AcademyMeasurement and Vectors
Academy
Measurement Uncertainty
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Uncertainty states measurement resolution, not a mistake to hide.
Notation
\(x\)
measured quantity
varies
\(x_{\mathrm{best}}\)
best estimate
same as x
\(\Delta x\)
absolute uncertainty
same as x
\(\Delta x/|x|\)
fractional uncertainty
none
\(q\)
calculated quantity
varies
Method
An uncertainty interval marks the values the measurement cannot distinguish.
Report interval
\[x=x_{\mathrm{best}}\pm\Delta x\]
Lower end
\[x_{\min}=x_{\mathrm{best}}-\Delta x\]
Upper end
\[x_{\max}=x_{\mathrm{best}}+\Delta x\]
Fractional scale
\[\text{fractional}=\frac{\Delta x}{|x_{\mathrm{best}}|}\]
Product spread
\[q=ab\Rightarrow \frac{\Delta q}{|q|}\approx\frac{\Delta a}{|a|}+\frac{\Delta b}{|b|}\]
Rules
Reported value
\[x=x_{\mathrm{best}}\pm\Delta x\]
Fractional form
\[\mathrm{fractional}=\frac{\Delta x}{|x_{\mathrm{best}}|}\]
Product estimate
\[\frac{\Delta q}{|q|}\approx\frac{\Delta a}{|a|}+\frac{\Delta b}{|b|}\]
Power estimate
\[\frac{\Delta q}{|q|}\approx |n|\frac{\Delta a}{|a|}\quad(q=a^n)\]
Checks
- Absolute uncertainty carries units.
- Fractional uncertainty has no unit.
- Relative uncertainty grows near zero.
- Do not report more precision than the interval supports.