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Section 2.1
Density Curves & the
Normal Distributions
AP Statistics
Berkley High School/CASA
Density Curve

“Mathematical
Models”
 The
area under the
curve is related to the
distribution of values
 “Idealized description”
AP Statistics, Section 2.1, Part 1
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Density Curves

Characteristics
 Always
above the x-
axis
 Area always equal to 1

The area under the
curve and above any
range of values is the
proportion of all
observations that fall
in that range.
AP Statistics, Section 2.1, Part 1
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Median and Mean

Since the area
represents
portions of the
population, the
Median is the
spot where the
area to the left
is the same as
the area to the
right.
AP Statistics, Section 2.1, Part 1
4
Median and Mean
The Mean
represents the
“balance
point”.
 Imagine a
“see-saw”

AP Statistics, Section 2.1, Part 1
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Median and Mean

The Mean is
always pulled
towards the
tail in a
skewed
distribution
AP Statistics, Section 2.1, Part 1
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“Idealized Distributions”

We use different
notation for
density curve
x  mean of a sample
(which represent
s  standard deviation of a sample
entire populations)
  mean of a population
as compared to
data sets (which   standard deviation of a population
represent
samples)
AP Statistics, Section 2.1, Part 1
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AP Statistics, Section 2.1, Part 1
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AP Statistics, Section 2.1, Part 1
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AP Statistics, Section 2.1, Part 1
10
Normal Distribution

A special, bell
shaped, symmetric,
single-peaked
distribution
AP Statistics, Section 2.1, Part 1
11
Normal Distribution

Because of the
symmetry, the mean
and median are the
same and at the line
symmetry
AP Statistics, Section 2.1, Part 1
12
Normal Distribution

The inflection points
(where the curve
starts to flatten out)
represent the width of
the standard deviation
μ-σ
AP Statistics, Section 2.1, Part 1
μ
μ+σ
13
68-95-99.7 Rule
AP Statistics, Section 2.1, Part 1
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68-95-99.7 Rule with N(64.5,2.5)
AP Statistics, Section 2.1, Part 1
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Assignment

Exercises 2.1 – 2.15 odd, The Practice of
Statistics
AP Statistics, Section 2.1, Part 1
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