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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
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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
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fairtoyourfriend.Sowhatdoyoudonow?Theanswerliesinknowingsomethingabouttheaccuracyofeachinstrument.Tohelpanswerthesequestions,wefirstdefinethetermsaccuracyandprecision:
Accuracyistheclosenessofagreementbetweenameasuredvalueandatrueoracceptedvalue.Measurementerroristheamountofinaccuracy.Precisionisameasureofhowwellaresultcanbedetermined(withoutreferencetoatheoreticalortruevalue).Itisthedegreeofconsistencyandagreementamongindependentmeasurementsofthesamequantity;alsothereliabilityorreproducibilityoftheresult.
Theaccuracyandprecisioncanbepicturedasfollows:
highprecision,lowaccuracy lowprecision,highaccuracy
Figure1.AccuracyvsPrecision
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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
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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
−=−=−
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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
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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.
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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.
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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
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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 −++−+−=
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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.
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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
... 222221
−=
−
+++=
∑Nx
Nxxx
s iNδδδδ
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
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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
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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
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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.
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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.
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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.
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Measurements & Uncertainty Analysis
18 University of North Carolina
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.
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Measurements & Uncertainty Analysis
19 Department of Physics and Astronomy
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
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Measurements & Uncertainty Analysis
20 University of North Carolina
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:
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Measurements & Uncertainty Analysis
21 Department of Physics and Astronomy
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.
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22 University of North Carolina
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.
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Measurements & Uncertainty Analysis
23 Department of Physics and Astronomy
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
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Measurements & Uncertainty Analysis
24 University of North Carolina
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 ⎟⎠
⎞⎜⎝
⎛=
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Measurements & Uncertainty Analysis
25 Department of Physics and Astronomy
Usingthedefinitionofσyields:
xf dxdfσσ =
Examplesforpowerlaws: f=√x f=x2
dfdx
= 12 x
dfdx
= 2x
σ f =σ x2 x
orσ ff= 12σ xx
σ ff= 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):
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Measurements & Uncertainty Analysis
26 University of North Carolina
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 x
2 + x2σ y2
Dividingtheaboveequationbyf=xyyields:
Whenadding(orsubtracting)independentmeasurements,theabsoluteuncertaintyofthesum(ordifference)istherootsumofsquares(RSS)oftheindividualabsoluteuncertainties.Whenaddingcorrelatedmeasurements,theuncertaintyintheresultissimplythesumoftheabsoluteuncertainties,whichisalwaysalargeruncertaintyestimatethanaddinginquadrature(RSS).Addingorsubtractingaconstantdoesnotchangetheabsoluteuncertaintyofthecalculatedvalueaslongastheconstantisanexactvalue.
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27 Department of Physics and Astronomy
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.
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28 University of North Carolina
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 ff= n2 σ x
x⎛⎝⎜
⎞⎠⎟2
+m2σ yy
⎛⎝⎜
⎞⎠⎟
2
TheproofofthisisshownintheAppendix.
Theuncertaintyestimatefromtheupper-lowerboundmethodisgenerallylargerthanthestandarduncertaintyestimatefoundfromthequadraturemethod,butbothmethodswillgiveareasonableestimateoftheuncertaintyinacalculatedvalue.Note:Onceyouhaveanunderstandingofthequadraturemethod,itisnotrequiredtoperformthepartialderivativeeverytimeyouarepresentedwithapropagationofuncertainty
problem in any of the above forms! Instead, simply apply the
correctformulafortherelativeuncertainties.
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29 Department of Physics and Astronomy
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.
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Measurements & Uncertainty Analysis
30 University of North Carolina
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
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31 Department of Physics and Astronomy
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
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32 University of North Carolina
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
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33 Department of Physics and Astronomy
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.
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34 University of North Carolina
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.
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35 Department of Physics and Astronomy
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
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36 University of North Carolina
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.
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37 Department of Physics and Astronomy
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.
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38 University of North Carolina
UsingLINESTinExcelLINESTisanalternativelinearleastsquarefittingfunctioninExcel.TheresultsoftheLINESTanalysisarevirtuallyidenticaltothelineartrendlineanalysisdescribedabove;however,LINESTprovidesasingle-stepcalculationofboththeslopeandinterceptuncertainties,insteadofthemulti-stepproceduredescribedabove.
• Startwithatablefortimeandvelocity(right).
•
TheLINESTfunctionreturnsseveraloutputs;toprepare,selecta2by5arraybelowthedata,asshown.
•
Note,velocitywasmistakenlylabeledashavingunitsofm/secinthetabletotheright.
•
UndertheInsertmenu,selectionFunction,thenStatistical,andfinallyLINESTasshownbelow(andhitOK).
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39 Department of Physics and Astronomy
Selectthey-valuesandx-valuesfromthetableandwrite‘TRUE’forthelasttwooptions(below).Again,pressOK.
TheLINESTfunctionisanarrayfunction;therefore,youmusttellExcelyouaredonewiththearray.Highlighttheformulaintheformulabarasshownbelow.PressCtrl+ShiftsimultaneouslywithEnter(Macusers,pressCommandandEnter)
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40 University of North Carolina
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.
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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 ff= ∂ 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 ff= nxn−1ym( )2 σ x
2
xnym( )2+ mxnym−1( )2 σ y
2
xnym( )2
Thisexpressionlookscomplicatedbutitsimplifiestosomethingrathersimple:
σ ff= n2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+m2σ yy
⎛
⎝⎜⎞
⎠⎟
2
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42 University of North Carolina
Theresultisthattherelative(fractional)uncertaintyinfistherootofthesquaredsum(RSS)ofindividualuncertaintiesinxandy.Examplesinthebodyofthisdocumentinclude:
f=xy σ ff= (1)2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+ (1)2σ yy
⎛
⎝⎜⎞
⎠⎟
2
f=x/y σ ff= (1)2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+ (−1)2σ yy
⎛
⎝⎜⎞
⎠⎟
2
f=xy2 σ ff= σ x
x⎛⎝⎜
⎞⎠⎟
2
+ 4σ yy
⎛
⎝⎜⎞
⎠⎟
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.