1.1: Measurement Error
- Page ID
- 34674
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The result of a single measurement should be reported in the format \[(\textrm{estimate})\, \pm\, (\textrm{measurement error}).\] The estimate is your best guess for the true value, while the measurement error states the range where the true value might lie. By convention, the estimate and measurement error are formatted according to the these rules :
- The measurement error has one significant figure.
- The estimate has the same precision as the measurement error.
Suppose you use a digital multimeter to measure the current in a circuit, and the readout is stable (i.e., not fluctuating). Then you should report a result like this:
\(\;\;=\; \left(0.320 \pm 0.005\right) \, \mathrm{A}\)
Why? According to the readout, the value is between \(0.315\,\mathrm{A}\) (rounded up to \(0.32\,\mathrm{A}\)) and \(0.324999\dots\mathrm{A}\) (which is rounded down). So the measurement error is \(\pm 0.005 \, \mathrm{A}\). Note that the estimate is reported as \(0.320 \,\mathrm{A}\) to have the same precision as the error.
When using a device with hatch marks, such as a ruler or analog oscilloscope display, the measurement error is determined by the smallest markings. For example, if the smallest markings on a ruler have \(1\,\mathrm{mm}\) spacing, the measurement error is \(\pm 0.5 \,\textrm{mm}\), so a reading should be reported like this:
\(\;\;=\; \left(6.60 \pm 0.05\right) \, \mathrm{cm}\)
In more complicated situations, you must exercise your judgment. For instance, suppose you have a digital multimeter reading that is not stable: the last digit changes constantly, so that the reading fluctuates between \(0.32\), \(0.33\), and \(0.34\,\mathrm{A}\). The value is between \(0.315\,\mathrm{A}\) and \(0.344999\dots\mathrm{A}\), which is a range of \(\pm0.015\,\mathrm{A}\). Since we use one significant figure for errors, the result is reported like this:
\(\;\;=\; \left(0.33 \pm 0.02\right) \, \mathrm{A}\)
Alternatively, suppose the last digit is changing so fast that you can’t make out its values at all. Then you can report the result like this:
\(\;\;=\; \left(0.35 \pm 0.05\right) \, \mathrm{A}\)
Measurement uncertainties can also come from other aspects of an experiment. Suppose you use a ruler to measure the distance to an object, but the object wobbles by \(\pm 2\,\textrm{mm}\), larger than the \(1\,\textrm{mm}\) hatch marks of the ruler. In that case, you should report a measurement error of \(\pm 2\,\textrm{mm}\), not \(\pm 0.05\,\textrm{mm}\).