Plotting Logarithmic Error Bars（如何在log log plot中绘制误差条）

Plotting Logarithmic Error Bars（如何在log log plot中绘制误差条）

Suppose that one has a sufficient number of measurements to make an estimate of a measured quantity y y y and report its absolute error, ± δ y pmdelta y ±δy. The absolute error ± δ y pmdelta y ±δy is represented on a Cartesian plot by extending lines of the appropriate size above and below the point y y y.

If plotted on a logarithmic plot, however, absolute error bars that are symmetric on a y y y vs. x x x plot become asymmetric; the lower portion is longer than the upper portion.

This gives a misleading view of measurement precision, especially when measured quantities vary by several orders of magnitude. To represent error bars correctly on a log plot, one must recognize that the quantity being plotted, which we call z z z, is different than the measured quantity y y y. z = log ⁡ ( y ) z=log(y) z=log(y) The error δ z delta z δz is δ z = δ [ log ⁡ y ] delta z=delta[log y] δz=δ[logy] On the assumption of small errors, a differential analysis can be used δ z ≈ d z = d [ log ⁡ 10 e ⋅ ln ⁡ y ] ≈ 0.434 δ y y delta zapprox dz=d[log_{10}e cdotln y]approx0.434frac{delta y}{y} δz≈dz=d[log10​e⋅lny]≈0.434yδy​ The error δ z delta z δz is thus given by the relative error in y y y: δ z ≈ 0.434 δ y y delta zapprox 0.434frac{delta y}{y} δz≈0.434yδy​ The error bars now display correctly on a logarithmic plot.

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