Transcript Chapter 2: Describing Location In a Distribution

Chapter 2: Describing Location
In a Distribution
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Section 2.1
Measures of Relative Standing
And Density Curves
Case Study
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• Read page 113 in your textbook
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Where are we headed?
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Analyzed a set of
observations
graphically and
numerically
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Consider individual
observations
Consider this data set:
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How
good is
this
score
relative
to the
others?
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Measuring Relative Standing:
z-scores
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• Standardizing: converting scores from the
original values to standard deviation units
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Measuring Relative Standing:
z-scores
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A z-score tells us how many standard deviations away
from the mean the original observation falls, and in
which direction.
Practice: Let’s Do p. 118 #1
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Measuring Relative Standing:
Percentiles
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• Norman got a 72 on the test.
Only 2 of the 25 test scores
in the class are at or below
his.
• His percentile is 2/25 =
0.08, or 8%. So he scores in
the 8th percentile.
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Mathematical
Model
For the
Distribution
Density Curves
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Histogram of the scores of
all 947 seventh-grade
students in Gary, Indiana.
The histogram is:
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•Symmetric
•Both tails fall off smoothly
from a single center peak
•There are no large gaps
•There are no obvious outliers
Density Curves
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Density Curves: Normal Curve
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This curve is an
example of a
NORMAL CURVE.
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More to come later….
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Describing Density Curves
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• Our measure of center and spread apply to
density curves as well as to actual sets of
observations.
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Proportions in a Density Curve
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Describing Density Curves
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• MEDIAN OF A DENSITY CURVE:
– The “equal-areas point”
– The point with half the area under the curve to
its left and the remaining half of the area to its
right
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Describing Density Curves
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• MEAN OF A DENSITY CURVE:
– The “balance point”
– The point at which the curve would balance if
made of solid material
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Mean of a Density Curve
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Usually
Notation
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• Use English letters for statistics
– Measures on a data set
– x = mean
– s = standard deviation
• Use Greek letters for parameters
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– Measures on an idealized distribution
– µ = mean
– σ = standard deviation