[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 -------------
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