The Normal Distribution
Distributions exist in the theoretical
world. A number of distributions have been defined mathematically that prove
useful in real applications. The most familiar is the normal
distribution. Many physical phenomenon have characteristics that can be
described by the normal distribution. Also, and more importantly, the
normal distribution describes the behavior of the mean value of a sample as
given by the central limit theorem.
The normal distribution is described by the
well-known bell shaped curve. There are an infinite number of normal
distribution curves:
The normal distribution curves can be
categorized by the mean value and standard deviation. Probabilities for
the normal distribution are available for the standard normal curve, which is
a normal distribution with mean of zero and standard deviation of 1. To
find a probability for a particular problem, we first have to convert our
problem to a problem on the standard normal curve:
We use the variable Z to denote units
of measure on the standard normal curve.
Example
As an example, suppose that height is
normally distributed. The mean height of American women 18-24 is 65.5"
with a standard deviation of 2.5". What is the probability that a randomly
selected woman is less than 70" tall?
First, we draw the picture:
We can use the form of the standard normal curve that is
provided in this web site by splitting the problem into two halves:
The probability that a woman is below
average in height (P(x < 65.5) is 50%. To find the
probability that a woman is between 65.5 and 70 inches tall, we have to convert
the problem to standard normal units:
To find this probability on the standard normal curve we have to
look down the first column with the heading Z until we find
1.8. We look under the next column over since our Z value
is exactly 1.80. This value in this column, 0.4641, is the probability of
an observation being between a Z value of zero and 1.8.
Thus, the probability that a woman is under 70 inches tall is 0.5 + 0.4641 =
0.9641, or a little over 96%.
Any problem can be solved using the standard normal curve, a little
geometry, the fact that the normal distribution curve is symmetric and that the
area under the entire curve is 1 (i.e. all things are possible). For
example, suppose we want to know the probability that a women is greater than
68 inches tall. We can draw the picture of the problem as follows:
To solve this problem, we note that the
probability that a woman is greater that 68 inches tall is equal to the
probability that a women is greater than 65.5 inches tall (which is 0.5) minus
the probability that a women is between 65.5 and 68 inches tall. We can
find the probability that a women is between 65.5 and 68 inches tall by first
converting the problem into standard normal units and looking up the value for Z
on the :
So the probability that a randomly selected
woman is greater than 68 inches tall is 0.5 - 0.3413 = 0.1587 or 15.87%.
Note that, because of symmetry, this is also the probability that a woman is
less than 63 inches tall.
Excel
Excel provides the normsdist() function to
calculate values from the standard normal distribution, however, it uses the
cumulative standard normal. The values on the standard normal curve used in this
web site give values for the area under the standard normal curve from Z =
0 to some positive value of Z. The cumulative standard
normal curve starts at Z = -infinity.
For example, in solving the problem above,
we looked up the value of Z = 1 and got 0.3413. If we enter
=normsdist(1)
in cell A1, we will get the result
0.8314, which is 0.5 larger than what we got from the table.