brain sizes: Einstein's and women's

Cary Kittrell cary at afone.as.arizona.edu
Mon Jul 15 19:40:25 EST 2002

```In article  <Y2lY8.59633\$P%6.3955165 at news2.west.cox.net> "John Knight" <johnknight at usa.com> writes:
<
<http://christianparty.net/timssl10.htm
<
<Question L10 on the 12th Grade TIMSS Math test given to 12th graders around
<the world in 1995 reveals an astounding difference in math skills between
<the sexes in all the countries who participated.  The average difference in
<all countries was 8.3%, with 31.9% of boys and 23.6% of girls answering
<correctly, but the difference in the US was 12.3% (14.9% of girls and 27.2%
<of boys).   In countries like Sweden where 59.8% of the boys answered
<correctly, guesses on the test would not have influenced the scores by that
<much, but where only 14.9% of American girls answered correctly, guesses
<must be taken into account.
<
<Since this was a multiple choice question with five possible choices, the
<probability of getting the correct answer just by guessing is 20%.  In other
<words, for every five students who guessed, one of them would have gotten
<the correct answer by chance.  The maximum score would have been achieved
<had all the students who didn't understand the problem guessed at the
<answer, so where 14.9% of American girls answered the problem correctly, 20%
<of them would have gotten the correct answer if all of them had just guessed
<at the question.  It's not clear how they managed to score lower than if
<they had just guessed, but discovering why may go a long way towards
<understanding what has gone wrong with American "education".
<

Oh, it's trivially clear, isn't it?  The composite score is derived
from three fractional populations:

P1, who knew the answer cold
P2, who thought they did -- incorrectly
P3, who guessed

so obviously any given score may be described as:

P1 * 100 + P2 * 0 + P3 * 20

where P1 + P2 + P3 = 1.

You can only apply a correction for guessing across the board
if you assume that P3 is one hundred percent of the test population.
In point of fact the problem is under-determined, and so there
is no way of to determine the relative sizes of P1, P2, and P3
from the data given.  You make an implicit assumption that all
girls are members of P3, and this leads to subtracting a correction
for random guessing to the entire population resulting in
mathematical absurdities like your claim that in eight of
the twenty-eight questions, girls scored less than zero.

Little speed bumps like "It's not clear how they managed to score lower
than if they had just guessed", or the conclusion that more girls
got the question wrong than took the test, should act as reality
checks, telling you something is wrong with your method.  Applying
some reality checks to my little equation:

if all girls knew, P1 = 1.0 and the composite score is 100.
if all girls knew wrong, P2 = 1.0 and the composite score is 0.
if all girls just guessed, P3 = 1.0 and the composite score is 20.

You assume p3 = 1, which incorrectly penalizes not only the girls who
knew the right answer, but even the girls who knew the wrong answer,
giving the former population less than their correct score of 100 for the
question, and  the latter population less than their correct score of zero.

-- cary

```