### Math 225 Course Notes

### Section 4.6: The Normal Distribution

Many naturally occuring variables have distributions that are well-approximated
by a "bell-shaped curve", or a normal distribution.
In addition, the sampling distributions of important statistics
such as the sample mean is approximately normal
for moderately large samples.
All normal curves share certain
characteristics.
It is possible to find areas under any normal curve
by working with only
the standard normal curve,
which has a mean of 0 and a standard deviation of 1,
which is
tabulated
in the back of the textbook.
All normal curves are described by two parameters:
the mean, mu, and the standard deviation, sigma.

These characteristics hold for all normal curves where the mean is mu
and the standard deviation is sigma.

- The bell-shaped normal curve is symmetric and centered at mu.
- The total area under the curve is 1.
- About 68% of the area is within one standard deviation of the mean,
about 95% of the area is within two standard deviations of the mean,
and almost all (99.7%) of the area is within three standard deviations of the
mean.
- The places where the normal curve is steepest are a standard deviation
below and above the mean
(mu - sigma and mu + sigma).

The standard normal curve has mean mu = 0
and standard deviation sigma = 1.
Any normal curve can be converted to a standard normal curve by the process
of *standardization*,
first subtracting the mean and then dividing by the standard deviation.
z = (x - mu) / sigma

The z-score tells how many standard deviations an observation x
is from the mean.
Positive z-scores are greater than the mean,
and negative z-scores are below the mean.
It is also useful to use this formula in reverse.

x = mu + z*sigma

This says explicitly (in algebra) that x is z standard deviations
above the mean.

The standard normal table is located on pages 688 and 689 of your textbook.
The standard normal table gives the cumulative distribution function
for a standard normal random variable.
In other words, it will tell you the area under the curve
to the left of z for any number z.
You will need to be able to use the table to find areas
when the numbers on the axis are known,
and to be able to use the table to find numbers on the axis
when areas are known.

Always draw a sketch of a normal curve in working out problems.

**Examples**

To find the area to the left of 0.57.

Use the table:

z | 0.00 0.01 0.07
-------------------------------------------------------
0.00 |
0.10 |
0.50 | .7157

Since the total area is 1, the area to the right of 0.57
must be 1 - .7157 = .2843.
The table may also be used backwards.
Suppose that you want to find a value z so that the area to the right of z
is .6000.

Then the area to the left of z is .4000.
Areas are in the middle of the table.
Try to find the area closest to .4000.

z | -0.09 -0.05
------------------------------------------
-3.80 |
-0.20 | .4013
.4013 is the closest, so z is about -0.25.

Last modified: Feb 19, 1996

Bret Larget,
larget@mathcs.duq.edu