Transcript Chapter 2: The Normal Distributions

CHAPTER 2: THE NORMAL
DISTRIBUTIONS
SECTION 2.1: DENSITY CURVES AND
THE NORMAL DISTRIBUTIONS

Chapter 1 gave a strategy for exploring data on a
single quantitative variable.
 Make a graph.
 Usually a histogram or stemplot
 Describe the distribution.
 Shape, center, spread, and any striking
deviations.
 Calculate numerical summaries to briefly
describe the center and spread.
 Mean and standard deviation for symmetric
distributions
 Five-number summary for skewed
distributions
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DENSITY CURVES

Chapter 2 tells us the next step.
 If the overall pattern of a large number of observations
is very regular, describe it with a smooth curve.

This curve is a mathematical model for the distribution.
 Gives a compact picture of the overall pattern.
 Known as a density curve.
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DENSITY CURVES

A density curve describes the overall pattern of a distribution.
 Is always on or above the horizontal axis.
 The area under the curve represents a proportion.
 Has an area of exactly 1 underneath it.

The median of a density curve is the equal-areas point.
 Point that divides the area under the curve in half.
 The quartiles divide the area into quarters
 ¼ of the area is to the left of Q1
 ¾ of the area is to the left of Q3

The mean of a density curve is the balance point.
 Point that the curve would balance at if made of solid
material.
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MATHEMATICAL MODEL

A density curve is an idealized description of the
distribution of data.
 It gives a quick picture of the overall pattern ignoring
minor irregularities as well as outliers
 Since a density curve is an idealized description of the
data (not the actual data), we need to differentiate
between the mean and standard deviation of the curve
and the mean and standard deviation of the actual
observations.
Population
Mean
Standard Deviation

Greek letter sigma 
Greek letter mu
Sample
”x-bar”
x
s
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NORMAL DISTRIBUTIONS:
 Normal



curves
Curves that are symmetric, single-peaked,
and bell-shaped. They are used to describe
normal distributions.
The mean is at the center of the curve.
The standard deviation controls the spread
of the curve.
The bigger the St Dev, the wider the curve.
 There are roughly 6 widths of standard
deviation in a normal curve, 3 on one side of
center and 3 on the other side.

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NORMAL CURVE
3  2  1 

1  2  3 
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HERE ARE 3 REASONS WHY NORMAL
CURVES ARE IMPORTANT IN STATISTICS.
 Normal
distributions are good descriptions for
some distributions of real data.
 Normal distributions are good approximations
to the results of many kinds of chance
outcomes.
 Most important is that many statistical
inference procedures based on normal
distributions work well for other roughly
symmetric distributions.
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THE 68-95-99.7 RULE OR
EMPIRICAL RULE:
 68%
of the observations fall within one
standard deviation of the mean.
 95%
of the observations fall within two
standard deviation of the mean.
 99.7%
of the observations fall within three
standard deviation of the mean.
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10
61.5
64
66.5
69
71.5
74
76.5
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2.5%
95%
2.5%
12
61.5
64
66.5
69
71.5
74
76.5
2.5%
64 to 74 in
95%
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61.5
64
66.5
69
71.5
74
76.5
2.5%
64 to 74 in
16%
68%
16%
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61.5
64
66.5
69
71.5
74
76.5
2.5%
64 to 74 in
16%
68%
84%
34%
50%
15
61.5
64
66.5
69
71.5
74
76.5
NORMAL DISTRIBUTION NOTATION
 Since
normal distributions are so
common, a short notation is helpful
 Abbreviate
the normal distribution with
mean  and standard deviation  as:
N ( , )
 The
distribution of men’s heights would be
N (69, 2.5)
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Find the proportion of observations
within the given interval
1.0
.75
.5
.25
0
0
.25
.5
P(0 < X < 2)
 P(.25 < X < .5)
 P(.25 < X < .75)
 P(1.25 < X < 1.75)
 P(.5 < X < 1.5)
 P(1.75 < X < 2)

.75
1.0
1.25
= 1.0
= .125
= .25
= .25
= .46875
= .15625
1.5
1.75
2.0
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SECTION 2.1 COMPLETE
 Homework:
p.83-91 #’s 2-4, 9, 12 & 14
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