Lognormal distribution from world bank quintiles PPP data I am not a stats person.  
The world bank has data giving PPP (personal purchasing parity, or something like that) for quintiles (actually first 10%, second 10%, 2nd, 3rd, 4th 20%, 9th 10% and 10th 10%) of a country's population.
I've been reading a good deal about the Gini index and also modeling income distribution curves for populations, and I would like to turn this quintiles data into a lognormal distribution representing the same properties (e.g. that if China has 1,300 USD PPP for the 1st 10%, that after modeling it, the weighted average PPP (integration under the curve?) for the poorest 10% of the lognormal distribution would come out to 1,300).
Any thoughts on how to do this, tactically?  I am cannot think my way through this  - but I am a reasonable programmer of simple scripts and would use python's scipy and numpy to fit curves.  Given some help on how to proceed.

@mpiktas, you are right, they do not give maximum income.  The quintiles/decile information must be averages.  I do not think you can straightforwardly fit this data as if it is raw data, to a distribution, if that were the case I would not have asked the question!  
@Michael, given that the quintile/decile data points are average values for the entire quintile/decile bracket, is it possible still to come up with a best fit?  Is this what you meant by least squares fitting?
 A: Here is the example of the quick and dirty R code to illustrate what Michael suggested:
Define quantiles available:
q<-c(0.1,0.2,0.4,0.6,0.8,0.9)

Create artificial data and add some noise
data <-jitter(qlnorm(q))

Create function to minimise
fitfun <- function(p)sum(abs(data-qlnorm(q,p[1],p[2])))

Run the optimiser with the initial guess of parameters of log-normal distribution:
opt <- optim(c(0.1,1.1))

The parameters fitted:
Display the fit visually:
aa<-seq(0,0.95,by=0.01)
plot(aa,qlnorm(aa,opt$par[1],opt$par[2]),type="l")
points(q,data)


Note, I intentionally plotted only 95%-quantile, since the log-normal distribution is unbounded, i.e. the 100%-quantile is infinity. 
Usual caveats apply, real life example might look much uglier than this one, i.e. fit might be much worse. Also try Singh-Maddala distribution instead of log-normal, it works better for income distributions.
A: I'm giving another answer, since more details about data were given. From the initial question it seemed that some quantiles are observed but that is not the case. The data is calculated in the following form.


*

*Calculate the total income of all population

*Divide population into income groups

*Calculate the total income of population in groups defined in previous step.

*Report for each group the proportion of total income in the group relative to total income of all population.


Suppose the population income is distributed according to unknown distribution function $F$. For the data the following income groups are defined:


*

*Population with income in a range of $[0,F^{-1}(0.1))$

*Population with income in a range of $[F^{-1}(0.1),F^{-1}(0.2))$

*Population with income in a range of $[F^{-1}(0.2),F^{-1}(0.4))$

*Population with income in a range of $[F^{-1}(0.4),F^{-1}(0.6))$

*Population with income in a range of $[F^{-1}(0.6),F^{-1}(0.8))$

*Population with income in a range of $[F^{-1}(0.8),F^{-1}(0.9))$

*Population with income in a range of $[F^{-1}(0.9),\infty)$


For each of this range the following proportion is reported:
$$\frac{n_r\int_{l_r}^{u_r}xdF(x)}{N\int_{0}^{\infty}xdF(x)},$$
where $n_r$ is the number of people in a range $[l_r,u_r)$ and $N$ is the total population. The nominator of the fraction is number of people in a range times the average income in the range. Denominator has total number of people in the range times the average income. 
Since the ranges defined are quantiles, proportions $n_r/N$ are known, i.e. for the first two and the last two ranges the proportion is equal to 0.1, for the rest 0.2. 
The integral in nominator can be expressed in more convenient form:
$$\int_{l_r}^{u_r}xdF(x)=\int_{F(l_r)}^{F(u_r)}F^{-1}(u)du$$
The most obvious way to fit the data would be to integrate $F^{-1}$ numerically to a given range (or calculate the integrals analytically, which might be a challenge). Then calculate the proportions and fit them using your criterion of choice, least squares, least absolute deviations, etc. Note that one proportion is redundant since the proportions sum to one. Another caveat is that you need to know average income of the population, which is not given in the data.  
A: A lognormal distribution is determined by two parameters, the mean and the variance of the related normal distribution. If you have raw data you could fit a lognormal distribution by maximum likelihood.  If not you can use a fit criterion such as least squares or minimum sum of absolute errors to fit the given percentiles (quantiles) to values of a lognormal fit for these percentiles.
A: A log-normal distribution is fully defined by the pair of parameters $\mu$ and $\sigma$. Since you want to fit this distribution to your data, it's sufficient to estimate these two values. Normally, you would have access to the raw data, and would apply the standard the maximum likelihood estimators (MLEs) for $\mu$ and $\sigma$, which are straightforward:
$$\hat{\mu} = \frac{1}{n}\sum_i \ln(y_i) = \langle \ln y \rangle\\
\hat{\sigma}^2 = \frac{1}{n}\sum_i (\ln(y_i)-\hat{\mu})^{2} \enspace .$$
That is, $\mu$ is the mean of the logarithm of your observed data $\{y_i\}$, and $\sigma$ is the standard deviation of the logarithm of the data.
But in this case, you don't have the raw data. Instead, you have some sketchy information about the cumulative distribution function (CDF). Very roughly, what you know the fraction of the distribution $\Pr(y)$ that is below some $y$ for some set of values $\{y_i\}$. You can still estimate the log-normal parameters (or those of any other distribution) from this kind of information, but there are subtleties. 

Two approaches come to mind. The first is a quick and dirty one that will not produce entirely accurate parameter estimates, but will get you close enough to get a sense of what the distribution looks like and, if you want, roughly what the Gini coefficient would be. The second is more complicated and more accurate for the kind of data you have.

Quick and dirty approximation
Here's the quick and dirty solution. The information you have is a "binned" version of the CDF, represented by a set of pairs $(q_i,y_i)$, where $q_i$ is the fraction of the distribution at or below the value $y_i$ (note: you said that the PPP is an average within the bin, which is technically distinct from the CDF, but for our calculation, that distinction doesn't make a difference).
Now, recall that the definition of the mean is
$$\langle x \rangle = \sum_i x_i \Pr(x_i)\enspace ,$$
where $\Pr(x_i)$ is the probability of observing $x_i$. We don't have $\Pr(x)$, but we can approximate it using the binned CDF information, like this
$$\hat{\mu} \approx \sum_{i=1}^k \Delta q_i \ln x_i$$
where $\Delta q_i=q_{i+1} -q_i$ is the size or width of the $i$th bin, out of $k$ bins. Similarly, for the standard deviation, the definition is
$$\sigma = \sum_i (x_i-\langle x \rangle)^2 \Pr(x_i)\enspace,$$
which becomes
$$\hat{\sigma} \approx \sum_{i=1}^{k} \Delta q_i (x_i-\hat{\mu})^2  \enspace .$$
To apply these to your data, you'll need to let $x_i=\ln y_i$ since you're working with the log-normal distribution, rather than the normal (or Gaussian) distribution. Coding up these estimators should be fairly easy.
In my numerical experiments with these estimators, I consistently get slight errors in the estimates relative to the underlying or "population" values I used to generate synthetic log-normal data. If you use these with your data, you should not treat the estimated values as being highly accurate. To get those, you'd need to apply a more mathematically sophisticated approach, which I'll sketch for you now.
Maximum likelihood approach
The more complicated and more accurate solution is to derive the maximum likelihood parameter estimate for the particular representation of the log-normal distribution you have, i.e., the binned CDF. The definition of the log-normal PDF is
$$\Pr(x) = \frac{1}{x\sigma\sqrt{2\pi}}{\rm e}^{-\frac{(\ln x - \mu)^2}{2\sigma^2} } \enspace ,$$
and the CDF is
$$\Pr(x<X) = F(x) = \frac{1}{2}\left(1+{\rm erf}\left( \frac{\ln x - \mu}{\sigma\sqrt{2}} \right) \right) \enspace ,$$
where $\textrm{erf}()$ is the error function, and where we let $F(x)$ be a short-hand representation for the CDF. (Normally, we would say $F(x\,|\,\mu,\sigma)$ to indicate that $F$ depends on your parameter choices, but I'm going to drop that notation henceforth; just remember that it's implied.) This is relevant because you want to assume your quantile data were drawn from a binned version of this distribution. If $F(x)$ is the CDF, i.e., the integral of $\Pr(x)$ from $-\infty$ to $x$, then let $F(x\,|\,a,b)$ be the integral of $\Pr(x)$ from $a$ to $b$. (Mathematically, $F(x\,|\,a,b)=F(b)-F(a)$.)
The log-likelihood of your observed quantile information is then
$$\ln \mathcal{L} = \sum_{i=1}^k \ln F(x_i\,|\,q_i,q_{i+1})\enspace .$$
The more sophisticated approach would be to estimate $\mu$ and $\sigma$ by maximizing this function over these parameters. This would give you the maximum likelihood estimate for your log-normal model, given the observed information you have. For arbitrary choices of $\{q_i\}$, an analytic solution for the MLE is not possible, but for regularly spaced choices of the bin boundaries, it may be. Regardless, however, you may always numerically maximize the function (which many numerical software packages can do for you, if you whisper the right words to them).
What makes this approach more complicated is that you need to get the mathematics correct when you code up the numerical routine to do the estimation with the data. If the accuracy of your answers is really important, then this approach might be worth the extra effort.
