"Numbers are like people;
torture them enough and they'll tell you anything."
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
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 left of the point z. To find the area to the right of a z value follow the procedure above but subtract the table value from 1 (the total area under the density curve is ALWAYS = 1). BE CAREFUL that you answer the question by sketching the normal curve, marking the z value, and shading the area of interest. The VALUE of the table is that we can use it to answer any question about proportions of observations in a normal distribution by standardizing and then using the standard normal table.
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
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.