and the Normal Distribution
- 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:
- It can be shown that many characteristics of interest,
such as IQ, height and weight of people, etc., have
a normal population distribution.
- It can be shown mathematically that this shape is
guaranteed in certain situations that will be
important to us in inferential statistics.
- The normal distribution:
- is symmetrical (the left side is a mirror image of
the right side).
- has 50% of the scores below the mean and 50% above.
(Mean = Median)
- 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
The standard normal distribution
- has a mean of 0 and a standard deviation of 1, just
like any other standardized distribution.
- has a normal shape, just like any other normal distribution.
- For a standard normal distribution it can be shown that
- 34.13% of the scores are between the mean and +1.00.
- 34.13% of the scores are between the mean and -1.00.
- 13.59% of the scores are between +1.00 and +2.00.
- 13.59% of the scores are between -1.00 and -2.00.
- 2.28% of the scores are above +2.00.
- 2.28% of the scores are below -2.00.
- Answering Probability Questions with the Unit Normal
- 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:
- Sketch the distribution, showing the mean and standard
deviation in raw scores.
- On the sketch, locate the specific score identified
in the problem, and draw a vertical line through the
distribution at this location.
- 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.
- Shade the appropriate portion of the distribution
(to the right or left of your line).
- 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.
- 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.
- 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:
where z is replaced with the z-score value in which you
(* 100 (normal-cdf z))
Multiply this by 100 for the percentile. You can do this
Subtract the value returned by the function from one
to get the proportion to the right of z. This can be done
(- 1 (normal-cdf z))