# Lognormal parameters knowing GDP per-capita, Gini coefficient and quintile shares

How can i recover μ and σ for the lognormal distribution (income) knowing mean GDP per-capita (which should be my μ) and the Gini coefficient with data on 5 quintile income shares? Thanks to all.

σ is a number. In the formula for σ = 2/ erf (g), erf is a function defined for values of x that I do not know.

• The parameter $\mu$ of a log-normal distribution is usually not its mean but the mean of its logarithm – Henry Jun 19 '17 at 16:03
• By the way, $\sigma = 2 \, {\rm erf}^{-1}(g)$ in my answer below does not mean $\sigma = 2 / {\rm erf}(g)$. In this context ${\rm erf}^{-1}$ means the inverse function, not one over the value of the function. Just like $\sin^{-1}$ means arcsine, not one over sine. – David Wright Jun 20 '17 at 20:47

So you know the mean $m$ and Gini $g$ of a distribution, and, assuming it is a lognormal distribution, you want to get the lognormal parameters $\mu$ and $\sigma$? That is pretty straightforward.
The mean and Gini of a lognormal are: $$m = \exp(\mu + \sigma^2 /2) \qquad g = {\rm erf}(\sigma / 2)$$ Just invert these equations to obtain: $$\sigma = 2 \, {\rm erf}^{-1}(g) \qquad \mu = \ln(m) - \sigma^2 / 2$$
• @a.russolT: ${\rm erf}$ and ${\rm erf}^{-1}$ are just special functions like log or exp; you can find calculators for them online (keisan.casio.com/exec/system/1180573448). You said you know the Gini. Suppose the Gini is 0.40; ${\rm erf}^{-1}(0.40) = 0.37$, so $\sigma = 2 \times 0.37 = 0.74$. – David Wright Jun 20 '17 at 20:43