Can quantiles be calculated for lognormal distributions? I just talked to someone who stated that quantiles cannot be computed for lognormal distributions. or it does not make sense.
Is this true?
 A: 
Let's start with definitions and notation. A random variable $X$ is lognormal if its natural logarithm, $Y = \log(X)$, is normal.
Denote with $M$ and $S$ the mean and standard deviation of $X$. Denote with $m$ and $s$ the mean and standard deviation of $Y$. Given $M$ and $S$, you can calculate $m$ and $s$ as: $m = \log[M^2/(M^2 + S^2)^{(1/2)}]$ and $s = (\log[(S/M)^2+1])^{(1/2)}$.
To calculate a quantile of $X$, we use the fact that the exponential function (inverse of the log function) is monotone increasing -- it maps quantiles of $Y$ into quantiles of $X$. Suppose we want to calculate the .95-quantile of $X$ (nothing special about .95, substitute any quantile you like). Let $Q$ denote the .95 quantile of $X$. Let $q$ denote the .95 quantile of $Y$. We know the mean and standard deviation, $M$ and $S$, of $X$. From these, we calculate the mean and standard deviation, $m$ and $s$, of $Y$. Since $Y$ is normal, we can easily calculate its .95 quantile $q$. The .95 quantile $Q$ of $X$ is then simply: $Q = \exp[q]$.

here is the original post by Glyn Holton: http://www.riskarchive.com/archive02_4/00000622.htm
A: I am not a statistician, but I am quite sure that the quantile function for the log-normal distribution is well-defined because it is the inverse of the cumulative distribution function, which is strictly increasing.

For all continuous distributions, the ICDF exists and is unique if 0 < p < 1.
  (source)

There is a software library (distributions-lognormal-quantile) I have used in some applications to evaluate that function, and I believe it uses this equation:

This function is also available in Microsoft Excel as LOGNORM.INV.
A: Here is the proof. Take $\log X \sim \mathcal{N}(\mu, \sigma)$. Then $X$ is log-normally distributed with CDF:
$$
  F(x) = \frac{1}{2}\left(1 + erf \left(\frac{\log x - \mu}{\sigma \sqrt{2}} \right) \right) 
$$
we can now solve:
\begin{align}
  x &= \frac{1}{2}\left(1 + erf \left(\frac{\log F^{-1}(u) - \mu}{\sigma \sqrt{2}} \right) \right) \\
  erf^{-1} \left(2x-1\right) &=  \frac{\log F^{-1}(u) - \mu}{\sigma \sqrt{2}} \\
  \sigma \sqrt{2} erf^{-1} \left(2x-1\right) +\mu &=  \log F^{-1}(u) \\
   \exp\left(\sigma \sqrt{2} erf^{-1} \left(2x-1\right) +\mu\right) &=  F^{-1}(u) \\
\end{align}
which is what iX3 got.
