Managing Errors and Uncertainty - Astronomy › sites › default › files › Managing Erro… · Managing Errors and Uncertainty It is inevitable that experiments will vary from

Department of Physics & Astronomy Lab Manual Undergraduate Labs

ManagingErrorsandUncertaintyItisinevitablethatexperimentswillvaryfromtheirtheoreticalpredictions.Thismaybeduetonaturalvariations,alackofunderstandingoftheprocess,orasimplifiedmodelinthetheory.Inthisdocument,wediscussafewofthesecauses,andmore importantly,howwecanaccountfortheseproblemsandovercomethemthroughcarefulcollection,handling,andanalysis.

TypesofUncertaintyTherearethreetypesoflimitationstomeasurements:

1) InstrumentallimitationsAnymeasuringdeviceislimitedbythefinenessofitsmanufacturing.Measurementscanneverbebetterthantheinstrumentsusedtomakethem.

2) SystematicerrorsThesearecausedbyafactorthatdoesnotchangeduringthemeasurement.Forexample,ifthebalanceyouusedwascalibratedincorrectly,allyoursubsequentmeasurementsofmasswouldbewrong.Systematicerrorsdonotenterintotheuncertainty.Theycaneitherbeidentifiedandeliminated,orlurkinthebackground,producingashiftfromthetruevalue.

3) RandomerrorsThese arise from unnoticed variations in measurement technique, tiny changes in theexperimental environment, etc. Random variations affect precision. Truly random effectsaverageoutiftheresultsofalargenumberoftrialsarecombined.Note:“Humanerror”isaeuphemismfordoingapoorqualityjob.Itisanadmissionofguilt.

Precisionvs.Accuracy

• A precise measurement is one where independent measurements of the same quantity closelyclusteraboutasinglevaluethatmayormaynotbethecorrectvalue.

• Anaccuratemeasurementisonewhereindependentmeasurementsclusteraboutthetruevalueofthemeasuredquantity.

Systematicerrorsarenotrandomandthereforecannevercancelout.

Theyaffecttheaccuracybutnottheprecisionofameasurement.

A. Low-precision,Low-accuracy:

Theaverage(theX)isnotclosetothecenter

B. Low-precision,High-accuracy:

The average is close to the true value, but datapointsarefarapart


C. High-precision,Low-accuracy:

Data points are close together, but he average isnotclosetothetruevalue

D. High-precision,High-accuracy

Alldatapointsclosetothetruevalue

WritingExperimentalNumbersUncertaintyofMeasurements

Measurementsarequantifiedbyassociatingthemwithanuncertainty.Forexample,thebestestimateofa length𝐿 is2.59cm,butduetouncertainty,thelengthmightbeassmallas2.57cmoras largeas2.61cm.𝐿canbeexpressedwithitsuncertaintyintwodifferentways:

1. AbsoluteUncertaintyExpressedintheunitsofthemeasuredquantity:𝑳 = 𝟐.𝟓𝟗 ± 𝟎.𝟎𝟐cm

2. PercentageUncertaintyExpressedasapercentagewhichisindependentoftheunitsAbove,since0.02/2.59 ≈ 1%wewouldwrite𝑳 = 𝟕.𝟕cm ± 𝟏%

SignificantFigures

Experimentalnumbersmustbewritteninawayconsistentwiththeprecisiontowhichtheyareknown.Inthiscontextonespeaksofsignificantfiguresordigitsthathavephysicalmeaning.

1. Alldefinitedigitsandthefirstdoubtfuldigitareconsideredsignificant.

2. Leadingzerosarenotsignificantfigures.Example:𝐿 = 2.31cmhas3significantfigures.For𝐿 = 0.0231m, thezerosservetomovethedecimalpointtothecorrectposition.Leadingzerosarenotsignificantfigures.

3. Trailingzerosaresignificantfigures:theyindicatethenumber’sprecision.

4. Onesignificantfigureshouldbeusedtoreporttheuncertaintyoroccasionallytwo,especiallyifthesecondfigureisafive.

RoundingNumbers

Tokeepthecorrectnumberofsignificantfigures,numbersmustberoundedoff.Thediscardeddigitiscalledtheremainder.Therearethreerulesforrounding:

ü Rule1:Iftheremainderislessthan5,dropthelastdigit.Roundingtoonedecimalplace:5.346 → 5.3

ü Rule2:Iftheremainderisgreaterthan5,increasethefinaldigitby1.Roundingtoonedecimalplace:5.798 → 5.8


ü Rule 3: If the remainder is exactly 5 then round the last digit to the closest even number.Thisistopreventroundingbias.Givenalargedataset,remaindersof5areroundeddownhalfthetimeandroundeduptheotherhalf.Roundingtoonedecimalplace:3.55 → 3.6,also3.65 → 3.6

Examples

Theperiodofapendulumisgivenby𝑇 = 2𝜋 𝑙/𝑔.

Here,𝑙 = 0.24misthependulumlengthand𝑔 = 9.81m/s2istheaccelerationduetogravity.

ûWRONG:𝑇 = 0.983269235922s

üRIGHT: 𝑇 = 0.98s

Yourcalculatormayreportthefirstnumber,butthereisnowayyouknow𝑇tothatlevelofprecision.Whennouncertaintiesaregiven,reportyourvaluewiththesamenumberofsignificantfiguresasthe

valuewiththesmallestnumberofsignificantfigures.

Themassofanobjectwasfoundtobe3.56gwithanuncertaintyof0.032g.

ûWRONG:𝑚 = 3.56 ± 0.032g

üRIGHT:𝑚 = 3.56 ± 0.03g

Thefirstwayiswrongbecausetheuncertaintyshouldbereportedwithonesignificantfigure

Thelengthofanobjectwasfoundtobe2.593cmwithanuncertaintyof0.03cm.

ûWRONG:𝐿 = 2.593 ± 0.03cm

üRIGHT:𝐿 = 2.59 ± 0.03cm

Thefirstwayiswrongbecauseitisimpossibleforthethirddecimalpointtobemeaningfulsinceitissmallerthantheuncertainty.

Thevelocitywasfoundtobe2. 45m/swithanuncertaintyof0.6m/s.

ûWRONG:𝑣 = 2.5 ± 0.6m/s

üRIGHT:𝑣 = 2.4 ± 0.6m/s

Thefirstwayiswrongbecausethefirstdiscardeddigitisa5.Inthiscase,thefinaldigitisroundedtotheclosestevennumber(i.e.4)

Thedistancewasfoundtobe45600mwithanuncertaintyaround1m

ûWRONG:𝑑 = 45600m

üRIGHT:𝑑 = 4.5600×10!m

Thefirstwayiswrongbecauseittellsusnothingabouttheuncertainty.Usingscientificnotationemphasizesthatweknowthedistancetowithin1m.


StatisticalAnalysisofSmallDataSetsRepeatedmeasurementsallowyoutonotonlyobtainabetterideaoftheactualvalue,butalsoenableyou to characterize theuncertaintyof yourmeasurement.Belowareanumberofquantities that areveryusefulindataanalysis.Thevalueobtainedfromaparticularmeasurementis𝑥.Themeasurementisrepeated𝑁 times.Oftentimes in lab𝑁 issmall,usuallynomorethan5to10. In thiscaseweusetheformulaebelow:

Mean(𝒙avg) Theaverageofallvaluesof𝑥(the“best”valueof𝑥) 𝑥avg =𝑥! + 𝑥! +⋯+ 𝑥!

𝑁

Range(𝑹) The “spread” of the data set. This is the differencebetweenthemaximumandminimumvaluesof𝑥 𝑅 = 𝑥max − 𝑥min

Uncertaintyinameasurement

(∆𝒙)

Uncertainty in a single measurement of 𝑥. Youdetermine this uncertainty by making multiplemeasurements.Youknowfromyourdatathat𝑥 liessomewherebetween𝑥maxand𝑥min.

∆𝑥 =𝑅2 =

𝑥max − 𝑥min2

UncertaintyintheMean

(∆𝒙avg)

Uncertaintyinthemeanvalueof𝑥.Theactualvalueof 𝑥 will be somewhere in a neighborhood around𝑥avg. This neighborhood of values is the uncertaintyinthemean.

∆𝑥avg =∆𝑥𝑁=

𝑅2 𝑁

MeasuredValue(𝒙m)

The final reported value of a measurement of 𝑥containsboththeaveragevalueandtheuncertaintyinthemean.

𝑥m = 𝑥avg ± ∆𝑥avg

The average value becomesmore and more precise as the number of measurements𝑁 increases.Althoughtheuncertaintyofanysinglemeasurement isalways∆𝑥, theuncertainty inthemean∆𝒙avgbecomessmaller(byafactorof1/ 𝑁)asmoremeasurementsaremade.


Example

Youmeasurethelengthofanobjectfivetimes.Youperformthesemeasurementstwiceandobtainthetwodatasetsbelow.

Measurement DataSet1(cm DataSet2(cm)𝑥! 72 80𝑥! 77 81𝑥! 8 81𝑥! 85 81𝑥! 88 82

Quantity DataSet1(cm) DataSet2(cm)𝒙avg 81 81𝑹 16 2∆𝒙 8 1∆𝒙avg 4 0.4

ForDataSet1,tofindthebestvalue,youcalculatethemean(i.e.averagevalue):

𝑥avg =72cm + 77cm + 82cm + 86cm + 88cm

5= 81cm

The range, uncertainty and uncertainty in the mean for Data Set 1 are then:

𝑅 = 88cm − 72cm=16cm

∆𝑥 =𝑅2= 8cm

∆𝑥avg =𝑅2 5

≈ 4cm

DataSet2yieldsthesameaveragebuthasamuchsmallerrange.

Wereportthemeasuredlengths𝑥mas:

DataSet1:𝒙m = 𝟖𝟏 ± 𝟒cm

DataSet2:𝒙m = 𝟖𝟏.𝟎 ± 𝟎.𝟒cm

NoticethatforDataSet2,∆𝑥avgissosmallwehadtoaddanothersignificantfigureto𝑥m.


StatisticalAnalysisofLargeDataSetsIfonlyrandomerrorsaffectameasurement,itcanbeshownmathematicallythat inthe limitofan infinitenumber ofmeasurements (𝑁 → ∞), the distributionof values follows anormal distribution (i.e. the bellcurveontheright).Thisdistributionhasapeakatthemean value𝑥avg and awidth given by the standarddeviation𝜎.

Obviously, we never take an infinite number ofmeasurements. However, for a large number ofmeasurements,say,𝑁~10!ormore,measurementsmay be approximately normally distributed. In thateventweusetheformulaebelow:

Mean(𝒙avg)Theaverageofallvaluesof𝑥(the“best”valueof𝑥).Thisisthesameasforsmalldatasets. 𝑥avg =

𝑥!!!!!

𝑁

Uncertaintyinameasurement

(∆𝒙)

Uncertainty in a single measurement of 𝑥. Thevast majority of your data lies in the range𝑥avg ± 𝜎

∆𝑥 = 𝜎 =(𝑥! − 𝑥avg)!!

!!!

𝑁

UncertaintyintheMean

(∆𝒙avg)

Uncertainty in the mean value of 𝑥. The actualvalueof𝑥willbesomewhereinaneighborhoodaround𝑥avg. This neighborhood of values is theuncertaintyinthemean.

∆𝑥avg =𝜎𝑁

MeasuredValue(𝒙m)

The final reported valueof ameasurementof𝑥contains both the average value and theuncertaintyinthemean.

𝑥m = 𝑥avg ± ∆𝑥avg

Mostofthetimewewillbeusingtheformulaeforsmalldatasets.However,occasionallyweperformexperimentswithenoughdatatocomputeameaningfulstandarddeviation.Inthosecaseswecantakeadvantageofsoftwarethathasprogrammedalgorithmsforcomputing𝑥avgand𝜎.

𝑥avg𝑥

Frequency

𝜎


PropagationofUncertaintiesOftentimeswecombinemultiplevalues,eachofwhichhasanuncertainty,intoasingleequation.Infact,wedothiseverytimewemeasuresomethingwitharuler.Take, forexample,measuringthedistancefromagrasshopper’s front legs tohishind legs. For rulers,wewill assume that theuncertainty in allmeasurementsisone-halfofthesmallestspacing.

Themeasureddistanceis𝑑m = 𝑑 ± ∆𝑑where𝑑 = 4.63cm − 1.0cm = 3.63cm.Whatistheuncertaintyin𝑑m?Youmightthinkthatitisthesumoftheuncertaintiesin𝑥and𝑦(i.e.∆𝑑 = ∆𝑥 + ∆𝑦 = 0.1cm).However,statisticstellsusthatiftheuncertaintiesareindependentofoneanother,theuncertaintyinasum or difference of two numbers is obtained by quadrature:∆𝑑 = (∆𝑥)! + (∆𝑦)! = 0.07cm. Thewaytheseuncertaintiescombinedependsonhowthemeasuredquantityisrelatedtoeachvalue.Rulesforhowuncertaintiespropagatearegivenbelow.

Addition/Subtraction 𝑧 = 𝑥 ± 𝑦 ∆𝑧 = (∆𝑥)! + (∆𝑦)!

Multiplication 𝑧 = 𝑥𝑦 ∆𝑧 = 𝑥𝑦∆𝑥𝑥

!

+∆𝑦𝑦

!

Division 𝑧 =𝑥𝑦 ∆𝑧 =

𝑥𝑦

∆𝑥𝑥

!

+∆𝑦𝑦

!

Power 𝑧 = 𝑥! ∆𝑧 = 𝑛 𝑥!!!∆𝑥

MultiplicationbyaConstant

𝑧 = 𝑐𝑥 ∆𝑧 = 𝑐 ∆𝑥

Function 𝑧 = 𝑓(𝑥,𝑦) ∆𝑧 =𝜕𝑓𝜕𝑥

!

(∆𝑥)! +𝜕𝑓𝜕𝑦

!

(∆𝑦)!

𝑦 = 4.63cm ± 0.05cm𝑥 = 1.0cm ± 0.05cm


Examples

Addition

Thesidesofafencearemeasuredwithatapemeasuretobe124.2cm,222.5cm,151.1cmand164.2cm.Eachmeasurement has an uncertainty of 0.07cm. Calculate themeasured perimeter𝑃m including itsuncertainty.

𝑃 = 124.2cm + 222.5cm + 151.1cm + 164.2cm = 662.0cm

∆𝑃 = (0.07cm)! + (0.07cm)! + (0.07cm)! + (0.07cm)! = 0.14cm

𝑃m = 662.0 ± 0.1cm

Multiplication

The sides of a rectangle are measured to be 15.3cm and 9.6cm. Each length has an uncertainty of0.07cm.Calculatethemeasuredareaoftherectangle𝐴mincludingitsuncertainty.

𝐴 = 15.3cm×9.6cm = 146.88𝑐𝑚!

∆𝐴 = 15.3cm×9.6cm0.0715.3

!+

0.079.6

!= 1.3cm2

𝐴m = 147 ± 1cm2

Power/MultiplicationbyConstant

Aball drops from rest from an unknownheightℎ. The time it takes for the ball to hit the ground ismeasured to be 𝑡 = 1.3 ± 0.2s. The height is related to this time by the equation ℎ = !

!𝑔𝑡! where

𝑔 = 9.81m/s2.Assumethatthevaluefor𝑔carriesnouncertaintyandcalculatetheheightℎincludingitsuncertainty.

ℎ =129.8m/s2 (1.3s)! ≈ 8.281m

∆ℎ =129.8m/s2 2×1.3s×0.2s ≈ 2.5m

ℎm = 8 ± 3m

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