A workshop micrometer gives the diameters of nine ball bearings from the same batch
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Data analysis and model fitting in the laboratory
Measurement and Method, Ch Monte Carlo and the bootstrap: uncertainty by sampling
Problem
A workshop micrometer gives the diameters of nine ball bearings from the same batch, in millimetres (illustrative data, not a measured batch): 6.02,6.05,6.01,6.08,6.03,6.06,6.04,6.09,6.02. No formula free of an assumed distribution shape is available for the uncertainty of a median, so it is obtained by the bootstrap: resample nine diameters with replacement from this list, recompute the median, repeat, and take the spread of the replicate medians. A spreadsheet carried out B=10 such resamples (illustrative, and kept to ten so that the arithmetic can be done by hand; seed stated as 7) and returned these ten replicate medians, in millimetres: 6.02,6.04,6.05,6.03,6.04,6.06,6.02,6.05,6.04,6.03.
((a)) Find the median of the original nine diameters.
((b)) Treating the ten replicate medians as the readings in hand, compute their mean and their sample standard deviation with the divisor B−1; this is the bootstrap standard uncertainty of the median.
((c)) Report the diameter's median with its bootstrap uncertainty in the standard xˉ±u(x) format.