What can we do given percentile values of the data? We are given 20 percentile values say equally spaced percentiles 0 to 100 and also the sample size. Basic things we could do is figure out if this is normal using qq plot, other than that what else can we do with this data ? can we generate random sample ? if its not normal, can we estimate distribution ? If we are approximating it as normal, how can we get mean and standard deviation ?
 A: Knowledge of some of the percentiles of the data (plus the sample size) is equivalent to partial knowledge of the empirical quantile function for the sample.  From this we can estimate the true quantile function of the data, which is equivalent to estimating its true distribution.  This can be done via non-parametric estimation, or it can be done via parametric estimation methods, assuming a particular distributional family.  In the latter case, the parameters of the distribution can be estimated via standard techniques (e.g., maximum likelihood estimation), treating the data as censored values falling within percentile intervals, or approximated them as imputed values in the intervals.
A QQ plot is a way to graphically compare the empirical quantiles to the theoretical quantiles under an assumed distribution (e.g., the normal family), but we can also use formal estimation methods to fit the data to an assumed distributional form.  Once the distribution is estimated, we can generate a random sample from this estimated distribution.  If you would like more specific information on how do maximum likelihood estimation for data from a normal distribution that is censored into intervals, please pose this as a new (specific) question.
