Formula 1:

\[ P = \frac{2(2T - G)}{mtan} \]

with \(T = 0.5 - L(0.5)\) and \(mtan = \frac{md}{\mu}\), where \(L(0.5)\) is the income share of the bottom 50%, \(mtan\) is the slope of tangent to the Lorenz curve at the 50th percentile (median), \(md\) is the median, \(\mu\) is the mean, and \(G\) is the Gini coefficient (Wolfson 1994).

Formula 2:

\[ P = \frac{2({\mu}^* - {\mu}^L) }{md}, \]

where \(\mu^*\) is the distribution-correction mean (i.e. \(\mu(1 - G)\)), \(\mu^L\) is the mean of the bottom 50% of the population, and \(md\) is the median (Ravallion and Chen 1996).

\[\begin{align} P = &\frac{2(2T - G)}{mtan} \\ \\ = &\frac{2(2(0.5 - L(0.5)) - G)}{md} * \mu \\ \\ = &\frac{2(1 - G -2L(0.5))}{md} * \mu \\ \\ = &\frac{2(\mu(1 - G) -2\mu L(0.5))}{md} \\ \\ = &\frac{2( {\mu}^* - {\mu}^L)}{md} \end{align} \]

Thus, \({\mu}^* = \mu(1 - G)\) and \({\mu}^L = 2\mu L(0.5)\).

Alternatively, \({\mu}^L = md*(1 - P_1(md)/0.5)\).

1. Illustrate \({\mu}^L = 2\mu L(0.5)\) with a synthetic distribution, \(y = [1, 2, 3,…,100]\)

   

2. With the same distribution, \({\mu}^L = md*(1 - P_1(md)/0.5) = 50.5*(1 - 0.2475/0.5) = 25.5\)

Source: Wolfson (1994)