 2-significant-figure number, 10, also suggests anuncertainty of±1,which in this case is a fractionaluncertainty of±1/10=±10%.The ranges for other numbers of significant figures can be reasoned in a similarmanner.Warning: thisprocedure isopen toawide rangeof 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 withmore significantfiguresthanarereliablyknown, theuncertaintyvalueshouldalsonotbereportedwithexcessiveprecision.Forexample,itwouldbeunreasonabletoreportaresultinthefollowingway:

measureddensity=8.93±0.475328g/cm3 WRONG!Theuncertaintyinthemeasurementcannotpossiblybeknownsoprecisely!Inmostexperimentalwork, the confidence in theuncertainty estimate is notmuch betterthanabout±50%becauseofallthevarioussourcesoferror,noneofwhichcanbeknownexactly.

Therefore,unlessexplicitlyjustified,uncertaintyvaluesshouldbestated(rounded)toonesignificantfigure.



Tohelpgiveasenseoftheamountofconfidencethatcanbeplacedinthestandarddeviation as a measure of uncertainty, the following table indicates the relativeuncertainty associatedwith the standard deviation for various sample sizes.Notethat in order for a standard deviation to be reported to a precision of 3 significantfigures,morethan10,000readingswouldberequired!

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

*Therelativeuncertaintyinthestandarddeviationisgivenbytheapproximateformula:When an explicit uncertainty estimate is made using the standard deviation, theuncertaintytermindicateshowmanysignificant figuresshouldbereported inthemeasured value (not the otherway around!). For example, the uncertainty in thedensitymeasurementaboveisabout0.5g/cm3,whichsuggeststhatthedigitinthetenthsplaceisuncertain,andshouldbethelastonereported.Theotherdigitsinthehundredthsplaceandbeyondareinsignificant,andshouldnotbereported:

measureddensity=8.9±0.5g/cm3 RIGHT!Anexperimentalvalueshouldberoundedtobeconsistentwiththemagnitudeofitsuncertainty. This generally means that the last significant figure in any reportedvalue should be in the same decimal place as the uncertainty, and unless a largenumberofmeasurementshavebeen taken, theuncertainty shouldbe reported toonlyonesigfig.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.

)1(21-

=Ns

ss



When conducting an experiment, it is important to keep inmind thatprecision isexpensive (both in termsof timeandmaterial resources).Donotwasteyour timetrying toobtainaprecise resultwhenonlya roughestimate isrequired.The costincreases exponentially with the amount of precision required, so the potentialbenefitofthisprecisionmustbeweighedagainsttheextracost.Important Notes about sig figs when interpreting both this guide and coursematerials:a) Onexams,homeworks,orsolutions,youmayseeaquantitycitedimprecisely

(forexample,“1200m”insteadof“1200.0m”).Thisistypicallydoneforreadability,anddoesnotimplyanythingaboutuncertaintyunlessthequantityistiedexplicitlytoanexperiment.Youmayconsidersuchvaluestobeexact.

b) Incontrast,youmayseeexamplesorsolutionsthatappeartoretainanexcessive

numberofsigfigs(includinginthisdocument).Suchexamplesmayhaveskippedthefinalroundingstepforeitherclarityorreadability.Often,theseexamplesretainhigherprecisiontohelpyouascertainthatyouhaveduplicatedacalculationcorrectly,ortodistinguishthemfromincorrectsolutionsthathappentobeclosetothecorrectanswer.Again,unlesssuchexamplesaretiedexplicitlytoanuncertainty,youmayignoresuchinconsistencies;however,whenyoupresentanansweronanassignmentorexam,youshouldminimizeyouruncertaintyprecisiontoonesigfigunlessotherwisedirectedorjustified.

c) Similarly,ifanexperimentdescribedonanexamorhomeworkclaimstohave

morethanonesigfigintheuncertainty,andyouarenotaskedtoaddressthisaspectoftheexperiment,taketheuncertaintyprecisionatfacevalueascorrect.

d) Normally,whenreviewingexperimentalquestions(suchasarefoundonthe

practicumorlabreports),graderscarefullyscrutinizeyouruseofmorethan1sigfigwhencitinguncertainty.Oneexceptionhasalreadybeendiscussed,andisembodiedinTable2:ifenoughsampleshavebeencollectedtojustify2sigfigs,thenthismustbeexplicitlydiscussedinyouranswers.Asecondexceptioncouldbewhentheleadingsignificantfigureisa“1”–itispossibletoinduceabiaswhenroundingtoonesigfiginthiscase.Considertheexampleofrounding±1.4asopposedto±9.4,toonesigfig.Thevalueof±1.0representsa40%discrepancyfrom±1.4(verylarge),whereasthevalueof±9.0representsa4%discrepancyfrom±9.4(muchsmaller,albeitnonzero).Thebestapproachhere,asforvirtuallyallexamandassignmentexamples,istociteanddiscussthehighprecisionanswer,whichindicatesthatyouknowthedetailsofthecalculation,andthenciteanddiscussyourfinalanswerwithsuitableroundingandjustification.

Practicaltipsformeasuringandcitinguncertainty,anduncertaintycalculationexamples,canbefoundinAppendixII.



PropagationofUncertaintySupposewewanttodetermineaquantityf,whichdependsonxandmaybeseveralothervariablesy,z,etc.Wewanttoknowtheuncertaintyinfifwemeasurex,y,etc,withuncertaintiessX,sY, 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.Asanexample,considerthedivisionoftwovariables–acommonexampleisthecalculationofaveragespeed:

Let’ssayanexperimentisdonerepeatedlyandmeasuresadistancetravelledof30±0.5mduringatimeof2±0.1sec.Tofindtheupperandlowerboundofvavg,theuncertaintiesmustbesettocreatethe“worstcasescenario”fortheuncertaintyinvavg:

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!Notethatitisnotcorrecttotakethedifferenceanddividebytwo: vavg=15.05±1.00m/s NOTPREFERRED!

vavg =ΔxΔt

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



Althoughthelastresultsatisfiessymmetrybetweenthebounds,itexplicitlycalculatesanincorrectvalueofthebest-knownexpectedvalueoftheaveragespeed.Manytimes,thedifferencebetweentheso-called“preferred“and“notpreferred”approachesisnotsignificantenoughtobeanissue.Citingtheuncertaintyinthisexampleto3sigfigswouldrequireastrongjustification.Theacceptedapproach(andtheonethatshouldbefollowedintheabsenceofsuchjustification)istoroundto15.0±1.1m/s,andthenroundagainto15±1m/s.Withthisapproach,argumentsaboutwhetherthe“actualanswer”is15.00or15.05areirrelevant,becausetheexperimentdoesnotjustifysuchprecision.This asymmetry arises in the case of non-linear functions as well. For example,supposeyoumeasureanangletobeq=25°±1°andyouneedtofindf=cosq:fmin=cos(26°)=0.8988 f=cos(25°)=0.9063 fmax=cos(24°)=0.9135

Thedifferencesaref–fmin=0.0075andfmax–f=0.0072.Whenroundingtheuncertaintyto1sigfig,themostconservativevalueshouldbechosen(0.008).Thustheanswerwouldbecitedasf=0.906±0.008.Notethefollowingaboutthisexample:

• themaximumqisassociatedwiththeminimumf,andviceversa• guarddigitsareretainedtomakeinformedchoicesaboutthefinalanswer• althoughqwasonlymeasuredto2sigfigs,fisknownto3sigfigs

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 thatweight the uncertainty non-uniformly. Themethod is derivedwithseveralexamplesshownbelow.Forasingle-variablefunctionf(x),thedeviationinfcanberelatedtothedeviationinxusingcalculus:

xdxdff dd ÷øö

çèæ=



Takingthesquareandtheaverageyields:

Usingthedefinitionofsyields:

Examplesforpowerlaws: f=Öx f=x2

or

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

Notethatinthissituation,sqmustbeinradians.Forq=25°±1°(0.727±0.017)

sf=|sinq|sq=(0.423)(p/180)=0.0074Thisisessentiallythesameresultasupper-lowerboundmethod.Thefractionaluncertaintyfollowsimmediatelyas:

22

2 xdxdff dd ÷øö

çèæ=

xf dxdfss =

dfdx

= 12 x

dfdx

= 2x

σ f =σ x

2 xσ f

f= 12σ x

xσ f

f= 2σ x

x

qq

sin-=ddf

qsqs sin=f

qsqs

tan=ff



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

Thepartialderivative meansdifferentiating fwith respect tox,whileholding

theothervariablesfixed.Takingthesquareandtheaverageyieldsthegeneralizedlawofpropagationofuncertaintybyquadrature:

(4)

Ifthemeasurementsofxandyareuncorrelated,then ,andthisreducestoitsmostcommonform:

AdditionandSubtractionExample: f=x±y

yyfx

xff ddd ÷÷

ø

öççè

æ¶¶

+÷øö

çèæ¶¶

=

xf¶¶

(δ f )2 = ∂ f∂x

⎛⎝⎜

⎞⎠⎟2

(δ x)2 + ∂ f∂y

⎛⎝⎜

⎞⎠⎟

2

(δ y)2 + 2 ∂ f∂x

⎛⎝⎜

⎞⎠⎟

∂ f∂y

⎛⎝⎜

⎞⎠⎟δ xδ y

0=yxdd

σ f =∂ f∂x

⎛⎝⎜

⎞⎠⎟2

σ x2 + ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2

σ y2

∂ f∂x

= 1, ∂ f∂y

= ±1 → σ f = σ x2 +σ y

2

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



Multiplicationexample: f=xy

Dividingtheaboveequationbyf=xyyields:

Divisionexample: f=x/y

Dividingthepreviousequationbyf=x/yyields:

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.

∂ f∂x

= y, ∂ f∂y

= x → σ f = y2σ x2 + x2σ y

2

22

÷÷ø

öççè

æ+÷

ø

öçè

æ=

yxfyxf sss

∂ f∂x

= 1y

, ∂ f∂y

= − xy2 → σ f =

1y

⎛⎝⎜

⎞⎠⎟

2

σ x2 + x

y2

⎛⎝⎜

⎞⎠⎟

2

σ y2

22

÷÷ø

öççè

æ+÷

øö

çèæ=

yxfyxf sss

( ) ( ) 3.1%or 031.0029.0010.02.11.0

8.91.0 22

2222

=+=÷øö

çèæ+÷

øö

çèæ=÷

øö

çèæ+÷

øö

çèæ=

tavtav sss

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



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.Timesavingapproximation:“Achainisonlyasstrongasitsweakestlink.”Ifoneof theuncertaintyterms ismorethan3timesgreater thantheotherterms,the root-squares formula can be skipped, and the combineduncertainty is simplythe largest uncertainty. This shortcut can save a lot of time without losing anyaccuracyintheestimateoftheoveralluncertainty.Thequadraturemethodcanbegeneralizedtoallpowerlawsinthefollowingway:

f=xnym

(5)

TheproofofthisisshowninAppendixI.

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

σ f

f= n2 σ x

x⎛⎝⎜

⎞⎠⎟2

+m2 σ y

y⎛⎝⎜

⎞⎠⎟

2

1.0% 3.1%

2.9%



CombiningandReportingUncertaintiesIn1993, the International StandardsOrganization (ISO)publishedthe 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.Thismethodisprimarilyassociatedwithuncertaintiesthatarerandomlydistributed.TypeBevaluationofstandarduncertainty–methodofevaluationofuncertaintybymeansotherthanthestatisticalanalysisofaseriesofobservations.Thismethodis primarily associated with uncertainties that are systematic, including theprecisionofthemeasuringinstrumentandanyotherfactorsthattheexperimenterbelievesareimportant.The individual uncertainty components ui should be combined using the law ofpropagation of uncertainties, commonly called the “root-sum-of-squares” or “RSS”method(verysimilartoquadrature).Thiscombinedstandarduncertaintyshouldbeequivalent to the standard deviation of the result, making this uncertainty valuecorrespondwitha68%confidenceinterval.Ifawiderconfidenceintervalisdesired,the uncertainty can be multiplied by a coverage factor (usually k = 2 or 3) toprovide an uncertainty range that is believed to include the true value with aconfidenceof95%(fork=2)or99.7%(fork=3).Ifacoveragefactorisused,thereshould be a clear explanation of itsmeaning so there is no confusion for readersinterpretingthesignificanceoftheuncertaintyvalue.Be aware that the ± uncertainty notation might be used to indicate differentconfidenceintervals,dependingonthescientificdisciplineorcontext.Forexample,apublicopinionpollmay report that the resultshaveamarginoferror of±3%,which means that readers can be 95% confident (not 68% confident) that thereportedresultsareaccuratewithin3percentagepoints.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, thenthemeasurementsaresaidtobediscrepant (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,andtrytodiscoverthesourceof 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), thesesamemeasurementsdonotagreesincetheiruncertaintiesdonotoverlap:If twomeasurements (with similar uncertainties, each represented by ±1 sigma)barely overlap, then they differ by approximately 2-sigma, whichmeans there isabout a 5% chance that the measurements agree. When two values differ by 3-sigmaormore, it ishighlyunlikely(less than1%chance) that theyagree,andwe

0 0.5 1 1.5 2 2.5

Measurements and their uncertainties

AB

0 0.5 1 1.5 2 2.5

Measurements and their uncertainties

A

B



would concludewith confidence that there is adiscrepancy.Further investigationwould be needed to determine the cause for the discrepancy. Perhaps theuncertainties were underestimated, there may have been a systematic error thatwasnotconsidered,ortheremaybeatruedifferencebetweenthesevalues.Analternativemethodfordeterminingagreementbetweenvaluesistocalculatethedifferencebetweenthevaluesdividedbytheircombinedstandarduncertainty.Thisratiogivesthenumberofstandarddeviationsseparatingthetwovalues.Ifthisratioisapproximately1orsmaller,thenitisreasonabletoconcludethatthevaluesagree.If the ratio is on the order of 2, ormore, then it is unlikely (less than about 5%probability)thatthevaluesarethesame.

Examplefromabovewithu=0.4: AandBlikelyagree

Examplefromabovewithu=0.2: AandBlikelydonotagree

Notethefollowingexampleofoverlap:

Thesetwomeasurementsagree,despitetheirlargedifferenceinerrorbarsandthefactthatthe“red”measurementisoutsideofthe“green”errorbars.Thereisnorequirementthateithermeasurementbeincludedintheothermeasurement’serrorbars–onlythattheerrorbarsofthetwomeasurementsoverlap.

1.157.0

|8.12.1|=

-

1.228.0

|8.12.1|=

-



MakingGraphsWhengraphsarerequiredinlaboratoryexercises,youwillbeinstructedto“plotAvs.B”(whereAandBarevariables).Byconvention,A(thedependentvariable)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.Theexperimentisameasurementoftimesandpositionsofacartmovingatconstantspeed.Tobegin,openExcel.Ablankworksheetshouldappear.Enterthesampledatafromtheexperiment(listedinthetablebelow)andcolumnheadingsshownbelowintocellsA1throughD6.Savethefiletoadiskortoyourpersonalfilespaceonthecampusnetwork.

Time (sec) Position (m) Time ± Position ± 0.64 1.15 0.05 0.20 1.10 2.35 0.07 0.30 1.95 3.35 0.05 0.20 2.45 4.46 0.06 0.40 2.85 5.65 0.10 0.40

Notethattheuncertaintiesforthetimeandposition(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:

(6)

Likewise,theuncertaintyinthey-interceptbis:

(7)

ThesevaluescanbecomputeddirectlyinExcelorbyusingacalculator.Forthissamplesetofdata,sm=0.1684m/s,andsb=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.

21)/1( 2

--

=nRmms

nx

mbå=

2

ss



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,pressCommand-EnterorShift-Control-EnterdependingonyourversionofExcel)



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.



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

Squaringandaveragingyields:

Foruncorrelatedmeasurements, iszero.Considertheaveragesquarechange

inquantitiestobetheuncertaintyineachofx,y,andf;thatis, ,etc.Then:

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:

Thepartialderivativesare and .Substitutingtheseyields:

Thisexpressionlookscomplicatedbutitsimplifiestosomethingrathersimple:

δ f = ∂ f∂x

δ x + ∂ f∂y

δ y

δ f( )2 = ∂ f∂x

⎛⎝⎜

⎞⎠⎟2

δ x( )2 + ∂ f∂y

⎛⎝⎜

⎞⎠⎟

2

δ y( )2 + 2 ∂ f∂x

∂ f∂y

δ xδ y

δ xδ y

δ f( )2 =σ f2

σ f =∂ f∂x

⎛⎝⎜

⎞⎠⎟2

σ x2 + ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2

σ y2

σ f

f= ∂ f

∂x⎛⎝⎜

⎞⎠⎟2 σ x

2

xnym( )2+ ∂ f

∂y⎛⎝⎜

⎞⎠⎟

2 σ y2

xnym( )2

∂ f∂x

= nxn−1ym ∂ f∂y

= mxnym−1

σ f

f= nxn−1ym( )2 σ x

2

xnym( )2+ mxnym−1( )2 σ y

2

xnym( )2

σ f

f= n2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+m2 σ y

y⎛

⎝⎜⎞

⎠⎟

2



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

f=xy

f=x/y

f=xy2

Note that the result formultiplication and division is the same (division is just apower lawwithanegativeexponent).Alsonote thatvariables that appearwithahigherpowerareweightedmoreheavilyinthepropagation.For some functions, especiallynon-linear trig functions, youmayhave toperformthe derivatives to find how the uncertainty propagates; however, for manyfunctions,performingthederivativeseachtime isnotrequired–merelyapplytheequationhighlightedinyellowabove.

σ f

f= (1)2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+ (1)2σ y

y⎛

⎝⎜⎞

⎠⎟

2

σ f

f= (1)2 σ x

x⎛⎝⎜

⎞⎠⎟

2

+ (−1)2σ y

y⎛

⎝⎜⎞

⎠⎟

2

σ f

f= σ x

x⎛⎝⎜

⎞⎠⎟

2

+ 4σ y

y⎛

⎝⎜⎞

⎠⎟

2



AppendixII.PracticalTipsandExamplesforMeasuringandCitingUncertaintyI.InstrumentPrecisionandAccuracyInstrumentprecisionisthesmallestreadinganinstrumentcanprovide.Forsomeinstruments,thisprecisionisunambiguous;forexample,adigitalbalancewhosereadoutisdisplayedas__._ghasaninstrumentprecisionof0.1g.Incontrast,ameterstick’sinstrumentprecisionisopentointerpretation.Ifthefinestmarkonthemeterstickis1mm,itcouldbeclaimedthattheinstrumentprecisionis0.001m,or1mm;however,adifferentobservermightclaimthatitispossibletointerpolatebetweenthefinestmarks,andthereforemightclaimthattheinstrumentprecisionis0.0005m,or0.5mm.Inallcases,thechoiceofinstrumentprecisionandthereasoningshouldbeincludedinanyreportthatincorporatesthatinstrument’suse.Incontrasttotheinstrumentprecision,theinstrumentaccuracyreferstotheabilitytocorrectlymeasurethevalueofaknownstandard.Ifadigitalbalancewithameasurementprecisionof0.1gisusedtomeasureaweightindependentlyknowntohaveamassofexactly500.0g,andtheresultis495.3g,theaccuracyisnobetterthanabout5g,eventhoughtheprecisionis0.1g.Severalstrategiesforaddressingthisdiscrepancyarelistedbelow:a)Citethevalueas495.3±4.7g.Unacceptable:citingtheuncertaintyto2sigfigsisnotwarrantedforlessthan50measurements(Table2).b)Citethevalueas495.3±5g.Unacceptable:citingthemeasurementtoahigherprecisionthantheuncertaintyisnotwarrantedwithoutsignificantjustification.c)Citethevalueas495±5g.Acceptablebutnotoptimal:themeasurementisclearlyincorrectbecauseitisknownthattheweightisexactly500g.Ifonlyasinglemeasurementispossible,citingasshownisallowed,butshouldbeaccompaniedbyadiscussionabouttheknowndiscrepancyandtheintroductionofasystematicerror.d)Citethevalueas495.3±0.1g.Allowedbutonlyifaknowncalibrationstandardorotherinformationabouttheaccuracyisunavailable:thecalibrationstatusoftheinstrumentshouldbeconsulted.Ifthisisnotpossible,itisnotunreasonabletoassumethattheinstrumenthasanimpliedaccuracythatmatchesitsprecision,asshowninthisexample.Itemd)isthemostcommonscenarioinPhysics118activities.Anysuchassumptionaboutsubstitutingprecisionforaccuracyshouldbeincludedinadiscussion,withanacknowledgmentthatasystematicerrormayhavebeenintroduced.Therecommendedproceduretoaddressthisissueistoacquiremultiple“identical”samplesofthe“known”weight,andcomputeanaveragevalueandstandarddeviation.Thiswouldresolvethequestionofwhethercitingtheprecisionasanuncertaintyiswarranted;however,itwouldnotresolvetheaccuracyquestion.The



bestapproachtotheaccuracyquestionisacalculationofthestandarderror,butitisextremelyimportanttoremoveanysystematicuncertainties–anymeasurementsperformedwithoutknowncalibrationstandardsavailablearesusceptibletosystematiceffects,andshouldbecitedwiththecombined(TypeAandTypeB)standarduncertainty.II.StackingExample1Assumeyouareaskedtomeasurethemassofatypicalpenny(accordingtotheUSMint,currentlymadepennieshaveanominalmassof2.5g)withascalewhoseaccuracyisknowntobe±0.2gbutreadstoaprecisionof0.1g.Onecanimmediatelydiscardthechoiceof0.1gastheaccuracy.Measuringonepennymightyieldameasurementof2.4±0.2g,andthiswouldbetheonlymeasurementpossibleforthatonepenny.Likewise,anotherpennymightyieldameasurementof2.5±0.2g.Isthereawaytogetamoreprecisemeasurement?Inthiscase,yes,becauseyouareaskedtofindthemassofatypicalpenny.Bystackingpenniesandmeasuringmorethanoneofthematthesametime,dividingbythenumberofpenniesmeasuredcanprovideamorepreciseanswer.Forexample,assumethatyoumeasurethemassvalueoffivepenniesseparately,withtheseresults(allwithanaccuracyof±0.2g):2.4,2.4,2.5,2.4,2.6.Therelativeuncertaintyofeachmeasurementisabout8%.Furtherassumethatwhenyoumeasureallfiveatthesametime,thevalueis12.3±0.2g,yieldingarelativeuncertaintyofabout2%forthestack.Themeanvalueforatypicalpennyistherefore(retainingguarddigits)2.460g.Butwhatdoweassignastheuncertainty?Onemightarguethattheuncertaintyisstill0.2g;however,wecanplausiblyrewritethesumas:sum=(2.4±0.04)+(2.4±0.04)+(2.5±0.04)+(2.4±0.04)+(2.6±0.04)=12.3±0.2gThisisdemonstrablythesameasthestackedvalueof12.3±0.2g,andsuggeststhat,basedontheupper-lowerboundmethod,theuncertaintycanalsobedividedbyfive.Therefore,asageneralrule,wecanreasonablydivideboththestackedvalueanditsuncertaintybyN,andassertthevalueforatypicalpennyas2.46±0.04g.NotethatasperTable2,theuncertaintyiscitedtoonlyonesigfig,becauseonlyfivemeasurementsweremade.Assumeinsteadthattheaccuracyofthescaleisunknown;aworkingpropositionisthatthescale’smechanismwillcorrectlyreportthatanobjectof0.1gissomewherebetween0.05gand0.15g.Thereadingof12.3gforall5penniestogethercouldthereforehavevaluesfallingbetween12.25and12.35g.Dividingthisby5yieldsanominalvalueof2.460gwithahighandlowof2.470gand2.450g,respectively.Thevalueof2.460±0.010gcouldthenbecitedas2.46±0.01gtoonesigfig;however,althoughtheresultsseems“better,”thereisnoconfidencethatthisvalueiscorrect,Inmeasurementcasesweretheaccuracyofthemeasuringdeviceisnotknown,theresultshouldexplicitlycitethisdiscrepancy.



III.StackingExample2Asanotherexample,assumethatastudentisaskedtomeasuretheperiodofapendulumanddiscussanysystematiceffectsthatcouldcausethemeasurementtobeinaccurate.Thestudentresponds:“Iheldthependulumbobstillatanangleof45°,andstartedastopwatchatthemomentIletgo.Iwaitedforthependulumtoswingbackandthenstoppedthestopwatchatthemomentthebobreachedthestartingposition.Thereadingonthestopwatchisthebestmeasurementoftheperiodofthependulumofthislength;however,frictionisasystematiceffectthatcouldcausethemeasurementtobeinaccurate.”Critiquethisresponse.

Thismeasurementcanbeimprovedgreatlybystacking.Inthisexample,“stacking”meanstoallowthependulumtoswingformultipleperiods,anddividebythenumberofperiods.Despitethisimprovement,multipleissuesremainwithsuchameasurement.Inparticular,theuseoftheterm“friction”istoovaguetobeinstructive.Intheintroductoryphysicscourse,“friction”isreservedtomeaneithera)twosurfacesincontactandslidingagainsteachother(kinematicfriction)orb)theresistancebetweentwosurfacesincontacttobeginsliding(staticfriction).Neitheroftheseapplytothependulumbob.Instead,thebobmaybesloweddownbydrag,whichisamoreappropriatetermthan“friction”whenfluids(e.g.air)arepresent–thephysicsofdragandthephysicsofkinematicslidingfrictionarequitedifferent;therefore,makingthedistinctionisasuperioranswerthatwillberewardedwithcredit.Inthecaseofdrag,stackingmayresultinaworseanswerfortheperiod,becauseasmoreswingsarecounted,drageffectsmultiply.Finally,wenotethatthesimplependulumisanon-lineardevice–theharmonicoscillatorperiodforasimplependulumappliesonlyforsmallangles(goodtoabout1%atanglesoflessthan10°).Foranglessuchas45°,theremaybepooragreementbetweenthemeasuredandtheoreticalperiodsunlessnon-linearityistakenintoaccount.



IV.ApproachtoUncertaintyWhenAccuracyisCitedasaPercentage.Instrumentaccuracymaybedescribedintermsofapercentage,suchas“±0.1%.”Forhighqualityinstruments,thisusuallymeans0.1%ofthefull-scale(FS)reading,acrosstheentiremeasurementrangeoftheinstrument.Forexample,ifadigitalbalancecanmeasureamaximumvalueof199.9g,theaccuracywouldbe0.1%ofFS(200g),or±2g(worsethantheinstrumentprecisionis0.1g).Inthiscase,followtheguidelinesinExampleI.However,theoriginofthepercentageuncertaintymaybeambiguous,andcouldimplyapercentageofthecurrentreading.Iftheinterpretationisambiguous,itcanbeevaluatedbycomparingtotheprecisionoftheinstrument.Asanexample,assumeitisclaimedthattheaccuracyofthebalanceis0.1%foranymass,andthatthemeasurementofoneobjectreads53.4g.Thepercentaccuracyof0.1%impliesandaccuracyof±0.05g,whichisbetterthantheinstrumentprecisionof0.1g!Thisinconsistencysuggeststhata“stackofone”itemisinappropriate.Stackingshouldbeused,butitcannotbeappliedarbitrarily;astackoftwowouldnotunambiguouslyresolvethequestion,butastackofthreewould.Thegeneralapproachinthiscaseistostackenoughobjectssothatthepercentageaccuracyexceedstheprecisionofthemeasurement.TheapproachinExampleIcanthenbeused;however,underallcircumstances,anyassertionofanuncertaintybeyond1sigfigisonlyjustifiedforN=50orabove(Table2).V.UseofContainersAssumeyouaregivenaquantityofwaterandyouareaskedtomeasureitsweightusingascale.Youneedacontainer,butthecontaineritselfhasaweight.Theprocedureistoweighthecontainerbyitself,andthenthecontainerandthewatertogether,andsubtracttogettheweightofthewater.However,inthiscase,theuncertaintiesofthetwomeasurementsmustbeadded.Thisshouldbeself-evidentfromanupper-lowerboundperspective;thequadratureanalysisisamoreformalstatementofthesameresult(seeQuadraturesection).



VI.ExplainingDiscrepanciesAsimplependulumisknowntohaveaperiodofoscillationT =1.55s.StudentAusesadigitalstopwatchtomeasurethetotaltimefor5oscillationsandcalculatesanaverageperiodT =1.25s.StudentBusesananalogwristwatchandthesameproceduretocalculateanaverageperiodforthe5oscillationsandfindsT =1.6s.

• Whichstudentmadethemoreaccuratemeasurement?ThemeasurementmadebyStudentBisclosertotheknownvalueandisthereforemoreaccurate.

• Whichmeasurementismoreprecise?ThemeasurementmadebyStudentAisreportedwithmoredigitsandisthereforemoreprecise.

• Whatisthemostlikelysourceofthediscrepancybetweentheresults?Althoughthedifferenceinperiodmeasurementsisonly0.3s,theoriginaltimingmeasurementsmayhavedifferedby5timesthisamountsincewearetoldthattheaverageperiodwascalculatedfromthetotaltimefor5oscillations.Soeventhoughreactiontime(typically~0.2s)isalikelysourceofsystematicerror,thiswouldnotexplainthediscrepancyofapproximately1.5s,oraboutoneperiod.Therefore,themostlikelysourceofthediscrepancyisthatStudentAmistakenlymeasuredonly4oscillationsinsteadof5.Thisexampleshowsthatameasurementwithgreaterprecisionisnotalwaysmoreaccurate.

VII.PropagatingUncertaintyandReportingValuesAstudentusesaprotractortomeasureanangleofθ=85° ±1°.Whatshouldshereportforsinθ?

sin(θ)=0.996±0.002.Theuncertaintycanbedeterminedfromeithertheupper/lowerboundmethodorpropagationofuncertaintyusingpartialderivatives.Bothmethodsyieldthesameresultwhenroundedtoonesignificantfigure.Notethatitwouldbeinappropriatetoroundthisfurtherto1.00±0.00,becauseitimpliesanuncertaintyofzero.



VIII.OutliersAgroupofstudentsaretoldtouseametersticktofindthelengthofahallway.Theytake6independentmeasurementsasfollows:3.314m,3.225m,3.332m,3.875m,3.374m,3.285m.Howshouldtheyreporttheirfindings,andwhy?

Nodatashouldbeexcludedwhenpresentingtherawnumbers.Therefore,allvaluesshouldappearinanydatatables,andallvaluesshouldappearonplots.Anexampleisshownbelowforthesedata:

Themeasurementof3.875misanoutlier(10standarddeviationsfromtheothervalues)andismostlikelyamis-measurement(see2ndbulletpointbelow).Inthiscase,thebestestimatewouldbetheaverageandstandarderrorforthe5“good”values(3.31±0.02m).Notetwoaspectsofthisapproach:

• Ifasystematicerrorissuspected,thenusingthestandarderrormayalsobesuspect.Thestandarddeviationis0.06m;therefore,amoreconservativeanswerwouldbe3.3±0.1m.

• Becarefulbeforethrowingoutdata–NobelPrizeshavebeenwonbecauseofoutliers(althoughprobablynotin118)!Alsothinkabouttheexperimentcarefully:inthisexample,thereislikelytobeamisusedmeterstick,butinthecaseofgoldprospecting,onlythehigheroutliersarekept,andeverythingELSEisthrownout!

IX.LimitingFactors AVerniercaliper(aninstrumentthatcanroutinelymeasurelengthsto0.001inch,orlessthan0.05mm),isusedtomeasuretheradiusofatennisball.ThevalueispresentedasR=3.2±0.1cm(halfofD=6.4±0.2cm).Whyisthisresultnotmoreprecise?

Theuncertaintyofthismeasurementisdeterminednotbytheresolutionofthecaliper,butbytheimprecisedefinitionoftheball'sdiameter(it’sfuzzy!).



X.UncertaintyPropagationIEquipmentisusedtomeasuretheaccelerationofaglideronaninclinedairtrackasaccuratelyaspossible.Thedistancebetweentwophotogatesismeasuredveryaccuratelyasd=1.500±0.005m.Theexperimentisrun5timeswiththegliderstartingfromrestatthefirstphotogate,andthephotogatetimesarerecordedast(insec)=1.12,1.06,1.19,1.15,1.08.Whatisthebestestimateoftheaccelerationoftheglider?

Theaveragetimeis1.12swithastandarderrorof0.02s.Therelationshipbetweenthedistanceandaccelerationisd=½at2;thebestestimatefortheaccelerationisthereforea=2.392m/s2(retainingguarddigits).Thequadratureresultfortheuncertaintyintheaccelerationis

Therelativeuncertaintyinthetimeismorethan5xtherelativeuncertaintyintheposition.Inaddition,thequadraticdependenceontimecreatesanadditionalfactorof4,sotheuncertaintyintimedominates,andtheuncertaintyinthedistancecanbeneglected.Therefore:

Thefinalresultisa=2.39±0.09m/s2,or2.4±0.1m/s2.

σ a

a=

σ d

d

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+4 σ t

t

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

σ a =2a

σ t

t

⎛

⎝⎜⎜

⎞

⎠⎟⎟=2(2.392m/s2)

0.021.12 =0.085m/s

2



XI.UncertaintyPropagationIIAstudentperformsasimpleexperimenttofindtheaverageaccelerationofafallingobject.Hedropsabaseballfromabuildingandusesastringandtapemeasuretomeasuretheheightfromwhichtheballwasdropped.Heusesastopwatchtofindanaveragetimeoffallfor3trialsfromthesameheightandreportsthefollowingdata:h=6.75±0.33m,t=1.14±0.04s.Determinetheaverageacceleration,andcommentontheaccuracyoftheresult(doyouthinkanyerrorsweremade?).Whatsuggestionswouldyoumaketoimprovetheresult?

Calculatingtheaccelerationasforthepreviousexampleyieldsa=2h/t2=10.387m/s2.Inthiscase,neitherrelativeuncertaintyintimeordistancedominates,sobothshouldbeincludedinaquadraturecalculation:

Thefinalanswerwouldbewrittenas10.4±0.9m/s2.Notethatthisanswerishigherthanthevalueforginavacuum.Althoughtheuncertaintyrangeincludesthevalue9.8m/s2,wewouldexpectthevalueforaccelerationtobeatleastsomewhatorwellbelowgbecauseoftheeffectofdrag.Therefore,thisresultshouldbeviewedwithsomesuspicion.Suggestionsincludedoingmoretrialsandimprovingtheaccuracyofeachmeasurement.Accuracyinthetimeiswherethemostgainscanbemade,becausetherelativeuncertaintyofthetimeisweightedtwicethatoftherelativeuncertaintyofthedistance.

σ a = a

σ h

h⎛

⎝⎜⎞

⎠⎟

2

+4 σ t

t⎛

⎝⎜⎞

⎠⎟

2

= (10.387m/s2) 0.336.75

⎛⎝⎜

⎞⎠⎟

2

+4 0.041.14

⎛⎝⎜

⎞⎠⎟

2

=0.888m/s2



XII.UncertaintyPropagation3Astudentusesametertomeasurethecapacitanceoftwoparallelplatesthatare22cmindiameterandseparatedbyapieceofcardboard(κ=3)thatis5mmthick.Whatcapacitancewouldyouexpectthisstudenttomeasure?Iftherelativeuncertaintiesinthediameterandplateseparationanddielectricconstantare2%,5%,and10%,respectively,whatuncertaintyinthemeasurementwouldyoucite?Whatsuggestionswouldyougivetoreducetheeffectofstraycapacitance?

Thenominalvaluewouldbe:

Note that since the area A depends on the diameter squared, the uncertainty in the diameter has a higher weight than the other variables and cannot be neglected in the uncertainty calculation, even though it is significantly smaller. The answer should be cited as 202 ± 24 pF. Toreducestraycapacitance,minimizethelengthofwireleadsusedinthemeasurementprocess.Ifpossible,zerothemeterwiththeleadsinplacebeforemakingthemeasurement.

C = QV=κε0

Ad=3(8.85×10−12 C2

Nm2 )π(0.11m)20.005m =202pF

σ C =C 4 σ D

D⎛

⎝⎜⎞

⎠⎟

2

+σ d

d⎛

⎝⎜⎞

⎠⎟

2

+σκ

κ⎛

⎝⎜⎞

⎠⎟

2

= (202pF) 4 0.02( )2 + 0.05( )2 + 0.10( )2 =23.99pF



XIII.UncertaintyfromLinearRegressionInaninvestigationtoempiricallydeterminethevalueofp,astudentmeasuresthecircumferenceanddiameterof5circlesofvaryingsize,andusesExceltomakealinearplotofcircumferenceversusdiameter(bothinunitsofmeters).Alinearregressionfityieldstheresultofy=3.1527x–0.0502,withR2=0.9967forthe5datapointsplotted.Howshouldthisstudentreportthefinalresult?DoestheempiricalratioofC/Dagreewiththeacceptedvalueofp?

ThetheoreticalresultisC=pD+0.Byinspection,theexperimentalvalueofpisgivenbytheslopeoftheregression,or3.1527.Theuncertaintyinthisvalueis:

Thefinalexperimentalvalueforπisthen3.15±0.10,or3.2±0.1.Notethattheexperimentalinterceptisnotzero,butnotenoughinformationhasbeengiventocalculatetheintercept’suncertainty.Thevalueforpcanthereforeonlybetrustedtotheextentthat–0.0502(theintercept)isexperimentallyequivalenttozero.Therefore,theuncertaintyintheinterceptmustbelargerthan0.0502;ifnot,thereismorediscussiontobewritten!

σ m =m

(1/R2)−1n−2 =(3.1527) (1/0.9967)−1

5−2 =0.1047



XIV. Analyzing the Effect of Multiple Variables Astudentusesaphotogatetodeterminethespeedofacartonatrack.A“flag”ontopofthecarthaslengthdandrunsinthedirectionoftravel(paralleltothelengthofthecart).Theflagispositionedtoblockthephotogatebeamasthecartcrosses.Whenthebeamisunblockedaftertheflagmoveon,theamountoftimeΔtthathaselapsedisrecorded.SincedandΔtareassumedtobeaccurate,anaccuratecalculationofthecart’sspeedcanbemade.Normallytheflagisorientedexactlyinlinewiththecartandparalleltothetrackasdescribed,butinsteadsupposethattheflagistilted(asviewedfromabove).Asviewedbythephotogate,theflagwillappeartobeshorterthanitreallyis.Supposethatanexperimenterdoesnotrealizethattheflagonthecartisangledawayfromthepathofmotion.Supposefurtherthatthephotogatetimerisinaccurate.Iftheflagisactuallytiltedatanangleof30°andthetimeractuallymeasures10%lesstimethanitshould,whatrelativeerrorinspeedwillresultfromthesetwosystematiceffects?Ignorefriction,rollingresistance,anddraginthisproblem.

Calculatingtherelativeerrorintheflaglengthisnotthecorrectsolution.Asfarasthestudentknows,theflagiscorrectlyoriented.Aflagerrorresultsinanincorrecttimemeasurement,andtheincorrectlyworkingtimerfurthercompoundsthiserror.Thecorrectspeed,usingthevariablesthataremeasured,issimplyv=d/Δt,wheredistheflaglengthandΔtisthetimethatthephotogateisblocked.Theincorrectspeedarisesentirelyfromerroneoustimemeasurements;themeasuredtimeisshortenedbyafactorofcos30°andthenisshortenedagainbyaseparatefactor(f)bywhichthetimerisdefective:

incorrectspeed= !"∗$%&'(°∆+

Therelativeerrorisgivenbyequation2: relativeerror=,-.&/0-!–-34-$+-!-34-$+-!

Anacceptablesolutiononapracticum,forcos(30°)=0.866andf=0.9,is:

Relativespeederror=5

6∗789:;°∆<–5∆<

5∆<

= ="∗$%&'(°

– 1=+28.3%(toofast)



Discussion

Thetimemeasuredbyaphotogateforaflagoflengthdmovingatspeedvist=(dcosθ)/v,whereθistheflaganglemeasuredfromthelineoftravel:foranyflaglengthd,thetimemeasuredbythephotogateapproacheszeroasθapproaches90°.Ifthephotogateitselfmeasuresincorrectlybyafactoroff,thenthetimemeasuredist=f(dcosθ)/v.Ifthephotogateismeasuring10%lessthanitshould,thenf=0.9.Wenowcalculatetherelativetimeerror:

Relativetimeerror=+?@A9–+@BC@7<@5+@BC@7<@5

=65789:;°

E –5E5E

=0.9𝑐𝑜𝑠30°– 1=–.22058

Thephotogatemeasuresatimethatis22.1%toolowintotal.Wenowask,forthespeedinquestion(whateveritis),whattherelativeerrorinthatspeedisfortheactuallengthoftheflag,whichisbothmeasuredandexpectedtobed:

𝑣,-.&–𝑣-34-$+-!𝑣-34-$+-!

=

𝑑,-.&𝑡,-.&

–𝑑-34-$+-!𝑡-34-$+-!

𝑑-34-$+-!𝑡-34-$+-!

=

𝑑0.77942𝑡-34-$+-!

– 𝑑𝑡-34-$+-!

𝑑𝑡-34-$+-!

= 1

. 77942− 1 = +.283

Thisisthesameanswerasbefore.Thereareavarietyofincorrectapproachestothisquestion.Welistbelowthemostcommonerrorsthathavebeenobservedonthepracticum:

a.Flaglengthrelativeerror:!$%&'(°–!

!=𝑐𝑜𝑠30°– 1=–13.4%(–13%to2sigfigs)

Notethewrongsign–theflagindeedappearsshorter(negative)buttheanswershouldresultinhigherspeed.

b.Missingthetimemeasurementerror:5

789:;°∆<–5∆<

5∆<

= =$%&'(°

– 1=+15.5%

c.Timemeasurementerror:5

V.V789:;°∆<–5∆<

5∆<

= ==.=$%&'(°

– 1=+4.97%

d.timemeasurementerror:789:;°5;.W∆< – 5∆<

5∆<

=1.1𝑐𝑜𝑠30°–1=–4.74%

Notethewrongsign–answershouldresultinhigherspeed ifthetimerreadslow.

e.incorrect:±13%erroronflaglength±10%errorontimer=±23%or±3%

f.incorrect:quadraturecalculation

(13%)2 + (10%)2 = ±16.4%



(e)and(f)areincorrectforthesamereason.Errorscannotbesummedexceptincasesofsimpleaddition–quadraturemustbeinvoked;however,evenquadratureisincorrect,becausethisisNOTanuncertaintypropagationquestion!Thisisanestimateofaone-sidedsystematiceffect.

Challengequestion:supposethetimeractuallyincorrectlymeasures10%moretimethanitshould–whatrelativeerrorinspeedresults?Answer:foravalueoff=1.1,theresultisidenticaltoexample(c)inthecommonerrorslistabove:

Relativespeederror=5

V.V∗789:;°∆<–5∆<

5∆<

= ==.=∗$%&'(°

– 1=+4.97%(toofast)

Thetake-homemessagehereistwo-fold.First,itcanbedifficulttodistinguishbetweenacarelessmistake(“ohwhenyousaid10%Ithoughtitwas1.1insteadof0.9”)andasystematiceffect.Anevenmoreimportantobservationisthatmeasurementsinanintroductoryphysicsclasstendnottobeveryaccurate–a5%uncertaintyisquitegoodinanintroclassandcouldeasilybedismissedas“probablyjustduetosloppymeasurementtechnique.”Becarefultopointoutthepossibilityofsystematiceffectsinyourlabreports,andcitespecificpotentialsourcesofsystematicerrorinyourlabreportdiscussion.



XV.InterpretationofPlottedDataStudentsconnectasolenoidtoavariablepowersupply.Theythenuseamagneticfieldsensortomeasurethemagneticfieldstrengthalongthecentralaxisatthecenterofthesolenoid.AplotofthemeasuredmagneticfieldstrengthBasafunctionoftheemfεofthepowersupplyisshowntotheright(errorbarsnotshown).Theoretically,thevalueofBshouldbeμ0nI,wherenisthenumberofturnsperunitlength,andIisthecurrent.OneofthestudentsnotesthatthehorizontalcomponentofthemagneticfieldoftheEarthis2.5×10–5 T.IsthereevidencefromthegraphthattheEarth’smagneticfieldsystematicallyaffectedtheirmeasurement?Whatcanyousayabouttheorientationofthesolenoid?

Thetheoreticalequationforthemagneticfieldis

whereεisthevoltage(emf)appliedtothesolenoidandRistheresistanceofthecoil.Therefore,theplotofBvsεisexpectedtobelinear,butthereshouldbenomagneticfieldwhennocurrentisflowingthroughthesolenoid(thatis,they-interceptshouldbezeroonthisgraph).However,itisnotdifficulttoseethattheplothasaninterceptatapproximatelythevalueoftheEarth’snaturalmagneticfield,asshowninred.SincetheEarth’smagneticfieldpresumablyisparalleltothesurfaceoftheEarthandappearsasasystemicoffsetinthegraph,itisnotunreasonabletoconcludethatthesolenoidwasdetectingtheEarth’smagneticfieldandwasthusorientedhorizontallyduringthissetofmeasurements.

B = µ0nεR

studio measurement and uncertainty analysis...2019/07/25 · measurement uncertainties may be...

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