Wednesday, October 3, 2007


All about the abuse of anecdotal math to falsify the truth and truthify falsehood.

This is the fourth part of my "presentation" on the topic of "Lies, ..." Click for Part 1, Part 2, and Part 3.

This part is about the "Normal Curve".

The height of young American women ranges from about 4' 9" to 6'. For young men it is 5' 2" to 6' 5". That's a difference of about five inches -- less than ten percent.

Therefore, in basketball and other sports where height is critical, you'd expect about ten percent fewer women than men. Right?

Anything less would be proof of discrimination against women. Right?


Actually, if you had a cut-off of six feet, over 100 men would qualify for every woman who qualified! Even if you had a cut-off of 5' 7", which is the average height of the population of young men and women combined, you'd find over five men for every woman who qualified.


Why are Our Expectations Wrong?

Glad you asked!

You have probably heard of the "Normal Curve" or the "Bell-Shaped Curve" and if you stay tuned for a bit you will understand what that is and why it is important. I promise to keep the math to a minimum and the understanding to a maximum.

The curve is called "Normal" because, when you make lots of measurements, such as the heights of a bunch of random people, you normally get a "Bell-Shaped Curve"!

Most of the measurements will be near the average value and you will get fewer and fewer as you go further away from the average.

[Click figure for larger view] The figure shows the Normal curve for the heights of young women (in red) and young men (in blue). Notice how each group of measurements resembles the shape of a bell?

Please look at the red bars that represent measurements of a thousand young women. Nearly all of them are between 57" and 72", a range of fifteen-inches. No more than five out of a thousand will be below or above that range. If you divide that range into six equal increments of two and a half inches each, about two-thirds of them will be in the two increments closest to the middle.

As indicated in the figure, in ordinary English, we would say women in that range are of "average" height.

On either side of the "average" are increments for "short" and "tall". Out of 1000, there will be about 136 "short" women and 136 "tall" women.

On either side of "short" and "tall" are "very short" and "very tall". Out of a 1000, there will only be about 21 "very short" and 21 "very tall".

The handful of women who fall outside the range would be called "extremely tall" and "extremely short".

The same situation prevails for young men, shown in blue. But note: the measurement results are shifted two increments to the right. A woman we'd call "tall" or "very tall" would be "average" if she were a man. Similarly, a man we'd call "short" or "very short" would be "average" if he were a woman.

None of the above is controversial. These are simply the facts that can be verified by anyone who would like to do the measurements.

Mathematical Terms (I'll keep this very short :^)

Mathematicians call the area that contains 68.3% of the measurements the "plus or minus one standard deviation" range. Since standard deviation is usually represented by the Greek letter “sigma”, this is called the “one-sigma” range.

A mathematician would analyze the height measurements and calculate the standard deviation as 2.5". He or she would note that the average for males is 5" above that for females and conclude that males are two standard deviations taller than females.

Please don't worry about the math terms "standard deviation" and “sigma" too much. These terms are just a fancy way of saying where to expect 68% of the measurements to be.

Representation of Women in Sports

For basketball, height is obviously a critical factor. If high schools and colleges insisted on having unisex teams, we'd find boys and young men outnumbering girls and young women by one-hundred to one! That would not be fair to girls and young women who want to play sports. That is why it makes total sense to separate basketball teams by gender.

Since height often correlates to strength and speed and other factors that are important in baseball, football, soccer and many other sports, it also makes sense to separate those sports by gender. In fact, only a small number of sports (gymnastics comes to mind) favor participants who tend to be shorter.

Bottom Line

1) In high school, college, and other amateur play, I favor separation by gender in the sports where males have a significant advantage. I would make an exception for the few girls and women who could qualify and allow them to join the male division if they wanted to.

2) For professional sports, I would make it illegal to exclude women from the highest level in any given sport. There are women who qualify, and, however few their number, it is unfair to exclude them. On the other hand, for the lower levels of professional sports, I would allow separation by gender to give highly qualified women a fair chance to play at their level.


OOPS - here is where we may get "politically incorrect".

Now that you understand all about the Normal curve and standard deviation and so on, let us apply our newfound knowledge to a different domain.

Lots of well-meaning people are misinformed about standardized tests, particularly those that are said to measure "intelligence".

I'll be the first to admit that some college graduates with advanced degrees don't have the intelligence to "rub two sticks together to save their lives". Some with the highest academic honors could not survive more than a few days in the woods or on the streets of a big city.

Some PhDs are at a total loss when it comes to doing carpentry or plumbing or fixing a TV set or PC or a car. They cannot grow fruits and vegetables and would be a total failure at "animal husbandry" (whatever that is :^) Some of them have no social intelligence at all and cannot sing on key or play a musical instrument. I would not want to "have a beer" with many of them.

The standardized so-called "Intelligence Quotient" (IQ) test does not measure any of the above talents.

However, IQ tests do a damn good job of evaluating "normal" people as to their ACADEMIC INTELLIGENCE.

People with high IQs generally excel in high school and college. They also excel at jobs that require lots of reading and writing and designing and science and math and so on.

People with low IQs generally do not do well in school and they find employment in fields that do not require academic-type talents.

Design of IQ Tests

IQ tests are designed to yield a score of 100 for the average person and to have a standard deviation of fifteen points. If you give IQ tests to a thousand people (in their native languages), all but a handful will fall between 55 and 145.

Out of a thousand people, about 683 will have IQs between 85 and 115, and will be said to have "average" intelligence. About 136 will be "high" and 136 "low". About 21 will be "very high" and 21 "very low".

A handful will be out of the range. People with IQs above 145 are considered "extremely intelligent".

In some jurisdictions, those below 70, with "very low" or "extremely low" intelligence, are exempted from things like the death penalty because their intelligence is so low they cannot be considered moral agents. A considerable portion of the prison population falls in the range of 80 and below.

Here is the Politically Incorrect Part

What if there was an ethnic or racial group that had an average IQ ten percent above or below 100? Say members of group "Beta" have an average IQ of 90 and members of group "Alpha" have an average of 110? (Actually, there are groups like that, or close to that. However, political correctness forbids me from mentioning their ethnic and/or racial descriptions.)

If the IQ difference between Alpha and Beta was only twenty percent, would you expect the Alpha group to have only twenty percent higher representation among professions that require high academic intelligence? Would you expect Alpha to have only twenty percent more scientists and engineers and accountants and so on? Would you expect Alpha to have only twenty percent more PhDs?

If you did you would be WRONG.

If the standard deviation for IQ is fifteen points, and the Alpha group is twenty points above the Beta group, that is a difference of over one standard deviation.

For example, if a Nobel Prize winner had to be "very intelligent" or "extremely intelligent" in the top two increments, there would be over ten people from the Alpha group for every one from the Beta group.

If you had to have an above-average IQ (100 or more) people in the Alpha group would outnumber those in the Beta group by three to one!

Bottom Line:

If members of some ethnic and racial groups are "over-represented" and other groups "under-represented" in professions requiring higher academic intelligence, that does not necessarily imply discrimination or favoritism.

If one group has an average IQ of 110 or more, you would expect them to be "over-represented" by at least three-to-one over the average American.

If another group has an average IQ of 90, you would expect them to be "under-represented" by a factor of three-to-one or more below the average American.


There is a great deal of overlap.

A "tall" woman is taller than 60% of all men and an "extremely tall" woman is taller than 90% of all the men.

A "very intelligent" member of the
Beta group is smarter than 60% of all members of the Alpha group and an "extremely intelligent" member of the Beta group is smarter than 90% of all the members of the Alpha group.

Do not judge a person by his or her group membership!

Ira Glickstein

The above is the fourth part of my "presentation" on the topic of "Lies, ..." Click for Part 1, Part 2, and Part 3.

1 comment:

joel said...

I've always found the presidential approval ratings interesting. No matter who the president is, the rating seems silly, especially when one tries to compare the ratings of various president under dissimilar circumstances. For instance, when the country is at war, a high disapproval rating can mean that lots of doves are unhappy plus lots of hawks are unhappy. The president might be too moderate! Without a set of statistics that tells us the nature of the disapproval, the highly publicized rating means nothing. With respect -Joel