Three positions along an optical bench are recorded as , ,

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

Three positions along an optical bench are recorded as x1=100000.0001 mx_1 = 100000.0001\,\mathrm{m}, x2=100000.0002 mx_2 = 100000.0002\,\mathrm{m}, x3=100000.0003 mx_3 = 100000.0003\,\mathrm{m}. Someone computes the population variance directly as Var=⟨x2⟩−⟨x⟩2,\mathrm{Var} = \langle x^2\rangle - \langle x\rangle^2, evaluating both terms in double-precision floating point, and reports Var=−1.9×10−6 m2\mathrm{Var} = -1.9\times10^{-6}\,\mathrm{m^2}, a negative number. (a) Find the error in this reasoning. (b) Compute the correct variance using deviations from the mean.

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