1 An Introduction to Measurement Uncertainty and Error Analysis New TA Orientation, Fall 2003 Department of Physics and Astronomy The University of North Carolina at Chapel Hill

2 Are these time measurements significantly different? t 1 = 1.86 s t 2 = 2.07 s  Yes  No  Can’t tell

3 Are these time measurements significantly different? t 1 = 1.86 s t 2 = 2.07 s Student responses (N = 44):  Yes 43%  No 27% ???  Can’t tell 30%

4 Purpose and Challenges Science relies on empirical data, which is inherently subject to measurement error Uncertainty estimates are necessary for: –assessing quality of data –comparison of data –verify/refute theoretical predictions Students often have difficulty analyzing errors Guidelines for reporting uncertainties vary: –Terminology and notation is not consistent –International standard exists but is not well known

5 Student Difficulties 1) Uncertainty is rarely estimated and stated, even when required (true for experts too!). 2) Even if found, most students do not use uncertainties to justify their conclusions. 3) Calculated values are often reported with too many (in)significant digits. 4) Students have difficulty identifying the primary source of error in an experiment.

6 Student Difficulties 1) Students often fail to report a quantitative uncertainty estimate, even when requested. Overall reporting rates from this study: 0 to 50% of students reported uncertainty 30 to 70% of TAs reported uncertainty

7 Student Difficulties Task: Use a ruler to measure the diameter of a penny as accurately as possible.

8 Student Difficulties 2) Even if stated, most students do not justify their conclusions based on the uncertainty Judgements are made based on arbitrary criteria: –“Our percent error was only 4%, so our experiment proved the theory.” –“I decide by how much two measurements differ in order to see if they agree.” –“The result should be accurate as long as the error is less than 10%.” Similar findings in other studies (Sere; Garratt)

9 Student Difficulties 3) Students tend to overstate precision (too many significant figures) of calculated values and to a lesser extent for directly measured values. L W Typical student values: L = 2.35 cm (± 0.05 cm) W = 1.85 cm (± 0.05 cm) A = L*W = cm 2 (Expert: A = 4.3 ± 0.1 cm 2 )

10 Student Difficulties 4) Students have difficulty identifying the primary source of error in an experiment. Are nickel coins made of nickel? Find density: Median density from 76 students = 7.1 g/cm 3 (standard deviation = 10 g/cm 3 ) Density of pure nickel = g/cm 3 Density of nickel coin = 8.9 ± 0.4 g/cm 3 (alloy of 25% nickel, 75% copper)

11 Sources of Error for Nickel Density

12 Students often focus on the details of error analysis and miss the big picture, losing sight of the forest for the sake of the trees.

13 “These time measurements agree with each other.” Student: t 1 (s) t 2 (s) Average = t1t1 t2t2 t (s) t 1 (s) t 2 (s) Std. Dev. = Std. Error = “The numbers are close, but different.” Expert:

14 Teaching Tips Remind students of the “big picture” view of why uncertainty estimates are important. Show examples of how to decide whether results agree or disagree within their uncertainty using error bars on number line. Require students to justify their conclusions based on uncertainty estimates, no general statements like: “Our results only had a 5% error, so this proved the theory.” - wrong, for more than one reason! Teach concepts consistent with ISO Guide

15 ISO Guide to the Expression of Uncertainty in Measurement (GUM) International Organization for Standardization published new guidelines in 1993 for industry and research NIST version: physics.nist.gov/cuu/Uncertainty Use standard uncertainty –Type A component: random, evaluated statistically –Type B component: systematic, judgement, a priori use terms “uncertainty” and “error” appropriately explain meaning of ± notation and uncert. value

16 “ It is better to be roughly right than precisely wrong.” - Alan Greenspan U.S. Federal Reserve Chairman For more on measurement uncertainty research, go to: