"Numbers are like people;

torture them enough and they'll tell you anything."

Chapter 2

Sec 2.2

All normal distributions share common properties and are the same IF we
measure in units of s
about the mean
m as center.
Changing to these units is called STANDARDIZING. To standardize a value,
subtract the mean from the value and divide by the standard deviation. If x is an observation from a
distribution that has mean m
and standard deviation
s, the standardized
value of x is called a z-score and is calculated by

z = ( x - m)/s

A "z-score" tells how many standard deviations the original observation falls away from the mean, and in which direction. Observations larger than the mean are positive when standardized, while observations smaller are negative. Standardizing a variable that has any normal distribution produces a new variable that has the standard normal distribution.

The standard normal distribution is the normal distribution
with mean 0 and standard deviation 1. If a variable x has any normal
distribution N(m,s)
then the standardized variable

z = ( x - m)/s

has the normal distribution.

An AREA under a DENSITY CURVE is a *proportion *of the observations in a
distribution. Any question about ** what proportion **of observations
lie in some range of values can be answered by finding the area under the
curve. We can find these areas from a table that gives areas under the curve
for the standard normal distribution. Using TABLE A in our text we can find the area to the left of a
value using these steps...locate the first two digits in the left-hand column, then locate the
remaining digit in the top row. The entry where the row and column
intersects is the area to the

We will discuss Ex. 2.7 (page 98) completely for explanation within application.

Review the steps first:

***State the problem** in terms of the observed variable x. Draw a
picture of the distribution and shade the are of interest under the curve.

***Standardize x** to restate the problem in terms of a standard normal
variable z. Draw a picture to show the area
of interest under the standard normal curve.

***Find the required area** under the standard normal curve, using the table
and the fact that the total area under the curve is 1.

***Write your conclusion **in the context of the problem.

Sometimes we may be asked to find the observed value with a given proportion of
the observations above or below it...SOooooo use the table backwards.

Deciding whether our data is approximately normal is necessary
before some statistical procedures can be invoked. There are two ways to
approach this question.

**1.** Construct a frequency histogram or stem-plot and see if the
graph is approximately bell-shaped and symmetric about the mean. You can
improve upon this analysis by including the points for the mean and for the mean
__+ __1 standard deviation and __+__ 2 standard deviations, then
compare the count in each interval with the

68-95-99.7 rule.

**2.** Construct a normal probability plot on the calculator (see page
105) for key strokes. If the distribution is close to a normal
distribution, the plotted points will lie close to a straight line.
Conversely, non-normal data will show a nonlinear trend. Outliers appear
as points that are far away from the overall pattern of the plot.

__To repeat some important points:__

The standardized score is called a z-score and is calculated by using the mean
and standard deviation along with the x value being converted.

All normal distributions are the same when they are standardized. to a standard
normal distribution with mean=0 and standard deviation = 1. Assessing
normality can be done by observation of the shape of the graph or by
constructing a normal probability plot.

Index