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 and report its absolute error, $\pm\delta y$±δy. The absolute error $\pm\delta y$±δy is represented on a Cartesian plot by extending lines of the appropriate size above and below the point $y$y.

If plotted on a logarithmic plot, however, absolute error bars that are symmetric on a $y$y vs. $x$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, is different than the measured quantity $y$y. $z=\log(y)$z=log(y) The error $\delta z$δz is $\delta z=\delta[\log y]$δz=δ[logy] On the assumption of small errors, a differential analysis can be used $\delta z\approx dz=d[\log_{10}e \cdot\ln y]\approx0.434\frac{\delta y}{y}$δz≈dz=d[log10​e⋅lny]≈0.434yδy​ The error $\delta z$δz is thus given by the relative error in $y$y: $\delta z\approx 0.434\frac{\delta y}{y}$δz≈0.434yδy​ The error bars now display correctly on a logarithmic plot.

Reference: https://faculty.washington.edu/stuve/log_error.pdf