Transcript TPS4e_Ch2_2.1

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Chapter 2: Modeling Distributions of Data
Section 2.1
Describing Location in a Distribution
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
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Chapter 2
Modeling Distributions of Data
 2.1
Describing Location in a Distribution
 2.2
Normal Distributions
+ Section 2.1
Describing Location in a Distribution
Learning Objectives
After this section, you should be able to…
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MEASURE position using percentiles
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INTERPRET cumulative relative frequency graphs
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MEASURE position using z-scores
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TRANSFORM data
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DEFINE and DESCRIBE density curves
One way to describe the location of a value in a distribution
is to tell what percent of observations are less than it.
Definition:
The pth percentile of a distribution is the value
with p percent of the observations less than it.
Describing Location in a Distribution
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Position: Percentiles
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 Measuring
“Percent correct” on a test does not measure the same thing as a percentile.
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Percentile should be whole numbers (decimals should be rounded to the
nearest integer)
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If observations have the same value, they will be at the same percentile. You
calculate percentile by finding the percent of the values in the distribution that
are below both values.
EX 1) The stemplot below shows the number of wins for each of the 30 Major
League Baseball teams in 2009.
Find the percentiles for the following teams:
a) The Colorado Rockies, who won 92 games.
5
6
7
8
9
10
9
2455
00455589
0345667778
123557
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b)The New York Yankees, who won 103 games.
c) The Kansas City Royals and Cleveland Indians,
who both won 65 games.
KEY: 5|9 represents a team with 59 wins.
Describing Location in a Distribution

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Reminders:
Age of First 44 Presidents When They Were
Inaugurated
Age
Frequency
Relative
frequency
Cumulative
frequency
Cumulative
relative
frequency
4044
2
2/44 =
4.5%
2
2/44 =
4.5%
4549
7
7/44 =
15.9%
9
9/44 =
20.5%
5054
13
13/44 =
29.5%
22
5559
12
12/44 =
27.3%
34
34/44 =
77.3%
6064
7
7/44 =
15.9%
41
41/44 =
93.2%
6569
3
3/44 =
6.8%
44
22/44 =
50.0%
44/44 =
100%
Cumulative relative frequency (%)
A cumulative relative frequency graph (or ogive)
displays the cumulative relative frequency of each
class of a frequency distribution.
100
80
60
40
20
0
40 45 50 55 60 65 70
Age at inauguration
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Relative Frequency Graphs
Describing Location in a Distribution
 Cumulative
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Interpreting Cumulative Relative Frequency Graphs
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Reminders
Use the context of the question to start on the correct axis.
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Find its paired value and interpret that number in context of the question.
A) Was Barack Obama, who was inaugurated at age 47, unusually
young?
B) Estimate and
interpret the
65
65th percentile
of the
distribution
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47
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Describing Location in a Distribution
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A z-score tells us how many standard deviations from the
mean an observation falls, and in what direction.
Definition:
If x is an observation from a distribution that has known mean
and standard deviation, the standardized value of x is:
x  mean
z
standard deviation
A standardized value is often called a z-score.
EX 4) Jenny earned a score of 86 on her test. The class mean
is 80 and the standard deviation is 6.07. What is her

standardized
score, and what does it mean?
x  mean
86  80
z

 0.99
standard deviation
6.07
Describing Location in a Distribution
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Position: z-Scores
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 Measuring
z-scores for Comparison
We can use z-scores to compare the position of individuals in different
distributions.
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 Using
EX 5) The single-season home run record for Major League Baseball has been
set just three times since Babe Ruth hit 60 home runs in 1927 (see table below).
In an absolute sense, Barry Bonds had the best performance of these four
players, since he hit the most home runs in a single season. However, in a
relative sense, this may not be true. Baseball historians suggest that hitting a
home run has been easier in some eras than others. This is due to many factors,
including quality of batters, quality of pitchers, hardness of the baseball,
dimensions of ballparks, and possible use of performance-enhancing drugs. How
can we make a fair comparison to see how these performances rate relative to
those of other hitters during the same year? Which player had the most
outstanding performance relative to his peers?
Year
Player
HR
Mean
SD
1927
Babe Ruth
60
7.2
9.7
1961
Roger Maris
61
18.8
13.4
1998
Mark McGwire
70
20.7
12.7
2001
Barry Bonds
73
21.4
13.2
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Data
Transforming converts the original observations from the original
units of measurements to another scale. Transformations can affect
the shape, center, and spread of a distribution.
Effect of Adding (or Subracting) a Constant
Adding the same number a (either positive, zero, or negative) to each
observation:
•adds a to measures of center and location (mean, median,
quartiles, percentiles), but
•Does not change the shape of the distribution or measures of
spread (range, IQR, standard deviation).
n
Example, p. 93
Mean
sx
Min
Q1
M
Q3
Max
IQR
Range
Guess(m)
44
16.02
7.14
8
11
15
17
40
6
32
Error (m)
44
3.02
7.14
-5
-2
2
4
27
6
32
Describing Location in a Distribution
 Transforming
a) Create the dotplot of the distribution.
12 18 22 25 29 31 32 32 35 35
35 36 36 37 37 37 38 38 38 38
40 40 41 42 43 44 45 45 45 48
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EX 1) The below data and summary statistics
is for a sample of 30 test scores. The
maximum possible score on the test was 50
points.
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Score
Score
+5
30
sx
35.8 8.17
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EX 1) b) Suppose the teacher was nice and added
5 points to each test score. How would this
change the shape, center and spread of the
distribution? Create the summary statistics for
the +5 scores in the table below:
Min
Q1
M
Q3
Max
IQR
Rang
e
12
32
37
41
48
9
36
1) c) Suppose the teacher wants to convert the
original test scores to percents. Since the test was
out of 50 points, the scores need to be multiplied
by 2 to make them out of 100. Plot the converted
percentage scores on a dotplot below.
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Score
Score
X2
30
sx
35.8 8.17
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 EX
Min
Q1
M
Q3
Max
IQR
Rang
e
12
32
37
41
48
9
36
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Data
Effect of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each observation by the same number b
(positive, negative, or zero):
•multiplies (divides) measures of center and location by b
•multiplies (divides) measures of spread by |b|, but
•does not change the shape of the distribution
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Example, p. 95
Mean
sx
Min
Q1
M
Q3
Max
IQR
Range
Error(ft)
44
9.91
23.43
-16.4
-6.56
6.56
13.12
88.56
19.68
104.96
Error (m)
44
3.02
7.14
-5
-2
2
4
27
6
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Describing Location in a Distribution
 Transforming
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In Chapter 1, we developed a kit of graphical and numerical
tools for describing distributions. Now, we’ll add one more step
to the strategy.
Exploring Quantitative Data
1. Always plot your data: make a graph.
2. Look for the overall pattern (shape, center, and spread) and
for striking departures such as outliers.
3. Calculate a numerical summary to briefly describe center
and spread.
4.
Sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve.
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Curves
Describing Location in a Distribution
 Density
Curve
A density curve is a curve that
•is always on or above the horizontal axis, and
•has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution.
The area under the curve and above any interval of values on
the horizontal axis is the proportion of all observations that fall in
that interval.
The overall pattern of this histogram of
the scores of all 947 seventh-grade
students in Gary, Indiana, on the
vocabulary part of the Iowa Test of
Basic Skills (ITBS) can be described
by a smooth curve drawn through the
tops of the bars.
Describing Location in a Distribution
Definition:
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 Density
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Our measures of center and spread apply to density curves as
well as to actual sets of observations.
Distinguishing the Median and Mean of a Density Curve
The median of a density curve is the equal-areas point, the
point that divides the area under the curve in half.
The mean of a density curve is the balance point, at which the
curve would balance if made of solid material.
The median and the mean are the same for a symmetric density
curve. They both lie at the center of the curve. The mean of
a skewed curve is pulled away from the median in the
direction of the long tail.
Describing Location in a Distribution

Density Curves
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 Describing
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Section 2.1
Describing Location in a Distribution
Summary
In this section, we learned that…
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There are two ways of describing an individual’s location within a
distribution – the percentile and z-score.
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A cumulative relative frequency graph allows us to examine
location within a distribution.
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It is common to transform data, especially when changing units of
measurement. Transforming data can affect the shape, center, and
spread of a distribution.
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We can sometimes describe the overall pattern of a distribution by a
density curve (an idealized description of a distribution that smooths
out the irregularities in the actual data).
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Looking Ahead…
In the next Section…
We’ll learn about one particularly important class of
density curves – the Normal Distributions
We’ll learn
The 68-95-99.7 Rule
The Standard Normal Distribution
Normal Distribution Calculations, and
Assessing Normality