Laboratory Methods
Laboratory Methods is about 6% of the GRE Physics Subject Test. It covers measurement, error analysis, counting statistics, instrument behavior, and experimental design - a small block where points are lost to sloppy uncertainty handling rather than to hard physics.
Core rules
- Random error shrinks with averaging: the standard error of the mean is σ/√N. Systematic error does not shrink with more data.
- Error propagation: for z = x ± y, δz = √(δx² + δy²). For products and quotients, fractional errors add in quadrature; for z = xⁿ, δz/z = n δx/x.
- Counting statistics: N recorded counts carry a standard deviation √N, so the fractional uncertainty is 1/√N. Background subtraction combines uncertainties in quadrature.
- Least-squares fits: slope uncertainty grows with the scatter of the points and falls as 1/√N. A χ² near the number of degrees of freedom indicates an acceptable fit.
- Instrument limits: resolution bounds precision; an oscilloscope's bandwidth and probe capacitance load fast circuits; voltmeters need high impedance and ammeters low impedance.
- Poisson statistics describe independent counting events such as radioactive decays or photon arrivals; Gaussian statistics describe repeated continuous measurements.
- For a fixed count rate, signal-to-noise improves as √(measurement time).
Worked example
A resistor is measured with V = 10.0 ± 0.1 V and I = 2.00 ± 0.02 A. Then R = V/I = 5.00 Ω. The fractional errors are 1% on V and 1% on I, so δR/R = √(0.01² + 0.01²) = 1.4%, giving δR = 0.07 Ω. Report R = 5.00 ± 0.07 Ω; quoting five decimal places would misrepresent the measurement.
Traps
Adding absolute errors when quantities are multiplied; forgetting to take √N for counting data; assuming repeated measurements remove a systematic offset; quoting more significant figures than the uncertainty supports; confusing the standard deviation of single measurements with the standard error of the mean; ignoring how the measuring device loads the circuit under test.