AE3051JMS090800-1Experimental Errors and Uncertainty:An IntroductionPrepared for students in AE 3051by J. M. Seitzmanadapted from material made availableby J. CraigAE3051JMS090800-2The Measurement System• Measurements– Direct/Indirect comparison (rulers, balance scale, interferometer– Calibrated system (odometer, spring scale, pressure gage)• Measurement System• Issues:– Detector/Sensor: device which detects and responds to measurand– Transducer: converts measurand to an analog more easily measured (force-displacement-resistance-voltage)– Signal Cond.: amplify, filter, integrate, differentiate, convert freq. to voltage, etc.– Computer: widely used todayDetector-SensorTransducerSignal ConditionerIndicator,Recorder,Controller,Computer,etc.PhysicalMeasurandAE3051JMS090800-3Example of Measurement System• These are simple mechanical systems• Issues– mechanical vs electrical output– analog vs digital– calibration– accuracy, precision, resolution, sensitivity, linearity, drift, backlash?detectortransducerreadoutdetectortransducerreadoutTire Pressure GageBourdon Type Pressure GageAE3051JMS090800-4Computer Readout Systems• Each channel is read in sequence (10 ms to 1 ms per reading).• ADC may output a reading that is from 8 to 16 bits in size.– 8 bits & bipolar range yields readings from –128 to +127 (28=256) corresponding to –Full Scale and + Full Scale– 16 bits & bipolar range yields reading from –32768 to +32767 (216=65,536) so resolution is about 4½ digits or about 0.01% of Full Scale (notof reading)– With ±1v range, an 8 bit ADC has a 7.9 mv resoultion (=1/127)– With ±10v range, a 16 bit ADC has a 0.31 mv resolution (=10/32767)S/H ADCControlVoltagesdigital wordsto computerComputerMUXMultichannel Data Acquisition (sequential sampling) Key:MUX = multiplexer (switch)S/H = sample & hold (hold voltage while ADC reads)ADC = analog to digital converter (voltmeter)VolttChan #1Chan #2skewsampleReadings are not recorded simultaneouslyAE3051JMS090800-5Experimental Error• Error: all measurements have some uncertaintyerror = ε = xmeas− xexact• Objectives1. Minimize error so that uncertainty, u, is:-u ≤ε≤+uat N:1 certainty (a statistical confidence)or xmeas− u ≤ xexact≤ xmeas+ u2. Estimate error to determine reliability, meaningfulness of datauexactumeas∆εAE3051JMS090800-6Accuracy 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 PrecisionAE3051JMS090800-7Other Related Terms• Resolution– Smallest increment of change in a system or property that a measurement device can reliably capture• Sensitivity– Change in a measurement device’s output for a unit change in the measured (input) quantity, e.g., volts/Torr for the Baratron• Dynamic Range– Maximum output of a measurement device divided by its resolutionAE3051JMS090800-8Accuracy/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 studiedAE3051JMS090800-9Some Systematic Errors of Measuring SystemsxModelActualxuModelActualtimeu(x=const)ModelActualxuModelbacklashhysteresisxuModelActualNonzero offset - BackgroundBacklash & HysteresisDrift (e.g., offset changing in systematic way with time)Quantization Error(digitized data –impacts resolution)NonlinearityuActualSystematic errors can be eliminated/removed if they are knownSystematic errors can be eliminated/removed if they are known•u=u(x)AE3051JMS090800-10Some Random Errors of Measuring SystemsxDetector “Noise” (random change in sensitivity of device)xuModelActualDisturbances – “Noise”(e.g., pickup electrical signals from other sourcesu0timeu(x=const)ModelActualBackground “Noise” (offset changing randomly with time)After data acquired, nearly impossible to separate random error (noise) sourcesAfter data acquired, nearly impossible to separate random error (noise) sourcesAE3051JMS090800-11Uncertainty: Statistical Approaches• Probability & statistics provide a way to deal with uncertainty– we will cover only a VERY limited introductionFrequencyBias (systematic) errorprecision (random) errorxmXavgXexacttotal errorXmnumber of timesreading falls inthis range of xtheoretical distributionBias error: systematic errors that could be removed by proper calibrationPrecision error: random error - not directlycontrollableBias:calibrationconsistent human errorbackgroundPrecision:disturbancesnoisevariable conditionsBlunders:human errorsoftware!Either:backlash, hysteresis, frictiondampingdriftvariations in test procedureAE3051JMS090800-12Basic Concepts in Probability• Sample Space: all possible events or outcomes of an experiment; also can be referred to as a population.• Event: subset of sample space• Sample: finite number taken from population (e.g., 15 turbine blades taken from 8600 produced for XX-300 engine)– sample must be randomly selected from population– sample may or may not be returned to population before resampling• Probability: likelihood of an event (measured as % of successful events in sample space or population). We can often analytically compute this likelihood based on mathematics applied to the population. 0 ≤ P(event) ≤ 1.• There are many rules for computing probabilities for different kinds of events; these are the subject of courses and books on probability theory.AE3051JMS090800-13Examples• These happen to be binomial distributions (repeat experiment N times; each trial is independent of others; each trial is successful or not; probability of success, p, is same for each trial).• These are discrete distributions but there are also continuous distributions that are defined by CDF and PDF curves (next)0 50 100NumberNumber of EventsHeads in 100 coin tosses1 2 3 4 5 6Number1Number of EventsNumbers on one die2 3 4 5 6 7 8 9 10 11 121Number of