Calculating actual quantile from poorly defined lognormal distributions I have a dataset in which the uncertainties in various parameters are modelled with a lognormal distribution. After the experiment I get a single number result. As part of a lookback to understand how good the uncertainty estimation is, I need to be able to calculate what quantile the result represents from our distribution, ie how good were the estimations. The problem is that I only get a 3 datapoints along the curve and a mean, no standard deviation.
For example:
Data about the pre-experiment distribution
90% chance of exceeding 0.10
10% chance of exceeding 0.45
median of 0.21
mean of 0.25
Experiment Result: 0.23
Is it possible to calculate where the 0.23 value would sit in that range (eg, before the experiment we had a 53% chance of exceeding 0.23)?
Is it possible to calculate the standard deviation of the distribution from this information?
I'm doing this in Excel currently, but any suggestions would be greatly appreciated.
 A: There are two possible situations*:


*

*the quantities you know are derived from the same set of parameter estimates, so any two of them should be enough to work out the two parameters of the lognormal from which the quantiles and mean were calculated

*the quantities are sample quantiles and sample mean
*(which two situations my question "How are the three quantiles and the mean obtained?" was meant to elicit the information necessary to tell which it was but unfortunately I couldn't discern this from the response)
I'll give an answer which will work for the first case (while involving more calculation than the minimum necessary that should present no difficulty if you're working in Excel), and which should be a good solution (though not quite optimal) in the second case.
Take logs of the three quantiles. Call the three quantiles on the log scale $q_{0.1}, m , q_{0.9}$.


*

*Find $\hat{\sigma}=\frac{q_{0.9}-q_{0.1}}{2.56}$

*Find $\hat{\mu}= \frac14 q_{0.1} + \frac12 m + \frac14 q_{0.9}$
This gives parameters (or parameter estimates) of the lognormal that should work in either of the above cases. The weights in 2 are rounded versions of optimal weights (they are very close). [Note that this doesn't use the mean at all.]


*

*the standard deviation of a lognormal is the $e^{\mu+\frac12\sigma^2}\sqrt{e^{\sigma^2}-1}$. Just substitute $\hat{\mu}$ for $\mu$ and $\hat{\sigma}^2$ for $\sigma^2$ (note the squaring of $\sigma$!).

*Now let $x_e$ be the experimental result. To work out $S(x_e)$ (the probability of exceeding $x_e$), find $1-\Phi(\frac{\log(x_e)-\hat{\mu}}{\hat{\sigma}})$ where $\Phi$ is the standard normal cdf.
