Wednesday, August 06, 2008

Small Differences in Variability of Ability Translate Into Big Differences 3-4 Std. Deviations from Mean

Results of a statistical experiment, to follow up on these two CD posts(link and link):

1. Generate a random sample of 1,000,000 observations in EVIEWS, distributed normally with zero mean, and variance of 1, representing female mathematical ability on a standardized test.

2. Generate a sample of 1,000,000 observations, distributed normally with zero mean, and variance of 1.21, to represent male mathematical abililty, with greater variability according to the table above for Grade 8.

3. Then look at the upper tail of each distribution and compare the M/F ratio for the "super-genius" level, many standard deviations above the mean.

4. For 3 standard deviations above the mean, the M/F ratio is 2.4 (3,188 "males" vs. 1337 "females").

5. For 4 standard deviations above the mean, the M/F ratio increases to 3.8 (111 males to 31 females).

6. For 5 standard deviations above the mean, there are 3 males and o females.

Assume that to be successful and get tenure in the math department at Harvard or MIT, you have to be 3-4 standard deviations above average, which is what
Larry Summers said - "We're talking about people who are three and a half, four standard deviations above the mean...."

In that case, wouldn't we expect females to be under-represented, as the experiment above demonstrates, where "men" represent 77% of those observations 4 or more standard deviations above the mean?


See Alex Tabarrok's related discussion here on Marginal Revolution.

11 Comments:

At 8/06/2008 5:12 PM, Anonymous QT said...

MP,

Thanks for your thoroughness on this subject. I believe that we are all pretty much on board with the conclusion.

Would very much like your opinion on another key question: what action should central banks in Asia take to address rising inflation?

 
At 8/06/2008 6:37 PM, Anonymous Anonymous said...

Well, since there's no reason that ability should follow a normal distribution (and it doesn't), I would disagree with your argument, although not necessarily with the conclusion.

 
At 8/06/2008 7:50 PM, Anonymous Anonymous said...

Nice work Mark. It's also worth noting that the lower tail would also be heavily male, a phenomenon that may explain other things we observe in society (e.g., the disproportionate involvement of men in crime).

 
At 8/07/2008 3:45 AM, Blogger Sophist said...

Summers was correct.

When someone is correct about such issues will receive attacks. The Sophist has tried for many years to explain this, i.e. the fact that when people talk about major truths based on facts they get viciously attacked. The Sophist has come to this conclusion: people attack facts because they want to live not in the best of all possible worlds but in the best of all thinkable worlds. This is taught to them by religion, parents, schools, fairy tales, etc.

This issue is extremely big. MP has the guts to discuss it but we must take it further with some potential conclusions:

(1) Anon @ 7:50 is absolutely correct. I would say that increased math ability that contributes to progress in science is counterbalanced by deviating behavior of men contributing to all sorts of crime and aggressive behavior that pushes society backwards.

(2) Insanity levels: someone should look into distributions of paranoia and schizofrtenia in men and women to see if they math the math ability ones. Recall that most great mathematicians were hospitalized, starting from Goedel and going to Nash.

(3) In general, which distribution does a modern society prefer? I am not sure.

 
At 8/07/2008 6:02 AM, Blogger Walt G. said...

Sophist said,

“Summers was correct.”

I don’t think you can go that far empirically. How about this, “The chances of Larry Summers being incorrect are extremely low”? I don’t think we are dealing with facts here, but we are dealing with an accepted method of analysis. To find your statement false, I would only have to find one possibility that it does not hold in the entire universe to refute it; even four standard deviations away from the mean does that. I didn't learn that from a fairy tale. Be careful where and what you attempt to pass off as "facts."

 
At 8/07/2008 7:57 AM, Anonymous Anonymous said...

walt g., you sweat too much over nothing. You type too much to say nothing. You say nothing thinking you say something.

 
At 8/07/2008 8:16 AM, Blogger Walt G. said...

Pretty tough talk from someone who is afraid to show his or her face.

 
At 8/07/2008 10:05 AM, Anonymous QT said...

Sophist,

The supposed correlation between insanity and genius is one of those fairy tales. The realities of the subject are much more complex than such simplistic generalizations.

The condition is Schizophrenia not "schizofrtenia".

Walt g,

Well said.

 
At 8/07/2008 10:27 AM, Blogger Sophist said...

qt,

where did you see in my post me alluding to a correletion between genious and insanity?

I only spoke about (in)sanity distributions of men and women and whether those match the math ability ones. I do not know the answer, I said someone must look, if haven't done so already.

If you are to be taken seriously, you must try not to interpret other posts in ways that suit your impulsive way of replying.

 
At 8/07/2008 5:34 PM, Anonymous qt said...

sophist,

What's the diff? It isn't as if you are going to actually address any comment whatever.

You're too busy being original.

 
At 8/11/2008 1:56 AM, Blogger OBloodyHell said...

Link added:

> where did you see in my post me alluding to a correletion between genious and insanity?

snicker...

sophist, the unintended humor that keeps on giving.

 

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