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 Statistics Part 1: Average and Standard Deviation by Mark Lawrence 
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People come in lots of different heights. Let's think about the height of American men.
The average American man is 5'10". This means half of all American men are taller than 5'10", and half are shorter than 5'10". 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:
Graph:  

Height  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"  6'4" 
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 5'7" and 6'. About 2/3 of all American men are 5'10" ± 3". About 1/3 of them are outside this range, with about half of those on each side. So, about 1/6 are 6'1" or taller, and about 1/6 are 5'6" 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.
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 AbdulJabbar 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 5'10" and the standard deviation is 3", we can predict how many of these men fall into various height categories.
Height Range  S.D.  Expected number 

4'7"  4'10"  4σ  3,200 
4'10"  5'1"  3σ  135,000 
5'1"  5'4"  2σ  2,100,000 
5'4"  5'7"  1σ  13,600,000 
5'7"  5'10"  average  34,000,000 
5'10"  6'1"  average  34,000,000 
6'1"  6'4"  1σ  13,600,000 
6'4"  6'7"  2σ  2,100,000 
6'7"  6'10"  3σ  135,000 
6'10"  7'1"  4σ  3,200 
7'1"  7'4"  5σ  28 
7'4"  7'7"  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 7'4" 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.
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.
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Copyright © 20022016 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.
Email me, mark@calsci.com, with suggestions, additions, broken links. Revised Thursday, 06Aug2009 11:19:05 PDT 
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