Table 2. How to convert between the mean, median, mode and standard deviation of log-thresholds and thresholds, assuming that log-thresholds are distributed normally (Fig 1).

Quantity	Units	Symbol	Formulae to convert between these
Mean of log-thresholds	log10 arcsec	M	$M={\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\mu -\frac{1}{2}{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}[1+{\left(\frac{\sigma }{\mu }\right)}^2]$
Standard deviation of log-thresholds	log10 arcsec	S	$S=\sqrt{\frac{{\log }_{10}\left[1+{\left(\frac{\sigma }{\mu }\right)}^2\right]}{\mathrm{l}\mathrm{n}10}}$
Median of log-thresholds	log10 arcsec		= M
Mode of log-thresholds	log10 arcsec		= M
Mean of thresholds	arcsec	μ	μ = 10Mexp(0.5(S ln 10)2)
Standard deviation of thresholds	arcsec	σ	$\mathrm{\sigma }=\mathrm{\mu }\sqrt{\exp ({(\mathrm{Sln}10)}^2)-1}$
Median of thresholds	arcsec		$={10}^M=\mu {\left[1+{\left(\frac{\sigma }{\mu }\right)}^2\right]}^{-0.5}$
Mode of thresholds	arcsec		$={10}^{M-S^2\mathrm{l}\mathrm{n}10}=\mu {\left[1+{\left(\frac{\sigma }{\mu }\right)}^2\right]}^{-1.5}$