Mean male height: 69.3 in.
Standard deviation: 2.8
Mean female height: 64 in.
Standard deviation: 2.8
Out of 100 million adult females in the US, less than 5 million of them are taller than 68.2 inches, or 5’ 8.2”.
Out of 100 million adult males, less than 5 million are shorter than 66 inches, or 5 ½’.
So about 5 million women are taller than 5 million men, while 95% of men are taller than 95% of women.
This is the same Gaussian distribution, standard deviation and gender gap as almost all standardized tests.
People come in lots of different
heights. Let's think about the height of American men.
The average American man is
5'9". This means half of all American men are taller than 5'9", and
half are shorter than 5'9". This one fact does not tell us much about how
height is distributed, however. One could ask what's the tallest American man?
The shortest? How many men are over 6'6"? Suppose you measured the height
of a hundred men chosen at random off the street. You would most likely measure
something much like the following table:
|
Measuring
the height of 100 American men |
||||||||||||||
|
Graph: |
|
|||||||||||||
|
Height |
5'2" |
5'3" |
5'4" |
5'5" |
5'6" |
5'7" |
5'8" |
5'9" |
5'10" |
5'11" |
6' |
6'1" |
6'2" |
6'3" |
|
Count |
1 |
3 |
4 |
6 |
7 |
12 |
17 |
17 |
12 |
7 |
6 |
4 |
3 |
1 |
It turns out that men's height falls
onto what's called a standard distribution, or a gaussian curve, or a bell
curve. Out of one hundred men, about 2/3 of them, about 68, are between
5'6" and 5'11". About 2/3 of all American men are 5'9" ±
3". About 1/3 of them are outside this range, with about half of those on
each side. So, about 1/6 are 6' or taller, and about 1/6 are 5'5" or
shorter. If we start looking for men who are much taller than 6' tall, we find
that as their height goes up, they get more and more rare.
|
Some
very famous very tall guys |
||
|
|
Players
|
US
population this tall |
|
3σ |
Michael
Jordan 6'6", Kobe Bryant 6'7" |
130,000 |
|
4σ |
Larry
Bird 6'9", Karl Malone 6'9" |
3,200 |
|
5σ |
Shaquille
O'Neal 7"1', Wilt Chamberlain 7'1", Kareem Abdul-Jabbar 7'2" |
28 |
|
6σ |
Yao
Ming 7'5" |
2 in the world |
Once we have graphed a
representative sample, as we have above, we can find the points which enclose 2/3
of the population. This is called the Standard Deviation range. Standard
Deviation is normally written as σ The standard deviation for American
men's height is about 3". Knowing that, we can figure out what the rest of
the population looks like too. Each time height increases by 3", by a
standard deviation, the population drops off considerably. There are just about
exactly 100,000,000 adult men in America. Now that we know their average height
is 5'9" and the standard
deviation is 3", we can predict how many of these men fall into
various height categories.
|
Population
of American Men in various height categories |
||
|
Height Range |
S.D. |
Expected number |
|
4'6" - 4'9" |
-4σ |
3,200 |
|
4'9" - 5'0" |
-3σ |
135,000 |
|
5'0" - 5'3" |
-2σ |
2,100,000 |
|
5'3" - 5'6" |
-1σ |
13,600,000 |
|
5'6" - 5'9" |
average |
34,000,000 |
|
5'9" - 6'0" |
average |
34,000,000 |
|
6'0" - 6'3" |
1σ |
13,600,000 |
|
6'3" - 6'6" |
2σ |
2,100,000 |
|
6'6" - 6'9" |
3σ |
135,000 |
|
6'9" - 7'0" |
4σ |
3,200 |
|
7'0" - 7'3" |
5σ |
28 |
|
7'3" - 7'6" |
6σ |
0 |
We see above that the number of men
at a given height drops off really quickly as you get away from the average height.
In fact, the expected number of men in the US who are over 7'3" is less
than 1. There actually is at least one guy in the US who is this tall: NBA star
Yao Ming. We had to import him from China, where they have four times as many
people as the US has.
The gaussian curve is a mathematical
curve, and does not fit population data perfectly. Height is subject to a lot
of things besides just statistics. There are chemical imbalances that can strongly
effect how people grow, and there are hormones and steroids you can take in
adolescence to effect your final height. One man, Robert Wadlow,
once grew to be 8'11". According to statistics, this is all but
impossible. But Robert had a pituitary problem, and pituitary glands don't know
anything about statistics.
.14% 2.1% 13.6% 68% 13.6% 2.1% .14%
Well, that was all a lot of fun.
What we're supposed to learn from this is that about 2/3 of the time a variable
with a gaussian distribution is within ±1σ. If we go out to ±2σ we
now have 95% of the values. At ±3σ it's 99.7%. After that, the numbers get
quite ridiculous until at 6σ we have all but about 1 in a billion. In
normal life, 6σ doesn't come up very much.