# brain sizes: Einstein's and women's

Bob LeChevalier lojbab at lojban.org
Wed Jul 17 03:22:30 EST 2002

```"John Knight" <johnknight at usa.com> wrote:
>Believe me, Parse, you don't need algebra or calculus to calculate the
>statistical average for American girls in TIMSS math.  Even adjusting for
>guesses doesn't require anything but some very basic probability theory.
>
>It's as simple as this:
>
>If you're asked a question which has four multiple choice answers, and you
>haven't got a clue what the answer is, what is the probability of getting a
>correct answer?  Since you have once chance in four of getting the right
>answer, your probability is 0.25.  If you guess on two questions, your
>probability is .5, and three it's .75, and four, it's 1.0.
>
>In other words, over the long run, or over millions of test takers, guessing
>on such a question will yield 25% correct answers, or conversely, every
>fourth answer will be correct.

This makes the assumption that those who know nothing guess randomly.  IN
reality, we don't know that people guess randomly when faced with a test
question they do not understand.  Indeed, we know that they do not.

But the assumption becomes totally meaningless if in fact they know
SOMETHING.  If 100% of them know something, but not enough to solve the
problem, then it is quite plausible that 100% of them will get the answer
wrong.  Thus someone knowing Newtonian physics perfectly will get the wrong
answer on a question that uses special relativity theory.  A good test
designer will know that the Newtonian approximation is a likely error, and
will include that answer among the incorrect alternatives.

>No algebra.  No calculus.  A bit of probability theory, and you already know
>that 25% of all students will get the correct answer if they only *guess* on
>a four part multiple choice question.

But you have no evidence that any kid "guessed" on any problem.

>Now here's the hard part:
>
>Question H04 on TIMSS had four multiple choice answers, so you would think
>that no country or age group or race or sex would answer less than 25% of
>them correct, right?

Wrong.  I would think that if the question were difficult and well designed,
that this would be quite possible.

>How do you think that's possible?
>
>You can probably figure this out with no knowledge of algebra or calculus,
>and you already know all the probability theory that might be needed, so
>what is your explanation?

I've given an explanation, and mine explains how on question D12, both boys
and girls in the US scored less than 17% and South Africans scored only 6.4%
correct.  You can look at the breakdown of the answers and see that the kids
did NOT guess randomly; they intentionally selected particular answers, which
were the wrong ones.

lojbab

```