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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…

MEASURE position using percentiles

INTERPRET cumulative relative frequency graphs

MEASURE position using z-scores

TRANSFORM data

DEFINE and DESCRIBE density curves
+ WARM UP
25
32
15
19
20
22
24
30
1.
Find the quartiles of the data.
2.
What percentage of the data is to the left
of Q1?
3.
What percentage of the data is to the
right of Q2?
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25
32
15
19
20
22
24
30
1. Find the quartiles of the data.
Q1 = 19.5, Q2 = 23 and Q3 = 27.5
2. What percentage of the data is to the left
of Q1? 25%
3. What percentage of the data is to the
right of Q2? 50%
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.
Example, p. 85
Jenny earned a score of 86 on her test. How did she perform
relative to the rest of the class?
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
Her score was greater than 21 of the 25
observations. Since 21 of the 25, or 84%, of the
scores are below hers, Jenny is at the 84th
percentile in the class’s test score distribution.
Describing Location in a Distribution

Position: Percentiles
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 Measuring
Example: Wins in Major League Baseball

The stemplot below shows the number of wins for each of
the 30 Major League Baseball teams in 2009.
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

Calculate and interpret the percentiles for the Colorado
Rockies who had 92 wins, the New York Yankees who had
103 wins, and the Cleveland Indians who had 65 wins.
Example
Key: 5|9 represents a
team with 59 wins.
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Relative Frequency
Cumulative Relative Frequency adds the counts in the
frequency column for the current class and all previous
classes.
Example: Let’s look at the frequency table for the ages of the 1st
44 US presidents when they were inaugurated.
Age
Frequency
Relative
Frequency
Cumulative
Frequency
Cumulative Rel.
Frequency
40 – 44
2
0.045
2
0.045
45 – 49
7
0.159
9
0.205
50 – 54
13
0.295
22
0.500
55 – 59
12
0.273
34
0.773
60 – 64
7
0.159
41
0.932
65 – 69
3
0.068
44
1.000
Total
44
1.00 or 100%
Describing Location in a Distribution
 Cumulative
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Relative Frequency Graphs
A cumulative relative frequency graph displays the
cumulative relative frequency of each class of a
frequency distribution.
Age
Frequency
Relative
frequency
Cumulative
frequency
Cumulative
relative
frequency
2/44 =
4.5%
2
2/44 =
4.5%
7/44 =
15.9%
9
13/44 =
29.5%
22
12/44 =
34%
34
4044
2
4549
7
5054
13
5559
12
6064
7
7/44 =
15.9%
41
41/44 =
93.2%
6569
3
3/44 =
6.8%
44
44/44 =
100%
9/44 =
20.5%
22/44 =
50.0%
34/44 =
77.3%
Cumulative relative frequency (%)
Age of First 44 Presidents When They Were
Inaugurated
100
80
60
40
20
0
40
45
50
55
60
65
Age at inauguration
70
Describing Location in a Distribution
 Cumulative
Interpreting Cumulative Relative Frequency Graphs

Was Barack Obama, who was inaugurated at age 47,
unusually young?

Estimate and interpret the 65th percentile of the distribution
65
11
47
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Describing Location in a Distribution
Use the graph from page 88 to answer the following questions.
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

Here is a cumulative relative frequency graph showing the
distribution of median household incomes for the 50 states
and the District of Columbia.
The point (50, 0.49) means that
49% of the states had a median
household income of less than
$50,000.
a) California, with a median household income of $57,445, is at what
percentile? Interpret this value.
b) What is the 25th percentile for this distribution? What is another
name for this value?
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Example: State Median Household Incomes
Example

A z-score is a value that 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.
Example: 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?
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
We can use z-scores to compare the position of individuals in
different distributions.
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.
Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry
Bonds hit 73 in 2001. 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. To make a fair comparison, we should see how these
performances rate relative to others hitters during the same year.
Calculate the standardized score for each player and compare.
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z-scores for Comparison
Describing Location in a Distribution
 Using
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Ruth: 60 – 7.2 = 5.44
McGwire: 70 – 20.7 = 3.87
9.7
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Maris: 61 – 18.8 = 3.16
13.4
12.7
Bonds: 73 – 21.4 = 3.91
13.2
Although all 4 performances were outstanding, Babe Ruth
can still lay claim to being the single-season home run
champ, relatively speaking.
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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. Roger
Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit
73 in 2001. 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. To
make a fair comparison, we should see how these performances rate
relative to others hitters during the same year. Calculate the
standardized score for each player and compare.
Example

 1.
Lynette, a student in the class, is 65 inches
tall. Find and interpret her z-score.
 2. Another
student in the class, Brent, is 74
inches tall. How tall is Brent compared with the
rest of the class?
 3.
Brent is a member of the school’s basketball
team. The mean height of the players is 76
inches. If Brent’s height translates to a z-score of
-0.85, what is the standard deviation of the team
member’s heights?
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WARM UP
Mrs. Navard measured the heights of all her
Statistics students in inches. She found that
their mean was 67 with a standard deviation of
4.29.
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1. Lynette, a student in the class, is 65 inches tall. Find and
interpret her z-score.
z = -0.47 ; Lynette’s height is
0.47 standard deviations below the mean
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2. Another student in the class, Brent, is 74 inches tall.
How tall is Brent compared with the rest of the class?
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Brent’s z-score is 1.63. He is about 1.63 standard
deviations above the mean.
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3. Brent is a member of the school’s basketball team. The
mean height of the players is 76 inches. If Brent’s height
translates to a z-score of -0.85, what is the standard
deviation of the team member’s heights?
2.35 inches
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Mrs. Navard measured the heights of all her Statistics
students in inches. She found that their mean was 67 with
a standard deviation of 4.29.
Answers
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a. A score of 60, where the mean score of the sample data values
is 40.
b. A score that is 30 points below the mean.
c. A score of 80, where the mean score of the sample data values
is 30.
d. A score of 20, where the mean score of the sample data values
is 50.
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Z-Scores Practice
1. A normal distribution of scores has a standard deviation of 10.
Find the z-scores corresponding to each of the following
values:
b. How many standard deviations is that?
c. If we consider “usual IQ scores to be those that convert z
scores between -2 and 2, is Einstein’s IQ usual or unusual?
Z-Scores Practice
a. What is the difference between Einstein’s IQ and the mean?
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2. IQ scores have a mean of 100 and a standard deviation of 16.
Albert Einstein reportedly had an IQ of 160.
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Z-Scores Practice
3. Women’s heights have a mean of 63.6 in. and a standard
deviation of 2.5 inches. Find the z-score corresponding to a
woman with a height of 70 inches and determine whether the
height is unusual.
b. A score of 90 on a test with a mean of 86 and a standard
deviation of 18.
c. A score of 18 on a test with a mean of 15 and a standard
deviation of 5.
Z-Scores Practice
a. A score of 144 on a test with a mean of 128 and a standard
deviation of 34.
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4. Three students take equivalent stress tests. Which is the
highest relative score?
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).
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Data
Describing Location in a Distribution
 Transforming

Suppose that the teacher is nice and wants to add 5
points to each test score. How would this affect the
shape, center and spread of the data?
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Example: Here is a graph and table of summary statistics
for a sample of 30 test scores. The maximum possible
score on the test was a 50.
Example
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Example
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The measures of center and position (mean, median,
quartiles, and percentiles) all increased by 5. However, the
shape and spread (range, IQR, standard deviation) of the
distribution did not change.
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
Example: Suppose that the teacher wants to convert each test
score to percents. Since the test was out of 50 points, he should
just multiply each score by 2 to make them out of 100. How
would this change the shape, center and spread of the data?
The measures of center, location and spread will all double. However,
even though the distribution is more spread out, the shape will not
change!!!
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Data
Describing Location in a Distribution
 Transforming
1. How would adding 50 points to each score affect the
mean?
2. The standard deviation was 100 points. What would
the standard deviation be after adding 50 points?
3. The times in the men’s combined event at the Winter
Olympics are reported in minutes and seconds. The
mean and standard deviation of the slalom times are
94.27 and 5.28 respectively. Suppose instead that we
report the times in minutes. What would the resulting
mean and standard deviation be?
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WARM UP
In 1995 the Educational Testing Service (ETS) adjusted
the scores of SAT tests. Before ETS re-centered the
SAT verbal test, the mean of all test scores was 450.
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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.
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So far our strategy for exploring data is :
 1. Graph the data to get an idea of the overall pattern
 2. Calculate an appropriate numerical summary to
describe the center and spread of the distribution.
Sometimes the overall pattern of a large number of
observations is so regular, that we can describe it by a
smooth curve, called a density 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
Here’s the idea behind density curves:
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No real set of data is described by a density curve. It
is simply an approximation that is easy to use and accurate
enough for practical use.
Describing
Density Curves
The
median of the density curve is the “equal-areas” point.
The
mean of the density curve is the “balance” point.
Density Curves
NOTE:
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Section 2.1
Describing Location in a Distribution
Summary
In this section, we learned that…

There are two ways of describing an individual’s location within a
distribution – the percentile and z-score.

A cumulative relative frequency graph allows us to examine
location within a distribution.

It is common to transform data, especially when changing units of
measurement. Transforming data can affect the shape, center, and
spread of a distribution.

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