Subject: minor study about variances at IMPs

I looked at 2959 boards played on OKbridge at IMPs.  
The average standard deviation of IMPs on a board 
was 5.435083.  The average variance was 31.695895.

The IMP scores were computed by IMPing across the 
field and averaging; no scores were thrown out.

After a little huffing and puffing, I remembered enough
statistics to work out the following: 

   In a 24-board match, the probability that the 
   better team will win is P (in %) if their expected
   result on a board was m (in IMPs/bd):

	m (IMPs/bd)	P (%)
	- ---------	- ---
	     0		 50
	     .25	 59
	     .50	 67
	     .75	 74
	     1.0	 81
	     2.0	 96
	     3.0	 99.6

Derivation: The central limit theorem says that 
n independent distributions with means m and 
variances s^2 added together approximate a 
normal distribution with mean nm and variance
ns^2.  That means that 
	P( ((X1+X2+...+Xn)-nm)/(s sqrt(n)) < x) 
	         is approximately
	P( Z < x), where Z is a unit normal.

So, P( Z < sqrt(n) m/s) is the probability that
the better team will win the match (assuming positive m.)

Letting s^2 = 31.695895,
	n   = 24,
	x   = 0,
we get: P( Z < .87 m) is the probability that the better
team will win, which gives rise to the table above.

Extending this simply to n = 7, 24, 32, and 64, gives us

	7 bd match: P( Z < .47 m)
	24 bds:     P( Z < .87 m)
	32 bds:     P( Z < 1.00 m)
	64 bds:     P( Z < 1.42 m)
	128 bds:    P( Z < 2.00 m)

Finally, we get the probability (in %) that a team with 
mean m (in IMPs/bd) on each board will win a match of n boards is:

	m\n	7	24	32	64	128
	---	-	--	--	--	---
	0	50	50 	50	50	50
	.25	55	59	60	64	69
	.50	59	67	69	76	84
	.75	64	74	77	86	93
	1.0	68	81	84	92	98
	2.0	83	96	98	99.77	99.99+
	3.0	92	99.6	99.87	99.99+	99.99+

Conclusions and handwaving:

A rough guess from experience at OKbridge tells me that
a national champion is only about 1 IMP/bd better than a
good flight A player.  Flight A players are, on average,
I think, about an IMP/bd better than Flight B players and
the difference between Flight B and Flight C is also about
one IMP.  The difference between the best player in the 
world and an awful one is probably not much bigger than 
about 4 IMPs.  That means that, in a seven board match, 
a team of average Flight C players will beat a team of
national experts about 8% of the time.  In a 24-board match,
those same Flight C players have almost no hope of beating
the experts and in a longer match, it is out of the question.
In a 32-board Spingold match, the chance of a team of random
Flight A players beating one of the top 20 seeds looks to be
in the neighborhood of 10-15%.  A team of Flight B players,
on the other hand, have 0-2% chance of beating a seed in the
Spingold.  A team of Flight C players have no chance to make
it past the first round; they would not usually win one in
their lifetimes, even if they played the Spingold every year.

The ACBL's handicap scheme only gives 1 IMP/bd to a Flight A/
Flight C matchup.  This is obviously too small.  1.5 IMPs/bd
would give the weaker team about 1/3 chance of winning, which
seems about right.  

Thanks to Matthew Clegg and okbridge for the sample data.
				--Jeff