An Introduction to Measurement Uncertainty and Error Analysis

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An Introduction to Measurement Uncertaintyand
Error Analysis

• New TA Orientation, Fall 2003
• Department of Physics and Astronomy

The University of North Carolina at Chapel Hill
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Are these time measurements significantly
different?

• t1 1.86 s
• t2 2.07 s
• Yes
• No
• Cant tell

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Are these time measurements significantly
different?

• t1 1.86 s
• t2 2.07 s
• Student responses (N 44)
• Yes 43
• No 27 ???
• Cant tell 30

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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

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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.

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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

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Student Difficulties

• Task Use a ruler to measure the diameter of a
  penny as accurately as possible.

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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)

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Student Difficulties

• 3) Students tend to overstate precision
  (too many significant figures) of calculated
  values and to a lesser extent for directly
  measured values.

Typical student values
L
L 2.35 cm ( 0.05 cm) W 1.85 cm ( 0.05 cm)
W
A LW 4.3475 cm2 (Expert A 4.3
0.1 cm2 )
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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/cm3
• (standard deviation 10 g/cm3 )
• Density of pure nickel 8.912 g/cm3
• Density of nickel coin 8.9 0.4 g/cm3
• (alloy of 25 nickel, 75 copper)

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Sources of Error for Nickel Density
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• 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.

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Student t1 (s) t2 (s) 1.86 2.07 1.74 1.89
2.15 2.20 Average 1.92 2.05
t1 (s) t2 (s) Std. Dev. 0.21 0.16 Std.
Error 0.12 0.09
The numbers are close, but
different.
Expert
These time measurements agree with each other.
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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

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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

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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 www.physics.unc.edu/deardorf/uncertainty