How do I fit a distribution to summary statistics of population data? I have a limited set of income data for CountryX. I have the mean, median and percentages (%) of the population that fall within various income brackets. I am trying to work out what percentage of the population earn more than 20k, which is in between a "bracket". I don't think this will fit on a normal distribution or log normal distribution but I'm a bit lost.
The distribution of wealth in this country is very unequal, so any curve used here is very skew. I don't have the standard deviation.
The data looks like this:
Median Income: 6,726
Mean Income:  22,191
Distribution of pop with income under 10k: 63.7%
Distribution of pop with income 10k - 100k: 33.1%
Distribution of pop with income 100k - 1 mil: 3.1%
Distribution of pop with income over 1 mil: 0.1%
 A: A simple approach would be to fit a "reasonable" distribution to the data you have. As you note, any "reasonable" distribution would need to account for the typical skew in income or wealth.
An example is to fit the lognormal distribution to your mean and median, via moment matching - simply solve the equations of the mean and the median for the parameters. The result is not too bad for the quantiles you have. In R:
> med <- 6726
> mm <- 22191
> 
> lognormal_mu <- log(med)
> lognormal_sigma_sq <- 2*(log(mm)-lognormal_mu)
> 
> plnorm(1e4,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq))
[1] 0.6012875
> 
> plnorm(1e5,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq))-
+ plnorm(1e4,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq))
[1] 0.3583857
> 
> plnorm(1e6,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq))-
+ plnorm(1e5,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq))
[1] 0.03972309
> 
> plnorm(1e6,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq),lower.tail=FALSE)
[1] 0.0006036879
> 
> plnorm(2e4,meanlog=lognormal_mu,sdlog=sqrt(lognormal_sigma_sq),lower.tail=FALSE)
[1] 0.2403169

The last line says that this distribution would yield a result of 24% for the quantity you are interested in.
Variations would include using a different distribution, like the Pareto, as cherub suggests. The Pareto is more common for income or wealth data. However, solving the mean and median equations for the two parameters is a bit more tricky. Alternatively, take a look at the gamma distribution, which can also handle skew - but here, there is no closed form for the median, so you would need to use an approximation (e.g., per Choi, 1994, ProcAMS).
Finally, you could also fit your distribution of choice not to the mean and median, but to other pieces of data you have. This would probably involve some sort of numerical optimization, and you would need to weight your information - is it more important that the mean fits, or that the lower quantiles are hit correctly?
You will need to decide whether to run with a simple approach like the lognormal, or whether a more complicated one is worth the additional effort. Keeping in mind that income statistics usually come with their own error, so there comes a point where any additional precision you get by investing in your process is spurious, and dominated by the imprecision in your input data.
