Normal%20Distribution

1
Normal Distribution

• The Bell Curve

2
Questions

• What are the parameters that drive the normal
  distribution? What does each control? Draw a
  picture to illustrate.
• Identify proportions of the normal, e.g., what
  percent falls above the mean? Between 1 and 2
  SDs above the mean?
• What is the 95 percent confidence interval for
  the mean?
• How can the confidence interval be computed?

3
Function

• The Normal is a theoretical distribution
  specified by its two parameters.
• It is unimodal and symmetrical. The mode, median
  and mean are all just in the middle.

4
Function (2)

• There are only 2 variables that determine the
  curve, the mean and the variance. The rest are
  constants.
• 2 is 2. Pi is about 3.14, and e is the natural
  exponent (a number between 2 and 3).
• In z scores (M0, SD1), the equation becomes

(Negative exponent means that big z values give
small function values in the tails.)
5
Areas and Probabilities

• Cumulative probability

6
Areas and Probabilities (2)

• Probability of an Interval

7
Areas and Probabilities (3)

• Howell Table 3.1 shows a table with cumulative
  and split proportions

z Mean to z Larger F(a) Smaller
0 0 .5 .5
.5 .1915 .6915 .3085
1 .3413 .8413 .1587
1.96 .4750 .9750 .0250
Graph illustrates z 1. The shaded portion is
about 16 percent of the area under the curve.
8
Areas and Probabilities (3)

• Using the unit normal (z), we can find areas and
  probabilities for any normal distribution.
• Suppose X120, M100, SD10. Then z(120-100)/10
  2. About 98 of cases fall below a score of
  120 if the distribution is normal. In the
  normal, most (95) are within 2 SD of the mean.
  Nearly everybody (99) is within 3 SD of the
  mean.

9
Review

• What are the parameters that drive the normal
  distribution? What does each control? Draw a
  picture to illustrate.
• Identify proportions of the normal, e.g., what
  percent falls below a z of .4? What part falls
  below a z of 1?

10
Importance of the Normal

• Errors of measures, perceptions, predictions
  (residuals, etc.) X Te (true score theory)
• Distributions of real scores (e.g., height) if
  normal, can figure much
• Math implications (e.g., inferences re variance)
• Will have big role in statistics, described after
  the sampling distribution is introduced

11
Computer Exercise

• Get data from class (e.g., height in inches)
• Compute mean, SD, StErr of Mean in Excel
• Compute same in SAS PROC UNIVARIATE
• Show plots (stem-leaf Boxplot)
• Show test of normality