Download The Normal Distribution

The Normal Distribution
Contrary to what you might think, the Normal Distribution was not invented by an evil
cabal of mathematics professors so that they could inflict it on their poor students. Quite
the contrary, the Normal Distribution was “invented” by nature and foisted upon
mathematics professors who spent hundreds of years trying to work out all the details.
But, before we go further, click on the link below and then re-size your windows so you
can read this and see the web page run at the same time
http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html
What are we looking at?
1) Each of the yellow balls starts from the same position. It drops from the exact
center of the top of the frame.
2) The green squares are “pins” When a ball drops on a pin it has an equal chance
of bouncing to the right or to the left.
3) The pins are staggered so that once a ball has dropped between two pins it falls
directly onto a pin in the next row, where, again, it has an equal chance of bouncing to
the right or the left.
4) After falling through the last row of pins, the balls drop straight down and form
vertical piles.
The link below gives a version of the same set-up but it runs much slower so that you can
see the details
http://javaboutique.internet.com/BallDrop/
Each time a ball hits a pin it has an exactly equal chance of going right or left regardless
of how many times it has gone right or left above. That is, each new bounce is
independent of what has happened before.
Many things which we observe in the world are the result of hundreds of such “bounces.’
Consider how many different “bounces” go into determining how tall a person will be.
We may have no clear idea of what all of the bounces are, but their cumulative effect is to
produce a normal curve. To see how powerful this force of randomness is, consider that
in the simulations we are looking at there are only eight rows of pins to bounce off of -in nature there may be hundreds.
Now for the Good News
A normal distribution can be described using only two numbers (we call them parameters
just so we sound smarter.) These two numbers are the mean and the standard deviation.
If you read about skew and kurtosis in this week’s lesson you know too much – these are
always zero for the normal distribution.
So the distribution for IQ scores is completely described by telling you that the mean is
100 and the standard deviation is 15. This is not quite a force of nature – the tests are
standardized to give this result. (Note: they are not standardized to give a normal
distribution, that is a fact of nature.)
The average height of an adult American male is 5’9” with a standard deviation of 3”
This is a natural phenomenon.
Once you have the mean and standard deviation you know everything that there is to
know. Even better – as we will learn in weeks to come, statisticians have studied the
characteristics of the Standard Normal Distribution – a normal distribution with a mean
of zero and a standard deviation of one. If we have any other normal distribution
(different mean and standard deviation) we can use a relatively simple calculation to
apply everything that has been worked out for the Standard Normal Distribution to the
specific normal distribution that we have. Stay tuned.