For the normal distribution, the mean, median, and mode are all equal
to value where the curve is highest. The mean 
is sometimes called a location parameter because increasing
(decreasing) it shifts the entire normal curve right (left). Likewise, 
is called a scale parameter because increasing (decreasing) it
causes the curve to be wider (narrower). See the ``Z, t, Chi-square, F''
concept lab for examples. The value of
is the distance from
to the inflection points of the normal curve. (The inflection
points are the locations where the curve changes from turning downward
to turning upward.) Note that the total area under any continuous curve
is equal to one.
In the figure below, we have drawn the standard normal (or Z)
curve, that is, the normal curve for 
and 
. We have also placed on the plot the percent of the area under the
curve between -3 and -2 (2.15%), between -2 and -1 (13.59%), and so on.
Note how the curve is symmetric about 0, how around 68% of the area is
between -1 and 1, and approximately 95.44% between -2 and 2. (Recall the
empirical rule from Week 2.)
The plot of any normal curve will look basically the same as this
figure except that the numbers on the horizontal axis instead of being
centered at 0 will be centered at
and the other numbers will be at multiples of
away from
. For example, the 1 would be replaced at 
, the 2 by 
, the -3 by 
and so on. The IQ histogram back in Week 1 gives an example of this
where the curve is centered at
= 100 and the labels on the horizontal axis are at multiples of
= 15 away from 100.
Since there are an infinite number of normal distributions (there is
one for any choice of
and 
), we will standardize all our normal distribution problems to a 
standard normal distribution. The notation we will use is 
. So for any 
,
where
.
 | Finding
Areas and Quantiles for Z Curve
|
Our method for finding probabilities for normal distribution problems
will be as follows:
 | Convert any normal distribution problem to a standard normal.
| Look up the probability in the standard normal (or Z) table
(or use StataQuest). |
|
EXAMPLE:\ Let X be the
blood platelet count (measured in thousands per cc of blood) in humans,
and suppose 
. What is the probability that an individual has a platelet count
between 185.4 and 360.2? It is a good idea to always draw a picture. The
solution is as follows. We have
is given by
which, in turn is P(-1.96<Z<1.96), which is
We might also ask what the value of X is equivalent to the 99%
percentile (99% of all individuals will have a platelet count below this
value). To solve this we reverse the process used above:
| Look up the Z value corresponding to our probability in the
standard normal table.
| Convert this value of Z to X by 
. |
|
From the Z table, we find that z=2.33 has area to the left
of 0.99. Now using the equation for Z we get
which means that
Thus, someone with a blood-platelet count of 376.72 has a higher
count than 99% of the population.
|
|
