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16 Mathematics of Normal Distributions
16.1 Approximately Normal Distributions
of Data
16.2 Normal Curves and Normal
Distributions
16.3 Standardizing Normal Data
16.4 The 68-95-99.7 Rule
16.5 Normal Curves as Models of RealLife Data Sets
16.6 Distribution of Random Events
16.7 Statistical Inference
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Excursions in Modern Mathematics, 7e: 16.1 - 1
Example 16.1 Distribution of Heights of
NBA Players
This table is a frequency table giving the
heights of 430 NBA players listed on team
rosters at the start of the 2008–2009 season.
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Excursions in Modern Mathematics, 7e: 16.1 - 2
Example 16.1 Distribution of Heights of
NBA Players
The bar graph for this data set is shown.
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Excursions in Modern Mathematics, 7e: 16.1 - 3
Example 16.2 2007 SAT Math Scores
The table on the next slide shows the scores
of N = 1,494,531 college-bound seniors on
the mathematics section of the 2007 SAT.
(Scores range from 200 to 800 and are
grouped in class intervals of 50 points.) The
table shows the score distribution and the
percentage of test takers in each class
interval.
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Excursions in Modern Mathematics, 7e: 16.1 - 4
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Excursions in Modern Mathematics, 7e: 16.1 - 5
Example 16.2 2007 SAT Math Scores
Here is a bar graph of the data.
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Excursions in Modern Mathematics, 7e: 16.1 - 6
Approximately Normal Distribution
The two very different data sets have one
thing in common–both can be described as
having bar graphs that roughly fit a bellshaped pattern.
These data sets have an approximately
normal distribution.
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Excursions in Modern Mathematics, 7e: 16.1 - 7
Normal Distribution
A distribution of data that has a perfect bell
shape is called a normal distribution.
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Excursions in Modern Mathematics, 7e: 16.1 - 8
Normal Curves
Perfect bell-shaped curves are called
normal curves.
Every approximately normal data set can be
idealized mathematically by a
corresponding normal curve
This is important because we can then use
the mathematical properties of the normal
curve to analyze and draw conclusions
about the data.
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Excursions in Modern Mathematics, 7e: 16.1 - 9
Normal Curves
Some bells are short and squat,others are
tall and skinny, and others fall somewhere
in between.
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Excursions in Modern Mathematics, 7e: 16.1 - 10
Essential Facts About Normal Curves
Symmetry.
Every normal curve has a vertical axis of
symmetry, splitting the bell-shaped region
outlined by the curve into two identical
halves.
This is the only line of symmetry of a normal
curve.
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Excursions in Modern Mathematics, 7e: 16.1 - 11
Essential Facts About Normal Curves
Median / mean.
We will call the point of intersection of the
horizontal
axis and the
line of
symmetry of
the curve the
center of the
distribution.
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Excursions in Modern Mathematics, 7e: 16.1 - 12
Essential Facts About Normal Curves
Median / mean.
The center represents both the median M
and the mean (average)  of the data.
In a normal distribution, M = .
The fact that in a normal distribution the
median equals the mean implies that 50%
of the data are less than or equal to the
mean and 50% of the data are greater than
or equal to the mean.
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Excursions in Modern Mathematics, 7e: 16.1 - 13
MEDIAN AND MEAN OF A
NORMAL DISTRIBUTION
In a normal distribution, M = .
(If the distribution is approximately
normal, then M ≈ .)
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Excursions in Modern Mathematics, 7e: 16.1 - 14
Essential Facts About Normal Curves
Standard Deviation.
The standard deviation is an important
measure of spread, and it is particularly
useful when dealing with normal (or
approximately normal) distributions.–
Denoted by the Greek letter  called sigma
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Excursions in Modern Mathematics, 7e: 16.1 - 15
Essential Facts About Normal Curves
Standard Deviation.
If you were to bend a
piece of wire into a bellshaped normal curve, at
the very top you would
be bending the wire
downward.
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Excursions in Modern Mathematics, 7e: 16.1 - 16
Essential Facts About Normal Curves
Standard Deviation.
But, at the bottom you
would be bending the
wire upward.
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Excursions in Modern Mathematics, 7e: 16.1 - 17
Essential Facts About Normal Curves
Standard Deviation.
As you move your hands down the wire, the
curvature gradually changes, and there is
one point on each side of the curve where
the transition from
being bent
downward to being
bent upward takes
place. Such a point
is called a point of
inflection of the
curve.
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Excursions in Modern Mathematics, 7e: 16.1 - 18
Essential Facts About Normal Curves
Standard Deviation.
The standard deviation of a normal
distribution is the horizontal distance
between the line of
symmetry of the
curve and one of the
two points of
inflection, P´ or P in
the figure.
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Excursions in Modern Mathematics, 7e: 16.1 - 19
STANDARD DEVIATION OF
A NORMAL DISTRIBUTION
In a normal distribution, the standard
deviation  equals the distance between
a point of inflection and the line of
symmetry of the curve.
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Excursions in Modern Mathematics, 7e: 16.1 - 20
Essential Facts About Normal Curves
Quartiles.
We learned in Chapter 14 how to find the
quartiles of a data set.
When the data set has a normal distribution,
the first and third quartiles can be
approximated using the mean  and the
standard deviation .
Multiplying the standard deviation by 0.675
tells us how far to go to the right or left of the
mean to locate the quartiles.
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Excursions in Modern Mathematics, 7e: 16.1 - 21
QUARTILES OF A
NORMAL DISTRIBUTION
In a normal distribution,
Q3 ≈  + (0.675)
and
Q1 ≈  – (0.675).
Pg. 604 # 6
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Excursions in Modern Mathematics, 7e: 16.1 - 22
Standardizing the Data
We have seen that normal curves don’t all
look alike, but this is only a matter of
perception.
All normal distributions tell the same
underlying story but use slightly different
dialects to do it.
One way to understand the story of any
given normal distribution is to rephrase it in
a simple language that uses the mean 
and the standard deviation  as its only
vocabulary. This process is called
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Excursions in Modern Mathematics, 7e: 16.1 - 23
z-value
To standardize a data value x, we measure
how far x has strayed from the mean 
using the standard deviation  as the unit of
measurement. A standardized data value is
often referred to as a z-value.
The best way to illustrate the process of
standardizing normal data is by means of a
few examples.
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Excursions in Modern Mathematics, 7e: 16.1 - 24
Example 16.4 Standardizing Normal
Data
Let’s consider a normally distributed data set
with mean  = 45 ft and standard deviation
 = 10 ft. We will standardize several data
values, starting with a couple of easy cases.
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Excursions in Modern Mathematics, 7e: 16.1 - 25
Example 16.4 Standardizing Normal
Data
■
x4 = 21.58 is ... uh, this is a slightly more
complicated case. How do we handle this
one? First, we find the signed distance
between the data value and the mean by
taking their difference (x4 – ). In this case
we get 21.58 ft – 45 ft = –23.42 ft. (Notice
that for data values smaller than the mean
this difference will be negative.)
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Excursions in Modern Mathematics, 7e: 16.1 - 26
Example 16.4 Standardizing Normal
Data
■
If we divide this difference by  = 10 ft, we
get the standardized value z4 = –2.342. This
tells us the data point x4 is –2.342 standard
deviations from the mean  = 45 ft (D in the
figure).
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Excursions in Modern Mathematics, 7e: 16.1 - 27
STANDARDING RULE
In a normal distribution with mean  and
standard deviation , the standardized
value of a data point x is z = (x – )/.
Pg. 605 # 18a,b,
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Excursions in Modern Mathematics, 7e: 16.1 - 28
Standardizing Normal Data
One important point to note is that while the
original data is given with units, there are no
units given for the z-value.
The units for the z-value are standard
deviations, and this is implicit in the very fact
that it is a z-value.
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Excursions in Modern Mathematics, 7e: 16.1 - 29
Finding the Value of a Data Point
The process of standardizing data can also
be reversed, and given a z-value we can go
back and find the corresponding x-value.
To do this take the formula
z = (x – )/ and solve for x in terms of z.
x =  +  •z
Pg. 605 # 22a,c
Groups Pg. 604 #4d, 8, 18c,d, 22b,d, 25
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Excursions in Modern Mathematics, 7e: 16.1 - 30
The 68-95-99.7 Rule
In a typical bell-shaped distribution, most of
the data are concentrated near the center.
As we move away from the center the
heights of the columns drop rather fast, and
if we move far enough away from the
center, there are essentially no data to be
found.
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Excursions in Modern Mathematics, 7e: 16.1 - 31
The 68-95-99.7 Rule
These are all rather informal observations,
but there is a more formal way to phrase
this, called the 68-95-99.7 rule.
This useful rule is obtained by using one,
two, and three standard deviations above
and below the mean as special landmarks.
In effect, the 68-95-99.7 rule is three
separate rules in one.
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Excursions in Modern Mathematics, 7e: 16.1 - 32
THE 68-95-99.7 RULE
1. In every normal distribution, about
68% of all the data values fall within
one standard deviation above and
below the mean.
In other words, 68% of all the data
have standardized values between
z = –1 and z = 1.
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Excursions in Modern Mathematics, 7e: 16.1 - 33
THE 68-95-99.7 RULE
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Excursions in Modern Mathematics, 7e: 16.1 - 34
THE 68-95-99.7 RULE
2. In every normal distribution, about
95% of all the data values fall within
two standard deviations above and
below the mean.
In other words, 95% of all the data
have standardized values between
z = –2 and z = 2.
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Excursions in Modern Mathematics, 7e: 16.1 - 35
THE 68-95-99.7 RULE
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Excursions in Modern Mathematics, 7e: 16.1 - 36
THE 68-95-99.7 RULE
3. In every normal distribution, about
99.7% (i.e., practically 100%) of all the
data values fall within three standard
deviations above and below the mean.
In other words, 99.7% of all the data
have standardized values between
z = –3 and z = 3.
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Excursions in Modern Mathematics, 7e: 16.1 - 37
THE 68-95-99.7 RULE
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Excursions in Modern Mathematics, 7e: 16.1 - 38
Practical Implications
Earlier in the text, we defined the range R of
a data set (R = Max – Min) and, in the case
of an approximately normal distribution, we
can conclude that the range is about six
standard deviations.
This is true as long as we can assume that
there are no outliers.
Pg. 606 # 32a, 34, 42, 52c
Groups # 34, 44a,d, 52a,b,d,e
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Excursions in Modern Mathematics, 7e: 16.1 - 39