24
Normal Curves
Properties of Normal Curves
:
Example
: Draw a Normal curve with a mean of 5, then
draw another Normal curve with a mean of 10 and a
standard deviation equal to that of the first curve.
25
Normal Curves
Properties of Normal Curves
:
4) The standard deviation controls the spread of the
Normal curve.
The larger
is, the more spread out the
σ
curve is.
26
Normal Curves
Properties of Normal Curves
:
5) We can measure
on a Normal curve by taking the
distance from the mean to the “change of curvature
points.” (Where the curve starts to flatten out)
σ
27
28
Normal Curves
Imagine that you are skiing down a mountain that has the shape of a Normal
curve.
At first, you descend at an eversteeper angle as you go out from the
peak:
Normal Curves
Properties of Normal Curves
:
Example
: Draw a Normal curve where =5 and =2.
μ
σ
29
Normal Distribution
Definition
: A Normal distribution
is a distribution that is
described by a Normal curve, and is described in full
by its mean and variance (
and ).
μ
σ
Remember that
and
alone do not specify the shape of
μ
σ
most distributions, and that the shape of
density
curves
in general does not reveal
.
These are special
σ
properties of
Normal distributions
.
30
Normal Distribution
Why are Normal distributions important
?
Three reasons:
1. Normal distributions are good descriptions for
real
data
.
–
Test scores
2. Normal distributions can approximate results of events
of chance (such as those in Vegas).
3. Many effective tools we will learn in this course are
based off of Normal distributions.
31
Normal distributions have a simple rule to follow the spread
of the data without making complex area calculations:
The 689599.7 Rule
(Also known as The Empirical Rule
)
32
689599.7 Rule for
Any Normal
Curve
•
68%
of the observations fall within
one
standard deviation
of the mean
•
95%
of the observations fall within
two
standard
deviations of the mean
•
99.7%
of the observations fall within
three
standard
deviations of the mean
33
689599.7 Rule for
Any Normal Curve
68%
+
σ

σ
µ
+2
σ
2
σ
95%
µ
+3
σ
3
σ
99.7%
µ
34
689599.7 Rule for
Any Normal Curve
35
•
689599.7 Rule
68%
[
μ
±
σ
]
95%
[
μ
±
2
σ
]
99.7%
[
μ
±
3
σ
]
689599.7 Rule for
Any Normal Curve
36
Health and Nutrition Examination
Study of 19761980
•
Heights of adult men, aged 1824
–
mean: 70.0 inches
–
standard deviation: 2.8 inches
–
heights follow a normal distribution, so we
have that heights of men are N(70, 2.8).
37
Health and Nutrition Examination
Study of 19761980
•
689599.7 Rule for men’s heights
68% are between 67.2 and 72.8 inches
[
μ
±
σ
=
70.0
±
2.8 ]
95% are between 64.4 and 75.6 inches
[
μ
±
2
σ
=
70.0
±
2(2.8)
= 70.0
±
5.6 ]
99.7% are between 61.6 and 78.4 inches
[
μ
±
3
σ
=
70.0
±
3(2.8)
= 70.0
±
8.4 ]