*Bounty: 50*

*Bounty: 50*

I have data on the **10th**, **25th**, **50th**, **75th**, and **90th** percentiles of a probability distribution, together with the **mean**, and **standard deviation**. I am interested in recovering a continuous distribution that would match or approximate well these data points with a flexible family of distributions.

One possibility is to let the density be flat between the percentiles and choose the 0th and 100th percentiles to match the mean and variance. This procedure, unfortunately, does not lead to reasonable results.

I have to do this many times, but just for concreteness here is one example:

$$p_{10}=-0.89, quad p_{25}= -0.20, quad p_{50}= 0.08, quad p_{75}= 0.33, quad p_{90}= 0.71,$$

with

$$ text{mean} = -0.21 quadtext{and}quadtext{std dev}=2.25.$$

I have tried the Generalized normal distribution, and the Exponentially modified Gaussian distribution, but they do not seem to be able to approximate the percentiles well enough.

I have also tried the method proposed in the answer to this question using many different distributions besides the standard normal, this allows to approximate the percentiles well, but then the standard deviation is always underestimated.

Any suggestions would be very welcome!