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

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

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