Variance of a subset of stocks vs. market variance

[DcifrThs]

The stock market is comprised of n different investments (stocks or whatever but i'll stick with stocks). Assuming it has an overall variance (i.e. there IS some central tendency from which to calculate a variance. it could be the average daily movement from time t0-the first day of trading of that market- to time tp-or the present day.), how does this variance relate to a random subset of investments?

specifically, let the subset Sk be comprised of [s1,s2,s3....sk] stocks. further, let this subset of stocks be randomly selected from all n stocks.

now, what is the probability that the VARIANCE of that random subset of stocks is LESS THAN the total market variance of all n stocks??

that is to say each subset has its own variance, i.e. average daily movement from initial public offering to the present day of each individual stock would comprise the mean and then the given day's difference in terms of % change would be the individual observation and this would obviously be summed over the lifetime of the shortest investment in the subset (for comparison the 'market variance' would be similarly computed).

given all that, what is the probability of selecting a subset whose variance as defined above is LESS than the total market variance over the lifetime of the shortest investment in the subset (clearly, if a subset was chosen where the shortest investment lifetime was too short to be significant then either a new subset would be drawn or that investment would be thrown out)

OR, more generally, what is the relationship between the variance of the subset and the variance of the whole over the time period dictated by the lifetime of the shortest lived stock in the subset??

Further, is there a better way to identify this problem? (i may be off base here with the comparisons and what not so somebody please correct me).

Questions? Comments?
-Barron

[adios]

A quick comment. The important factor to consider is the correlation between price movements of an indivual stock with price movements of the overall market. Mostly risk is evaluated in terms of beta. A link to the definition of beta:

Beta Explanation

Evaluating stock market risk and reward is wide open and subject to much debate.

[DcifrThs]

true. and that beta thing you sent me definately comprises some of what i was talking about except does it with the expressed purpose of maximizing utility by relating rates of return with standard deviations (with beta the variance is the denomenator) to optiumum allocations.

My question came from the more general interest in the nature of market variance as it relates to random subsets of its individual components.

specifically, i was reading richard epstein's 'the theory of gambling and statistical logic' where he generalizes many different forms of probability theory and information theory and relates them to gambling. in that work, many interesting relationships arise between different variances and this sparked my initial question.

-Barron