Experimental Methods, Data Analysis, and Measurement
Core concepts
Measurement uncertainty: distinguish random from systematic errors, report results with uncertainty and units, and use significant figures correctly. Combine independent uncertainties in quadrature: for z = x + y, σ_z = √(σ_x² + σ_y²); for products, fractional uncertainties add in quadrature. Mean and standard error: σ_mean = σ/√N. Least-squares fitting, residuals, and χ² assess model quality; R² describes variance explained. Calibration, instrument resolution, and repeat measurements reduce uncertainty. Experimental design: controls, blind protocols, and error propagation through formulas.
Traps
Do not add absolute uncertainties when multiplying-use fractional uncertainties. Systematic errors do not shrink with more trials. Significant figures reflect precision, not a count of digits. Standard deviation of the sample is not the standard error of the mean. A high R² does not prove causation or a correct model. Watch unit conversions and rounding too early in calculations.
Worked example
A length is measured as 5.0 ± 0.1 cm and width 2.0 ± 0.1 cm. Area = 10 cm²; fractional uncertainties 0.02 and 0.05 add in quadrature to ≈0.054, so σ_A ≈ 0.54 cm², reported as 10.0 ± 0.5 cm². Rule of thumb: keep one or two significant figures in uncertainty and match the decimal place of the value.