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