Table A-II.

Formulas for estimating summary statistics from OEMD data that include N.

Input values	Estimate	Formula
Method #1′: v = 5 values, m = 6 equations (eqs. A1a, A1b, A5, A8a, A8b & A9), s = −1 (over-determined solutions)
N, Min, Max & AM	$\widehat{AM}=$	AM or $\mathit{GME}\sqrt{{\widehat{\mathit{GSD}}}^{ln\widehat{\mathit{GSD}}}}$
	$\widehat{\mathit{GSD}}=$	${\left(\sqrt{\mathit{Max}/\mathit{Min}}\right)}^{1/E_N\left[z_{\mathit{Maz}}\right]}$
	$\widehat{GM}=$	GME or $AM/\sqrt{{\widehat{\mathit{GSD}}}^{ln\widehat{\mathit{GSD}}}}$
Method #2′: v = 4 values, m = 6 equations, s = 0 (exact solution)
N, Min & Max	$\widehat{GM}=$	GME
	$\widehat{\mathit{GSD}}=$	${\left(\sqrt{\mathit{Max}/\mathit{Min}}\right)}^{1/E_N\left[z_{\mathit{Maz}}\right]}$
	$\widehat{AM}=$	$\widehat{GM}\sqrt{{\widehat{\mathit{GSD}}}^{ln\widehat{\mathit{GSD}}}}$
Method #3′: v = 3 values, m = 6 equations, s = 1 semi-empiric parameters (GSD)
N & Max	$\widehat{GM}=$	$\mathit{Max}/{\overline{\mathit{GSD}}}^{E_N\left[z_{\mathit{Max}}\right]}$
	$\widehat{AM}=$	$\mathit{Max}\sqrt{Q^{-2E_N\left[z_{\text{Max}}\right]}}$, where $Q={\overline{\mathit{GSD}}}^{ln\overline{\mathit{GSD}}}$
	$\widehat{\mathit{GSD}}=$	$\overline{\mathit{GSD}}$