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Learn how to identify and analyze errors in experimental measurements, types of errors, and techniques to estimate uncertainties in measured values. This chapter covers error sources, uncertainty analysis methods, and propagation of uncertainties in engineering experiments.
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MECH 373Instrumentation and Measurements Lecture 16 Experimental Uncertainty Analysis (Chapter 7) • Introduction • Types of Errors • Propagation of Uncertainties
When we measure some physical quantity with an instrument and obtain a numerical value, we want to know how close this value is to the true value. The difference between the true value and the measured value is the error. Unfortunately, the true value is in general unknown and unknowable. Since this is the case, the exact error is never known. We can only estimate error or a range of probable error in the result from that measurement. This estimate is called the uncertainty in the measured value. This chapter discuss methods to combine uncertainties from all sources to estimate the uncertainty of the final results of an experiment. This “propagation of uncertainty” is an important aspect of any engineering experiment. Uncertainty analysis is performed during the design stage of an experiment to assist in the selection of measurement techniques and devices. Introduction
Difference between measured result and “true” value. Illegitimate Errors Blunders result from mistakes in procedure. We must be careful. Computational or calculation errors after the experiment. Systematic or Bias Errors An error that persists and cannot be considered to exist entirely by chance. This type of error tends to stay constant from trial to trial. (e.g. zero offset) Systematic errors can be corrected through calibration Faulty equipment - Instrument always reads 3% high or low Consistent or recurring human errors - observer bias This type of error cannot be studied theoretically but can be determined by comparison to theory or by alternate measurements. Types of Errors
Random or Precision Errors: The deviation of the measurement from the true value resulting from the finite precision of the measurement method being used. Instrument friction or hysteresis Errors from calibration drift Variation of procedure or interpretation of experimenters Test condition variations or environmental effects Reduce random errors by conducting more experiments/ take more data. Because of the varying nature of random errors, exact values cannot be probable estimates of the error that can be made through statistical analyses. Types of Errors (cont.)
• The relationship between the true value and the measured data set (containing both systematic and random errors) is illustrated in the figure below. Types of Errors (cont.)
Calibration Laboratory certification of equipment Data Acquisition Errors in data acquisition equipment Data Reduction Errors in computers and calculators Errors of Method Personal errors/blunders Grouping & Categorizing Error Sources
Propagation of Uncertainties • A general relationship between some dependent variable y and a measured variable x is y = f(x).
Propagation of Uncertainties • If the result R is dependent only on the product of the measured variable, that is
