Probability and the Normal Distribution

Definition

The normal distribution is defined by a complicated equation that we don't need to know or understand. Here is an example:

Why use Normal Distributions?

What is important is to understand that the normal distribution is used very frequently because:

1. It can be shown that many characteristics of interest, such as IQ, height and weight of people, etc., have a normal population distribution.
2. It can be shown mathematically that this shape is guaranteed in certain situations that will be important to us in inferential statistics.

Characteristics:

The normal distribution:

1. is symmetrical (the left side is a mirror image of the right side).
2. has 50% of the scores below the mean and 50% above. (Mean = Median)
3. has most scores are in the middle. Few scores are at the edges.

The Standard Normal Distribution

A normal distribution is a Standard(ized) normal distribution when its scores are expressed in standardized z-scores.

The standard normal distribution

1. has a mean of 0 and a standard deviation of 1, just like any other standardized distribution.
2. has a normal shape, just like any other normal distribution.

For a standard normal distribution it can be shown that

1. 34.13% of the scores are between the mean and +1.00.
2. 34.13% of the scores are between the mean and -1.00.
3. 13.59% of the scores are between +1.00 and +2.00.
4. 13.59% of the scores are between -1.00 and -2.00.
5. 2.28% of the scores are above +2.00.
6. 2.28% of the scores are below -2.00.

Answering Probability Questions with the Unit Normal Table:

The unit normal table provides a listing of proportions (probabilities) corresponding to many z-scores in the standard normal distribution. Take the following steps to answer probability questions using this table:

1. Sketch the distribution, showing the mean and standard deviation in raw scores.
2. On the sketch, locate the specific score identified in the problem, and draw a vertical line through the distribution at this location.
3. Make sure whether you need to find out about values greater than (to the right side of) or less than (to the left side of) the specific score.
4. Shade the appropriate portion of the distribution (to the right or left of your line).
5. Now transform the specific score into a z-score to identify the specific z-score in the standard normal distribution that appears in the Unit Normal Table.
6. Look at the shaded portion in your sketch to determine which column (B or C) in the table corresponds with the proportion you are trying to find.
7. Ignore the sign of your z-score and look it up in the table, taking the appropriate value from column B or C.

Percentiles and the Normal Distribution

You can use the normal distribution to determine percentiles (and percentile ranks).

Because a percentile (rank) of a score is the percentage of the scores that fall at or below the score, you will need to find the proportion of the distribution that is to the left of the score.

Practice Problem

For the normal distribution N(50,20) --- that is that has a mean of 50 and a standard deviation of 20 --- find X such that 71.57% of the area under the curve is to the left of X. (Hint: Use table B, starting on page A-24 in the back of the book). Answer

ViSta and the Normal Distribution

With ViSta, you can get the proportion of the scores that are below (to the left) of a given z-score by typing, in the listener window, the function:
(normal-cdf z) where z is replaced with the z-score value in which you are interested.

Multiply this by 100 for the percentile. You can do this by typing:

(* 100 (normal-cdf z))

Subtract the value returned by the function from one to get the proportion to the right of z. This can be done by typing:

(- 1 (normal-cdf z))