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Lecture 3, Section 1.3
As I have mentioned before, if the overall pattern of a large number of
observations is quite regular, we chose to describe it by a smooth curve called a
density curve
.
A density curve is an idealized model for the distribution of a
quantitative variable.
A Density curve
has the following properties:
1. Is on or above the horizontal axis.
2.
The total area under the curve is 1.
3.
The area under the curve and above any range of values is the relative
frequency of all observations that fall in that range (probability of
occurrence).
4. Because density curves are continuous distributions, the chance of any
exact value occurring is 0; only a range of values has a frequency, a
probability of occurring.
The
median
of a density curve is the equalareas point.
The
mean
of a density curve is the balance point.
The
median and the mean are always equal on a symmetric density curve.
We designate the mean of a density curve
as
μ
and the standard deviation as
σ
,
when we are dealing with the
population.
When we take actual observations of a
sample
,
we distinguish the mean of the distribution of these observations as
x
and
the standard deviation as
s
.
Lecture 3, Section 1.3
Page 1
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are a particularly important class of density curves.
These
density curves are symmetric, unimodal, and bellshaped.
They have the following properties:
•
They are all symmetric
•
Their mean is equivalent to their median
•
The standard deviation
σ
controls the spread of a normal curve.
We can
actually locate
by eye on a normal curve.
•
Changing the mean,
μ
,
without changing standard deviation,
,
moves the
normal curve along the horizontal axis without changing the spread.
•
Changing the
without changing
changes only the spread of the normal
distribution.
•
The Normal density curve can be fully described by giving its mean,
,
and
standard deviation,
.
The values
and
are parameters of the curve
and the Normal curve is completely determined by
and
.
Lecture 3, Section 1.3
Page 2
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 Spring '08
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