2010 Biostatistics 08

Normal Distribution



The Normal Distribution

The Normal Distribution, also known as the "Gaussian Distribution" or "bell-curve", is the most widely
employed function relating observations X with probabilty P(X) in statistics. Many natural populations
are approximately normally distributed, as are several important derived quantitities even when the
original population is not normally distributed.
P roperly speaking, the Normal Distribution is a continuous "probability density function" meaning that
values of a random variable X may take on any numerical value, not just discrete values. In addition,
because the values of X are infinite the "exact" probabiliy P(X) for any X is zero. Thus, in order to
determine probabilities one typically looks at invervals of X such as X >2.3 or 1< X < 2 and so forth. It is
interesting to note that because the probability P(X) = 0, we don't have to worry about correctly
interpreting pesky boundaries, as seen in discrete distributions, since X > 2 means the same thing as X  2
and X < 2 is the same as X  2.
As described previously, the Normal distribution N(,  2) consists of a family of curves that are specified by
supplying values for two parameters:  = the mean of the Normal population, and 2 = the variance of
the same population.

Prototyping the Normal Function using the Gaussian formula:
Making the plot of N(50,100):
< specifying mean (

  50
 


i  0  100

< Defining a bunch of X's ranging in value from 0 to 100. Remember that the
range of X is infinite, but we'll plot 101 point here. That should give us enough
points to give us an idea of the Gaussian function shape!

X  i

Y1 

< specifying variance ( 2)

  100


 2

  1  X   2
 2 i  

 e

< Formula for Normal distribution. Here we
have computed P(X) for each of our X's.
Zar 2010 Eq. 6.1, p. 66.

Now, let's compare...