Transcript Normal Distribution

Normal Distribution
The Bell Curve
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?
Function
• The Normal is a theoretical distribution
specified by its two parameters.
f ( x;  ,  ) 
2
1
2
2
e
 ( x   ) 2 / 2 2
• It is unimodal and symmetrical. The mode,
median and mean are all just in the middle.
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 (M=0, SD=1), the equation
becomes:
(Negative exponent means
1  z 2 / 2 that big |z| values give
f ( z) 
e
small function values in the
2
tails.)
Areas and Probabilities
• Cumulative probability:
F (a)  p( X  a)
Normal Curv e
probability density
Cumulative Probability
1  F (a)  p(a  X )
-3
-1
0
Z
a=X
2
3
Areas and Probabilities (2)
• Probability of an Interval
Normal Curv e
probability density
Interval Probability
F (2)  F (1)  p (1  X  2)
-4
-3
-2
-1
0
Z
1
2
3
4
Areas and Probabilities (3)
• Howell Table 3.1 shows a table with
cumulative and split proportions
z
Mean to
z
0
0
Graph illustrates
.1915
z = 1. The shaded .5
portion is about
1
.3413
16 percent of the
1.96 .4750
area under the
curve.
Larger
F(a)
.5
.6915
.8413
.9750
Smaller
.5
.3085
.1587
.0250
Areas and Probabilities (3)
• Using the unit normal (z), we can find
areas and probabilities for any normal
distribution.
• Suppose X=120, M=100, SD=10. 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.
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?
Importance of the Normal
• Errors of measures, perceptions, predictions
(residuals, etc.) X = T+e (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
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