AE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-1Experimental Errors and Uncertainty:An IntroductionPrepared for students in AE 3051by J. M. Seitzmanadapted from material by J. CraigOutline• Errors and types of error• Statistic/probability: confidence levels• Uncertainty analysisAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-2Experimental Error• Error: all measurements have some uncertaintyerror = ε = umeas− uexact• Objectives1. Minimize error so that-∆≤ ε ≤+∆within some uncertainty (statistical confidence)or umeas−∆≤ uexact≤ umeas+∆2. Estimate error (uncertainty) to determine reliability, meaningfulness of datauexactumeas∆εErrors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-3Accuracy and Precision• Accuracy: also called systematic or bias error– denotes something repeatably “wrong” with the measurement or experiment• Precision: also called random error or noise– denotes errors that change randomly each time you try to repeat experimentGood Accuracy Good PrecisionGood PrecisionPoor Accuracy(can calibrate)Good AccuracyPoor Precision(can average)Poor AccuracyPoor PrecisionErrors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-4Other Related Terms• Sensitivity– Change in a measurement device’s output for a unit change in the measured (input) quantity, e.g., volts/Torr for the Baratron• Resolution– Smallest increment of change in a system or property that a measurement device can reliably capture• Dynamic Range– Maximum output of a measurement device divided by its resolution (or minimum measureable signal)Resolution = 50 psiDyn. Range= 5000/50= 100Sensitivity= 5000 psi/270°= 18.5 psi/degree0°270°Errors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-5Accuracy/Systematic Errors• Sources– Measuring system errors• difference between model of measuring system and reality• could be corrected, e.g., with better model of measurement– Measured system “errors”• influence of uncontrolled or unaccounted for variables in the experiment• the measured data may be “correct”, but may lead to an incorrect model of the object/process being studied– Blunders• human errors - misunderstandingsErrors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-6Some Systematic Measurement Errors xuModelActualNonlinearityxModelActualNonzero offset - BackgrounduxuModelQuantization Error(digitized data –impacts resolution)Actual• Systematic errors can be eliminated/removed if they are known•u=u(x)xuModelActualbacklashhysteresisBacklash & Hysteresistimeu(x=const)ModelActualDrift (e.g., offset changing in systematic way with time)Errors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-7Some Random Measurement Errors timeu(x=const)ModelActualBackground “Noise” (offset changing randomly with time)xDetector “Noise” (random change in sensitivity of device)u0• After data acquired, nearly impossible to separate random error (noise) sources• Examine random error with statistical methodsErrors and types of errorAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-8Probability Distributions• When we make measurements (i.e., take samples) of a system a number of times, we will get a distribution of resultsFrequencySystematicuncertaintyRandomuncertaintyyyavgyexact# timesreading is ingiven Y rangetheoreticaldistribution015001000500Prob. Distrib. Function f()1=∫∞∞−dyyfStatistics and probability• We might even make just one measurement (sample) of a system that has a distribution of possible statesAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-9Statistics and Probability• Since we can not make an infinite number of measurements to determine the true probability distribution– we use statistics to make estimates based on assumed distribution function• Some useful distribution functions– normal (Gaussian)– student’s t– log normal– exponentialAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-10• Commonly used when measurements/measurement system:– made up from many independent systems,each with any kind of distribution– # samples taken is very large (e.g., sample means)– more…µxf(x)πσ21−−=222)(exp21)(σµπσxxfµ = meanσ2= varianceσ = standard deviationStatistics and probability2σNormal/Gaussian Probability DistributionAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-11Normal Distributions – Probability Range 997.0)33(obPr954.0)()22(obPr683.0)()(obPr22=+≤≤−==+≤≤−==+≤≤−∫∫+−+−σµµσµσµµσµσµµσµσµσµσµσµdxxfdxxfOne Sigma:Two Sigma:Three Sigma:Statistics and probability• What fraction of values (combined probability) lie within given range from mean for a normal distribution?xf(x)µAE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-12Sample Statistics• What if µ and σ are unknown (as is often the case)?– use estimates from measurements, – use (N-1) for sx2because we have N independent xibut if also know xmeanthen only need to know (N-1) xito compute last remaining xi⇒only (N-1) “degrees of freedom” for this calculation11)(2121221−−=−−====∑∑∑===NNxNxNxxsvarianceSampleNxxmeanSampleNiiNiixNiiMean SquareSquare of meanxsx andStatistics and probabilityxsx ≅≅σµ;AE3051Copyright  2002-2003,2008 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertaintyrevised-13Uncertainty Estimates• Question: If one takes N (large) readings and computes , how confident can you be that the average is really close to the true mean (µ)?• Confidence intervals are way to describe thisStatistics and probabilityxµxσµ−Prob = c% that µlies in