aez-notes

An unbiased instrument for measuring distances makes random errors whose distribution has standard deviation \(\sigma\). You are allowed two measurements all told to estimate the lengths of two cylindrical rods, one clearly longer than the other. Can you do better than to take one measurement on each rod? (An unbiased instrument is one that on the average gives the true measure)

Yes, let \(A\) be the length of the longer rod and \(B\) the length of the shorter and measure their total length end-to-end, \(A + B\), and the difference in their lengths, \(A - B\). Solving for \(A\) and \(B\) from the resulting measurements produces values with half the variance than what would be obtained measuring them directly.