2010 Biostatistics 08

Normal Distribution

ORIGIN 0

1

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

2

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

i

Y1

i

< specifying variance ( 2)

100

100

1

2

1 X 2

2 i

2

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...

Normal Distribution

ORIGIN 0

1

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

2

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

i

Y1

i

< specifying variance ( 2)

100

100

1

2

1 X 2

2 i

2

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...