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Reporting Measurement Uncertainties According to the ISO Guide Duane Deardorff Dept. of Physics and Astronomy The University of North Carolina at Chapel Hill Contributed Paper EK06 presented at 127 th National Meeting of the AAPT Madison, WI August 6, 2003

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How many of you: Have reported measurements from an experimental physics lab (teaching or research) Have reported measurements from an experimental physics lab (teaching or research) Use or encourage students to use SI units Use or encourage students to use SI units Are familiar with the book An Introduction to Error Analysis by John Taylor Are familiar with the book An Introduction to Error Analysis by John Taylor Are familiar with the ISO Guide to the Expression of Uncertainty in Measurement Are familiar with the ISO Guide to the Expression of Uncertainty in Measurement Know the difference between Type A and Type B components used to evaluate the standard uncertainty of a measurement Know the difference between Type A and Type B components used to evaluate the standard uncertainty of a measurement

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Motivation Physics relies on empirical data that is inherently subject to measurement uncertainties Physics relies on empirical data that is inherently subject to measurement uncertainties Reporting of uncertainties must be standardized in order for values to be interpreted correctly Reporting of uncertainties must be standardized in order for values to be interpreted correctly Students should learn these standards Students should learn these standards (just like they should learn and use SI notation) (just like they should learn and use SI notation) Outcome of my dissertation research on students’ treatment of uncertainties Outcome of my dissertation research on students’ treatment of uncertainties

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Different conventions have been used to report measurement uncertainties Difficult to compare results for agreement Difficult to compare results for agreement Confuses students (and experts!) Confuses students (and experts!) Many scientists may not even realize the differences in notation! Many scientists may not even realize the differences in notation!

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m = 75 ± 5 g What is the meaning of ± 5 ? Best guess by experimenter Best guess by experimenter Half the smallest division of measurement Half the smallest division of measurement Standard deviation: Standard deviation: Standard error: m = / n Standard error: m = / n Expanded uncertainty of ± 2 or ± 3 (95% or 99% confidence interval) Expanded uncertainty of ± 2 or ± 3 (95% or 99% confidence interval) Standard uncertainty: u Standard uncertainty: u Combined standard uncertainty: u c Combined standard uncertainty: u c

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What does x ± u mean? Physicists generally report ±1 (68% CI) Physicists generally report ±1 (68% CI) Chemists report ±2 or ±3 (95% or 99% CI) Chemists report ±2 or ±3 (95% or 99% CI) Survey/poll margin of error is 95% CI Survey/poll margin of error is 95% CI Accuracy tolerances are often 95% or 99% Accuracy tolerances are often 95% or 99% NIST Calibration certificate is usually 99% NIST Calibration certificate is usually 99% Conclusion: The interpretation of ± u is not consistent within a field, let alone between fields, and the meaning is generally not specified (except in NIST publications).

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ISO Guide to the Expression of Uncertainty in Measurement In 1993 the International Organization for Standardization published new guidelines for industry and research: “GUM” In 1993 the International Organization for Standardization published new guidelines for industry and research: “GUM” NIST version: physics.nist.gov/cuu/Uncertainty Use combined standard uncertainty u c that includes both Type A and Type B components Use combined standard uncertainty u c that includes both Type A and Type B components use term uncertainty not error use term uncertainty not error avoid use of ambiguous ± notation without explanation avoid use of ambiguous ± notation without explanation

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“The ± format should be avoided whenever possible because it has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused with an expanded uncertainty.” “The ± format should be avoided whenever possible because it has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused with an expanded uncertainty.” -ISO Guide, p. 7

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ISO Guide recommendation: Clearly define uncertainty values. Ex. m = 100.021 47 g with a combined standard uncertainty u c = 0.35 mg or: m = 100.021 47(35) g, where the number in parentheses is the numerical value of u c and refers to the corresponding last digits of the quoted result or: m = (100.021 47 ± 0.000 35) g, where the number following the symbol ± is the numerical value of u c and not a confidence interval

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Determination of combined standard uncertainty: u c Type A component: random, evaluated statistically Type A component: random, evaluated statistically (e.g. standard deviation or standard error) Type B component: scientific judgment based on all available information, a priori Type B component: scientific judgment based on all available information, a priori (e.g. instrument precision, rated accuracy of instrument, variation in previous data, physical factors, etc.) Combined standard uncertainty: Combined standard uncertainty:

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Example 1 A meter stick is used to measure the width of a table: Width (cm): 56.2, 56.7, 56.3, 56.9, 56.5 u A = 0.13 cm (standard error) u A = 0.13 cm (standard error) u B = 0.1 cm (resolution and assumed accuracy) u B = 0.1 cm (resolution and assumed accuracy) u C = 0.16 cm u C = 0.16 cm Average width = 56.52 cm with u C = 0.16 cm Typical intro physics: W = 56.6 ± 0.2 cm

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Example 2 A DMM is used to measure the current in a circuit. Type A component Type A component Meter readings (mA): 1.426 to 1.428 Meter readings (mA): 1.426 to 1.428 Uncertainty from fluctuations = 0.001 mA Uncertainty from fluctuations = 0.001 mA Type B component Type B component Accuracy rating of meter = ± 1% (assume 99% CI) Accuracy rating of meter = ± 1% (assume 99% CI) Corresponding uncertainty = (0.014 mA)/2.576 Corresponding uncertainty = (0.014 mA)/2.576 Combined standard uncertainty: u c = 0.006 mA Combined standard uncertainty: u c = 0.006 mA

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Conclusion When reporting a measured value and its estimated uncertainty, remember to include units and a similar explanation of the uncertainty.

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“Stick” with the ISO Guide (GUM)! ISO Guide to the Expression of Uncertainty in Measurements (1993) NIST: physics.nist.gov/cuu/Uncertainty For more information about the expression of measurement uncertainty by and for introductory physics students, go to: www.physics.unc.edu/~deardorf/uncertainty www.physics.unc.edu/~deardorf/uncertainty

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