Data Treatment, Error Analysis, and Graphical Methods
Data Treatment, Error Analysis, and Graphical Methods
This domain tests whether your reported uncertainty is derived, not guessed, and whether your graph supports the model you claim.
Core rules
- Carry extra digits through the calculation and round only the final result. For sums, add absolute uncertainties; for products and quotients, add relative uncertainties in quadrature for independent random errors.
- Separate random from systematic error. Replicates reduce random error, not bias; a certified reference material or blank reveals bias.
- Least-squares regression estimates slope and intercept, but inspect residuals for curvature and heteroscedasticity. Weighted regression is required when uncertainty varies with x. Do not force the intercept through zero unless the model demands it.
- Report n, standard deviation, standard error, and a confidence interval using t at the appropriate degrees of freedom. Use Grubbs’ test before rejecting an outlier, and state the reason.
- Linearise deliberately: ln k vs 1/T, ln A vs t, or 1/[A] vs t. Propagate uncertainty through the transformation.
Worked example
A stock solution is 0.1000 ± 0.0005 M. You pipette 1.00 ± 0.01 mL into a 100.0 ± 0.1 mL flask. Relative uncertainties are 0.50% for the stock, 1.0% for the pipette, and 0.10% for the flask. Combined relative uncertainty = √(0.50² + 1.0² + 0.10²) = 1.12%. The diluted concentration is 1.00 × 10⁻³ M, so the absolute uncertainty is 1.1 × 10⁻⁵ M; report (1.00 ± 0.01) × 10⁻³ M, not 0.001 M.
Traps
Adding absolute uncertainties when multiplying; quoting r² as proof of linearity; using an unweighted fit for data with unequal errors; over-rounding intermediate values; discarding an outlier without a statistical test.