Transcript 16.2 Normal Curves and Normal Distributions

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: 16.2 - 2
Normal Curves
The study of normal curves can be traced
back to the work of the great German
mathematician Carl Friedrich Gauss, and for
this reason, normal curves are sometimes
known as Gaussian curves. Normal curves
all share the same basic shape–that of a
bell–but otherwise they can differ widely in
their appearance.
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Excursions in Modern Mathematics, 7e: 16.2 - 3
Normal Curves
Some bells are short and squat,others are
tall and skinny, and others fall somewhere in
between.
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Normal Curves
Mathematically speaking, however, they all
have the same underlying structure. In fact,
whether a normal curve is skinny and tall or
short and squat depends on the choice of
units on the axes, and any two normal
curves can be made to look the same by
just fiddling with the scales of the axes.
What follows is a summary of some of the
essential facts about normal curves and
their associated normal distributions.These
facts are going to help us greatly later on in
the chapter.
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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, so we can refer to it without
ambiguity as the line of symmetry.
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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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Essential Facts About Normal Curves
Median / mean.
The center represents both the median M
and the mean (average)  of the data. Thus,
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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MEDIAN AND MEAN OF A
NORMAL DISTRIBUTION
In a normal distribution, M = .
(If the distribution is approximately
normal, then M ≈ .)
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Essential Facts About Normal Curves
Standard Deviation.
The standard deviation–traditionally
denoted by the Greek letter  (sigma)–is an
important measure of spread, and it is
particularly useful when dealing with normal
(or approximately normal) distributions, as
we will see shortly. The easiest way to
describe the standard deviation of a normal
distribution is to look at the normal curve.
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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: 16.2 - 11
Essential Facts About Normal Curves
Standard Deviation.
But, at the bottom you
would be bending the
wire upward.
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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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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: 16.2 - 14
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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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 . The
magic number to memorize is 0.675.
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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QUARTILES OF A
NORMAL DISTRIBUTION
In a normal distribution,
Q3 ≈  + (0.675)
and
Q1 ≈  – (0.675).
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Example 16.3 A Mystery Normal
Distribution
Imagine you are told that a data set of
N = 1,494,531 numbers has a normal
distribution with mean  = 515 and standard
deviation  = 114. For now, let’s not worry
about the source of this data–we’ll discuss
this soon.
Just knowing the mean and standard
deviation of this normal distribution will allow
us to draw a few useful conclusions about this
data set.
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Example 16.3 A Mystery Normal
Distribution
■
In a normal distribution, the median
equals the mean, so the median value is
M = 515. This implies that of the
1,494,531 numbers, there are 747,266
that are smaller than or equal to 515 and
747,266 that are greater than or equal to
515.
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Excursions in Modern Mathematics, 7e: 16.2 - 19
Example 16.3 A Mystery Normal
Distribution
■
■
The first quartile is given by
Q1 ≈ 515 – 0.675  114 ≈ 438.
This implies that 25% of the data set
(373,633 numbers) are smaller than or
equal to 438.
The third quartile is given by
Q1 ≈ 515 + 0.675  114 ≈ 592.
This implies that 25% of the data set
(373,633 numbers) are bigger than or
equal to 592.
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Excursions in Modern Mathematics, 7e: 16.2 - 20