OT - About average and distributions (Was: Kennyboy kills the Zone system for digital in Outdoor Photograpy)

Igor Roshchin str at komkon.org
Sat Aug 8 16:35:12 EDT 2009


Sat Aug 8 13:24:08 CDT 2009
John Francis wrote:

> On Sat, Aug 08, 2009 at 03:38:52AM -0700, Keith Whaley wrote:
> > Igor Roshchin wrote:
> >> I hear people loughing at this phrase and its variations.
> >> In principal, there is nothing contradictory in the statement itself.
> >> It is clear to everybody who knows the basics of statistics (at the 
> >> level of definition of "average", which for a distribution is the same 
> >> as an arithmetic mean).
> >>
> >> As a matter of fact, depending on the distribution you can get any
> >> number (percentage) X ,  0% < X < 100% that can be above (or below)
> >> average. If it is unclear, a simple example can demonstrate it.
> >> Imagine in a class of 100 students, 99 students getting 100% score,  
> >> and one student getting a smaller score, say, 99%.
> >> 99% of students will have their score above average, because average
> >> will be below 100%.
> >>
> >> One should not mix average (arithmetic mean) and median.
> >>
> >> Igor
> >
> > Well stated, Igor! Thanks...
> >
> > keith whaley
> 
> Anyone who knows the basics of statistics should also be well aware
> of the fact that in the real world (rather than in some hypothetical
> exercise) actual distributions generally look pretty similar - tending
> to the "normal" distribution (in both the mathematical and colloquial
> sense of the word).
> 

John, 

I respectfully disagree with the universality of your statement.

That's a typical (mis-)concept that is given in the _basics_ of
statistics. In reality, very often, it is not the case...
There are plenty of publications and specialists discussing this
in applications to economics, social behavior, etc.
To give you a few simple examples: 

1. Pareto (or Bradford) distributions, aka 80-20 rule 
("20% of the population controls 80% of the wealth")

2. Clay Shirky, "Here Comes Everybody: The Power of Organizing 
Without Organizations" talks about power-low distributions (related to
the 80-20 rule).
A good example from that book is given in his talk:
http://www.youtube.com/watch?v=sPQViNNOAkw
(watch the 9th minute (i.e. 8:xx) with the culmination directly
addressing the phrase from which it all started - at about 9:00)
and/or read here:
http://shirky.com/writings/powerlaw_weblog.html

3. "The Black Swan: The Impact of the Highly Improbable" by Nassim Nicholas
Taleb - talks about large impact of random effects that are not
described by the "normal" or Gaussian distribution.

4. From my personal experience:
I am teaching a physics course (introductory mechanics for engineers)
for a rather large class: 80-115 students per semester.
Very seldom the distribution of scores is Gaussian or even close
to it (if ever). I hear the same from my colleagues teaching the same 
or similarly-sized, general subject classes. One can argue that the 
statistical ensemble is not large enough or that it depends on style 
of teaching, etc. There are other factors that explain why the 
distribution is bi- or even tri-modal... 
But the bottom line, - these distributions are not Gaussian.

Cheers,

Igor






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