Types of error
- Systematic error: a consistent bias in one direction (zero offset, uncalibrated instruments, parallax). Repetition does not reduce it; calibration does.
- Random error: unpredictable scatter, reduced by averaging repeated readings (standard error = s/√n).
- Gross errors: human mistakes such as misreading a scale.
Accuracy and precision
Accuracy is closeness to the true value (linked to systematic error); precision is closeness of repeated readings to each other (linked to random error). Tightly grouped shots far from the bullseye are precise but inaccurate.
Uncertainty (GUM)
- Type A: statistical analysis of repeated measurements.
- Type B: other information such as specifications or resolution; a digital reading with resolution d contributes d/(2√3).
- Combined standard uncertainty: root-sum-of-squares of components.
- Expanded uncertainty: U = k·uc, usually k = 2 (about 95% coverage).
Error propagation
For sums and differences of independent quantities, add absolute uncertainties in quadrature; for products and quotients, add relative uncertainties in quadrature, e.g. q = xy gives (δq/q)² = (δx/x)² + (δy/y)². Round the result to match the uncertainty's decimal place. When areas or volumes are approximated by slicing, as in the method of exhaustion, the approximation error must be considered too.
References
- JCGM 100:2008. Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM). BIPM.
- Taylor, J. R. (1997). An Introduction to Error Analysis (2nd ed.). University Science Books.
- JCGM 200:2012. International Vocabulary of Metrology (VIM) (3rd ed.). BIPM.
- Bevington, P. R., & Robinson, D. K. (2003). Data Reduction and Error Analysis for the Physical Sciences (3rd ed.). McGraw-Hill.