Normal Distribution Curve

Date: 11/13/2000 at 10:17:09
From: Jaime Egars (Mrs. Watkins teacher)
Subject: Normal Distribution Curve

Dear Dr. Math,

I am a seventh grader at St. James Middle School, part of Horry County 
Public Schools, and my question has to do with the normal distribution 
curve:

When and where did it come into being and what are some of the 
applications?

Date: 11/13/2000 at 10:59:37
From: Doctor Mitteldorf
Subject: Re: Normal Distribution Curve

Dear Jaime,

A distribution curve is like a histogram where the bars get so close 
together that each one looks like a line rather than like a bar. You 
could make a histogram of all the heights of the girls in your class: 
How many girls are between 4'6" and 4'8", how many between 4'8" and 
4'10", etc. If you look at this bar graph when you're done, it will 
look like a skyline, in which the tallest building is in the middle, 
and shorter and shorter buildings are around it.

If you imagine doing the same curve for all the 7th grade girls in the 
United States, you could make the "width" of each bar much smaller: 
say, all the girls between 4'6" and 4'6.01", etc. Then you'd have a 
few thousand different bars, and to plot them all on the same page, 
you'd have to make each one a thin line. The tops of all the lines 
together would make a smooth curve, again with the highest point in 
the middle. This is the distribution curve for heights of 7th grade 
girls in the country.

In fact, the curve you drew would look approximately like the 
bell-shaped normal distribution curve that is becoming familiar to you 
now as you study it. You can take this as an experimental fact about 
the world. No one can prove it, but it seems to be true that a lot of 
biological measures like this are approximately normally distributed 
over a given population.

Now, there's a theorem from pure math that hints why the normal curve 
is so commonly seen in this and other areas. It's not a proof that 
heights of 7th grade girls are normally distributed - nothing of the 
sort. It's just an abstract mathematical theorem that seems to have 
something to do with a lot of real life situations.  

The theorem says, roughly, that if you have very many factors 
contributing to an outcome, and each factor contributes in the same 
way, then it doesn't matter how the factors are distributed: the net 
result will be a normal distribution.

The classic example of an application of the theorem is this: 
Experiment A is to flip 4 coins and count how many heads you get. If 
you repeat Experiment A thousands of times and count how many times 
you get 0 heads, how many times you get 1 head, etc., and make a bar 
graph, the bar graph will have a nice symmetric shape, with the 
highest bar in the middle, at 2 heads. (There are only 5 bars in the 
bar graph, for 0 heads through 4 heads.)

Experiment B is to flip 100 coins and count how many heads. Now there 
are 101 possible outcomes, and the bar graph starts to look smoother. 
Still, there's a tallest bar in the middle at 50 heads, flanked on 
either side by smaller bars. You can start to see the bell shape in 
the result. Remember that to collect this data you have to repeat the 
entire experiment very many times. Each experiment consists of 100 
flips, and you have to do the 100 flips thousands of times in order to 
get a fair sample of the shape of the histogram.

Experiment Z is to flip 100,000 coins. Repeat experiment Z many 
thousands of times, and make a histogram as you did before. The bars 
will be very, very close together now. The bars to the left of 48,000 
and to the right of 52,000 will be completely empty, because it's so 
unlikely that you will get a result that departs very much from 50,000 
heads. But the area between 48,000 heads and 52,000 heads still has 
4,000 bars in the bar graph, and it will look like a very nice outline 
of the classic normal distribution, or bell-shaped curve.

I don't recommend that you spend your day sitting at home flipping 
enough coins to do experiment Z. But on a computer, the equivalent 
experiment is quite feasible, and the programming job should be not 
too hard for anyone who knows a language like Basic or Pascal or C. 
LOGO will also work for this.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/