Download Density Curves and the Normal Distribution

The Normal Distribution
Chapter 2
Continuous Random Variable
• A continuous random variable:
– Represented by a function/graph.
– Area under the curve represents the proportions of
the observations
– Total area is exactly 1.
• How do we locate the median for a continuous
random variable? the mean?
• The median is the value that divides the graph
into equal area while the mean is the “balance”
point.
Continuous Random Variable 1
A= .4(1) =0.4
0.4
0.5
What percent of the observations lie below 0.4?
40%
Continuous Random Variable 2
A= 1.4(.5) =0.7
0.6
What proportion of the observations lie above 0.6?
Notice, to find proportion for observation above,
we can use the complement rule.
Continuous Random Variable 3
• Where is the mean and median?
• How will the curve change as s changes?
Normal Distributions
• Symmetric, single-peaked, and mound-shaped
distributions are called normal distributions
• Normal curves:
– Mean = median
– The mean m and standard deviation s completely
determine the shape
• Fathom
The Normal Curve
• Will finding proportions work different than
previous random variable examples?
• Empirical Rule Discovery
68% of observations fall within 1s of m
95% of observations fall within 2s of m
99.7% of observations fall within 3s of m
68-95-99.7 Rule
Applet
68-95-99.7 Rule
34%
.15%
2.35%
Applet
13.5%
34%
13.5%
2.35%
Percentiles?
16th
34%
2.35%
13.5%
50th
84th
34%
13.5%
2.35%
What’s Normal in Statistics?
• Normal distributions are good descriptions for real
data allowing measures of relative position to be
easily calculated (i.e. percentiles)
• Much of statistical inference (in this course)
procedures area based on normal distributions
• FYI: many distributions aren’t normal
Distribution of dates is approximately
normal with mean 1243 and standard
deviation of 36 years.
1135
1171
1207
1243
1279
1315
1351
Assume the heights of college women are
normally distributed with a mean of 65
inches and standard deviation of 2.5 inches.
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are taller than 65 in.?
50%
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are shorter than 65 in.?
50%
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are between 62.5 in.
and 67.5 in.?
68%
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are between 60 in. and
70 in.?
95%
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are between 60 and
67.5 in?
68%
13.5%
81.5%
57.5
60
62.5
65
67.5
70
72.5
What percentage of women are shorter than 70 in.?
50%
34%
13.5%
97.5%
57.5
60
62.5
65
67.5
70
72.5