### Section 4.6: The Normal Distribution

#### Key Concepts

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.

#### Characteristics

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

1. The bell-shaped normal curve is symmetric and centered at mu.
2. The total area under the curve is 1.
3. 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.
4. 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 Distribution

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.

#### Using the Normal Table

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