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 .