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CHAPTER 13A
Normal Distributions
SO FAR…
We always want to plot our data. We make a
graph, usually a histogram or a stemplot.
 We want to look for an overall pattern (shape,
center, spread) and for any striking deviations
from that pattern.
 We choose either the five-number summary or
the mean/standard deviation to describe the
center and spread of a distribution numerically.
 Sometimes, the overall pattern of a large number
of observations is so regular that we can describe
it by a smooth curve.

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SO FAR…
If we draw a curve
through the tops of the
bars in a histogram, we
get what is called a
density curve.
 Unlike histograms,
density curves don't show
counts but proportions.
 The total percentage
under the curve is always
100%.

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THE CENTER OF A DENSITY CURVE
As with histograms or stemplots, we can use the
median M, the quartiles Q1, Q3 and the mean x
to describe a density curve.
 We describe the shape of density curves with the
same vocabulary as for histograms (symmetric,
skewed).
 If we have a symmetric density curve, the mean
is roughly equal to the median.

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NORMAL DISTRIBUTION

Normal curves are symmetric, bell-shaped curves
that have the following properties:
A specific Normal curve is completely described by its
mean and its standard deviation.
 The tails of the distribution fall off quickly, so we do
not expect any outliers.

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NORMAL DISTRIBUTION
The mean determines the center of the
distribution.
 The standard deviation determines the shape of
the curve. It is the distance from the mean to the
change-of-curvature points on either side.

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http://www.math.psu.edu/dlittle/java/probability/plinko/index.html
THE 68-95-99.7 RULE

In any Normal distribution, approximately:
68% of the observations fall within one standard
deviation of the mean.
 95% of the observations fall within 2 standard
deviations of the mean.
 99.7% of the observations fall within 3 standard
deviations of the mean.


This is also known as the Empirical Rule.
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THE 68-95-99.7 RULE

Here is a picture of the rule.
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EXAMPLE 13.1

The distribution of heights of women aged 18 to
24 is approximately Normal with mean 65 inches
and standard deviation 2.5 inches. To use the 6895-99.7 rule, always start by drawing a picture.
The middle 68% of all women will be between
____ and ____ inches tall.
 The middle 95% of all women will be between
____ and ____ inches tall.
 The middle 99.7% of all women will be between ____
and ____ inches tall.

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EXAMPLE 13.2

Researchers in Chicago collected the daily high
temperature during the month of August from
1944 to 2000. After drawing a histogram, they
noticed it was approximately Normal with mean
80°F and standard deviation 8°F.

About what percent of the days in Chicago in August
will have a high temperature between 72°F and
88°F?

68%
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EXAMPLE 13.3

Heights of adults, ages 18-24

Men
mean: 70.0 inches
 standard deviation: 2.8 inches


So…
68% of men are between ____ and ____ inches
 95% of men are between ____ and ____ inches
 99.7% of men are between ____ and ____ inches

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EXAMPLE 13.3


What proportion of men are less than 72.8 inches
tall?
What proportion of men are more than 75.6
inches tall?
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EXAMPLE 13.4


The figure below is a stemplot of the IQ test scores of 74
seventh-grade students. This distribution is very close to
Normal with mean 111 and standard deviation 11. It
includes all the seventh-graders in a rural Midwest school.
Take the Normal distribution with mean 111 and standard
deviation 11 as a description of the IQ test scores of all
rural Midwest seventh-grade students. Use this
distribution and the 68-95-99.7 rule to answer the following
questions.
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9
10
11
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Key
8|6 represents 86
6,9
0,1,3,3,6,7,7,8
0,0,2,2,3,3,3,3,4,4,5,5,5,6,6,6,7,7,7,7,8,9
0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,5,5,6,8,8,9,9,9
0,0,3,3,4,4,6,7,7,8,8,8
0,2,6
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EXAMPLE 13.4



Between what values do the IQ scores of 95% of
all rural Midwest seventh-graders lie?
What percentage of IQ scores for rural Midwest
seventh-graders are less than 100?
What percentage of all students have IQ scores
144 or higher?

None of the 74 students in our sample school had a
score this high, does this surprise you?
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REMINDERS
We will discuss Chapter 13b next.
 Chapter 13 homework is posted online and is due
tomorrow.

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