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CHAPTER 2
Modeling
Distributions of Data
2.1
Describing Location in a
Distribution
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Describing Location in a Distribution
Learning Objectives
After this section, you should be able to:
 FIND and INTERPRET the percentile of an individual value within a
distribution of data.
 ESTIMATE percentiles and individual values using a cumulative
relative frequency graph.
 FIND and INTERPRET the standardized score (z-score) of an
individual value within a distribution of data.
 DESCRIBE the effect of adding, subtracting, multiplying by, or
dividing by a constant on the shape, center, and spread of a
distribution of data.
The Practice of Statistics, 5th Edition
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Measuring Position: Percentiles
One way to describe the location of a value in a distribution is to tell
what percent of observations are less than it.
The pth percentile of a distribution is the value with p percent of the
observations less than it.
Example
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.
No other student in this class is AT the same
percentile UNLESS they have the score.
The Practice of Statistics, 5th Edition
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Mr. Pryor’s first test.
Use the scores on the test to find the percentiles for the following
students.
• A. Norman who earned a 72.
– __________ student(s) scored lower than Norman so ____________
and he is at the _____________ percentile.
• B.
Katie who scored a 93.
– _________ student(s) scored lower than Katie so ____________
and she is at the _____________ percentile.
• C.
The two students who earned scores of 80.
– _________ student(s) scored lower than an 80
so ______________ and the two students are
both at the _____________ percentile.
6 7
7 2334
7 5777899
8 00123334
8 569
9 03
The Practice of Statistics, 5th Edition
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Cumulative Relative Frequency Graphs
A cumulative relative frequency graph displays the cumulative
relative frequency (the percentiles) of each class of a frequency
distribution. Age 45 is at the 4.5th percentile of the inauguration age
distribution.
100
Age
Frequency
Relative
frequency
Cumulative
frequency
Cumulative
relative
frequency
2/44 =
4.5%
2
2/44 =
4.5%
7/44 =
15.9%
9
4044
2
4549
7
5054
13
13/44 =
29.5%
22
22/44 =
50.0%
5559
12
12/44 =
34%
34
34/44 =
77.3%
6064
7
7/44 =
15.9%
41
41/44 =
93.2%
6569
3
3/44 =
6.8%
44
44/44 =
100%
The Practice of Statistics, 5th Edition
9/44 =
20.5%
Cumulative relative frequency (%)
Age of First 44 Presidents When They Were
Inaugurated
80
60
40
20
0
40
45
50
55
60
65
70
Age at inauguration
5
Cumulative Relative Frequency Graphs
Estimate the IQR of the
distribution.
• HINT: Q1 is at the 25th
percentile and Q3 is at
the 75th percentile.
100
Cumulative relative frequency (%)
Estimate the median age of
this distribution.
• HINT: the median is at the
50th percentile.
80
60
40
20
0
40
45
50
55
60
65
70
Age at inauguration
The Practice of Statistics, 5th Edition
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Measuring Position: z-Scores
A z-score tells us how many standard deviations from the mean an
observation falls, and in what direction.
If x is an observation from a distribution that has known mean and
standard deviation, the standardized score of x is:
x - mean
z=
standard deviation
A standardized score 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
The Practice of Statistics, 5th Edition
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Measuring Position: z-Scores
Brent is 74 inches tall. The mean height in his math class is 67
inches with a standard deviation of 4.29 inches.
• Find Brent’s standardized score.
Brent is a member of the school’s basketball team. The mean
height of the players on the team is 76 inches. Brent’s height
translates to a z-score of -0.85 in the team’s height distribution.
• Interpret the meaning of this z-score.
• Compare this z-score to his standardized score of his height.
• Find the standard deviation of the height distribution of the
basketball team
The Practice of Statistics, 5th Edition
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Transforming Data
Transforming converts the original observations from the original units
of measurements to another scale.
Effect of Adding (or Subtracting) a Constant
Adding the same number a to (subtracting a from) each observation:
• adds a to (subtracts a from) 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).
Effect of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each observation by the same number b:
• multiplies (divides) measures of center and location (mean,
median, quartiles, percentiles) by b
• multiplies (divides) measures of spread (range, IQR, standard
deviation) by |b|, but
• does not change the shape of the distribution
The Practice of Statistics, 5th Edition
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Describing Location in a Distribution
Section Summary
In this section, we learned how to…
 FIND and INTERPRET the percentile of an individual value within a
distribution of data.
 ESTIMATE percentiles and individual values using a cumulative
relative frequency graph.
 FIND and INTERPRET the standardized score (z-score) of an
individual value within a distribution of data.
 DESCRIBE the effect of adding, subtracting, multiplying by, or dividing
by a constant on the shape, center, and spread of a distribution of
data.
The Practice of Statistics, 5th Edition
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