Possible to estimate distribution around mean given #samples, mean, highest and lowest? For a project for which there are multiple bidders, the following is known:


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*Number of bidders: 24

*Mean bid: 104

*Highest bid: 356 

*Lowest bid: 20


Given the above, is it possible (however roughly) to estimate (i) the number of bids above and below the mean and (ii) the median?
 A: No. There is not enough information in the descriptive statistics you have provided to estimate 1) number of bids above and below mean and 2) median.
For 1), you need information regarding the overall shape of the distribution. At the very least you'd want information regarding whether the distribution is symmetrical around the mean or not.  This information could be expressed as a Bernoulli trial: $p = Pr(x > \bar x )$.  Obviously, if you knew $p$, you could multiply it by the total number of bids to estimate the number of bids above the mean, and the subtract from the total number of bids for the reverse.
Are $x_{min}$ and $x_{max}$ informative regarding $p$?  Not sufficiently informative, unfortunately. (EDIT: they do provide a sliver of information that allow you to rule the most extreme estimates for 1). But in most cases, this sliver of information is not very useful.)
For 2), you also need information regarding the overall shape of the distribution.  For example, it would help to know the number of modes and their positions, and some information about the skew of the distribution.

If you have out-of-band information (e.g. from simulations) regarding the shape or type of distribution (e.g. Beta distribution) then you can combine this prior information with these descriptive statistics to estimate 1) and 2).
