[rc5] Re: OS/2 ramdisk and spindown

wooledge at kellnet.com wooledge at kellnet.com
Mon Oct 13 23:52:16 EDT 1997

James Mastros wrote:

> (I
> think.  I can't describe the median of an infinite set, wheras I can
> describe the mean).

Hmm... I'm no mathematician, but perhaps this will work:

The median of a set of real numbers S is the number m such that for any
randomly chosen element n of set S, there is precisely a 0.5 chance that
n < m.

That's probably not rigorous enough for real math, because "randomly
chosen" isn't something I'd care to define except in vague terms.  But
I think it's pretty clear for messy engineers like me. :-)

> In any case, I'm fairly certian that as the sample size increases,
> the mean and median tend to converge.  I could be completly wrong here.  In
> fact, I wouldn't be at all surprised.

In "real life", that may be accurate, but it's not hard to construct
a counterexample.  Consider the sequence {1, 2, 6}.  The mean is 3,
but the median is 2.  Now double the sample size by using the sequence
{1, 1, 2, 2, 6, 6}.  The mean and median have not changed.  Extending
this to an arbitrary 3n-member sequence yields no convergence no matter
how large n is.

Well, that's a pretty ugly example.  Math is supposed to be prettier.
How about this: consider the sequence {2**0, 2**1, 2**2, ..., 2**n} for
n even.  The sum of this sequence is (2**(n+1) - 1).  The mean is thus
((2**(n+1) - 1) / n).  The median is of course 2**(n/2).  The ratio of
the mean divided by the median is therefore

     2**(n+1) - 1      2**(n/2)
    -------------- =~= -------- which increases with n.  Thus, for this
     n * 2**(n/2)         n

sequence, the mean and median diverge (rather dramatically, I'd say).

Maybe I'll stick to computers. :-/

------------                  Greg Wooledge                  -------------
-------                   wooledge at kellnet.com                     -------
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