Download Chapter 3 - 3.3,3.4,3.5

Section
3.3
Measures of
Central Tendency
and
Dispersion from
Grouped Data
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Approximate the mean of a variable from
grouped data
2. Compute the weighted mean
3. Approximate the standard deviation of a
variable from grouped data
3-2
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Objective 1
• Approximate the Mean of a Variable from
Grouped Data
3-3
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We have discussed how to compute descriptive
statistics from raw data, but often the only
available data have already been summarized in
frequency distributions (grouped data).
Although we cannot find exact values of the
mean or standard deviation without raw data, we
can approximate these measures using the
techniques discussed in this section.
3-4
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Approximate the Mean of a Variable from a
Frequency Distribution
Population Mean
Sample Mean
xf


f
xf

x
f
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
i i
i
i i
i
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
n is the number of classes
3-5
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EXAMPLE Approximating the Mean from a Relative
Frequency Distribution
The National Survey of Student Engagement is a survey
that (among other things) asked first year students at
liberal arts colleges how much time they spend
preparing for class each week. The results from the 2007
survey are summarized below. Approximate the mean
number of hours spent preparing for class each week.
Hours
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
3-6
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Time Frequency
0
0
1-5
130
6 - 10
250
11 - 15
230
16 - 20
180
21 - 25
100
26 – 30
60
31 – 35
50
 fi  1000
3-7
xi
xi fi
0
0
3.5
455
8.5
2125
13.5
3105
18.5
3330
23.5
2350
28.5
1710
33.5
1675
 xi fi  14,750
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
xf

x
f
i i
i
14,750

1000
 14.75
Objective 2
• Compute the Weighted Mean
3-8
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The weighted mean, xw , of a variable is found by
multiplying each value of the variable by its
corresponding weight, adding these products, and
dividing this sum by the sum of the weights. It
can be expressed using the formula
xw
wx


w
i i
i
w1 x1  w2 x2  ...  wn xn

w1  w2  ...  wn
where w is the weight of the ith observation
xi is the value of the ith observation
3-9
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE Computed a Weighted Mean
Bob goes to the “Buy the Weigh” Nut store and creates
his own bridge mix. He combines 1 pound of raisins, 2
pounds of chocolate covered peanuts, and 1.5 pounds
of cashews. The raisins cost $1.25 per pound, the
chocolate covered peanuts cost $3.25 per pound, and
the cashews cost $5.40 per pound. What is the cost per
pound of this mix?
1($1.25)  2($3.25)  1.5($5.40)
xw 
1  2  1.5
$15.85

 $3.52
4.5
3-10
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Objective 3
• Approximate the Standard Deviation of a
Variable from Grouped Data
3-11
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Approximate the Standard Deviation of a
Variable from a Frequency Distribution
Population
Standard Deviation
 x   
f
2

i
fi
Sample
Standard Deviation
 x  x  f
 f  1
2
s
i
i
i
i
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
3-12
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An algebraically equivalent formula for the
population standard deviation is
x f


f 
2
x
i
2
i i
f
f
i
3-13
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i
EXAMPLE Approximating the Standard Deviation
from a Relative Frequency Distribution
The National Survey of Student Engagement is a survey
that (among other things) asked first year students at
liberal arts colleges how much time they spend
preparing for class each week. The results from the 2007
survey are summarized below. Approximate the standard
deviation number of hours spent preparing for class each
week.
Hours
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
3-14
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Time
0
1-5
6 - 10
11 - 15
16 - 20
21 - 25
26 – 30
31 – 35
3-15
Frequ
ency
xi
0
0
130
3.5
250
8.5
230 13.5
180 18.5
100 23.5
60
28.5
50
33.5
 fi  1000
xi  x
0
–11.25
–6.25
–1.25
3.75
8.75
13.75
18.75
 
 
xi  x f i s 2   x i  x f i
0
 fi  1
16,453.125
65,687.5

9765.625
1000  1
 65.8
359.375
2531.25
7656.25 s  s 2  65.8
11,343.75  8.1 hours
17,578.125
2
 xi  x fi  65,687.5

2

2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.

Section
3.4
Measures of
Position and
Outliers
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1.
2.
3.
4.
Determine and interpret z-scores
Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile
range
5. Check a set of data for outliers
3-17
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Objective 1
•
3-18
Determine and Interpret z-Scores
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The z-score represents the distance that a data
value is from the mean in terms of the number of
standard deviations. We find it by subtracting the
mean from the data value and dividing this result
by the standard deviation. There is both a
population z-score and a sample z-score:
Population z-score
Sample z-score
x
xx
z
z

s
σ
The z-score is unitless. It has mean 0 and standard
deviation 1.
3-19
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EXAMPLE Using Z-Scores
The mean height of males 20 years or older is 69.1
inches with a standard deviation of 2.8 inches. The
mean height of females 20 years or older is 63.7
inches with a standard deviation of 2.7 inches. Data
is based on information obtained from National
Health and Examination Survey. Who is relatively
taller?
Kevin Garnett whose height is 83 inches
or
Candace Parker whose height is 76 inches
3-20
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83  69.1
zkg 
2.8
 4.96
76  63.7
zcp 
2.7
 4.56
Kevin Garnett’s height is 4.96 standard
deviations above the mean. Candace
Parker’s height is 4.56 standard deviations
above the mean. Kevin Garnett is
relatively taller.
3-21
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Objective 2
• Interpret Percentiles
3-22
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The kth percentile, denoted, Pk , of a set of
data is a value such that k percent of the
observations are less than or equal to the value.
3-23
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EXAMPLE
Interpret a Percentile
The Graduate Record Examination (GRE) is a test
required for admission to many U.S. graduate schools.
The University of Pittsburgh Graduate School of Public
Health requires a GRE score no less than the 70th
percentile for admission into their Human Genetics
MPH or MS program.
(Source:
http://www.publichealth.pitt.edu/interior.php?pageID=1
01.)
Interpret this admissions requirement.
3-24
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EXAMPLE
Interpret a Percentile
In general, the 70th percentile is the score such that
70% of the individuals who took the exam scored
worse, and 30% of the individuals scores better. In
order to be admitted to this program, an applicant must
score as high or higher than 70% of the people who
take the GRE. Put another way, the individual’s score
must be in the top 30%.
3-25
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Objective 3
• Determine and Interpret Quartiles
3-26
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Quartiles divide data sets into fourths, or four equal parts.
• The 1st quartile, denoted Q1, divides the bottom 25%
the data from the top 75%. Therefore, the 1st quartile is
equivalent to the 25th percentile.
• The 2nd quartile divides the bottom 50% of the data
from the top 50% of the data, so that the 2nd quartile is
equivalent to the 50th percentile, which is equivalent to
the median.
• The 3rd quartile divides the bottom 75% of the data
from the top 25% of the data, so that the 3rd quartile is
equivalent to the 75th percentile.
3-27
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Finding Quartiles
Step 1 Arrange the data in ascending order.
Step 2 Determine the median, M, or second
quartile, Q2 .
Step 3 Divide the data set into halves: the
observations below (to the left of) M and
the observations above M. The first
quartile, Q1 , is the median of the bottom
half, and the third quartile, Q3 , is the
median of the top half.
3-28
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EXAMPLE
Finding and Interpreting Quartiles
A group of Brigham Young University—Idaho students
(Matthew Herring, Nathan Spencer, Mark Walker, and
Mark Steiner) collected data on the speed of vehicles
traveling through a construction zone on a state
highway, where the posted speed was 25 mph. The
recorded speed of 14 randomly selected vehicles is
given below:
20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40
Find and interpret the quartiles for speed in the
construction zone.
3-29
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EXAMPLE
Finding and Interpreting Quartiles
Step 1: The data is already in ascending order.
Step 2: There are n = 14 observations, so the median,
or second quartile, Q2, is the mean of the 7th and 8th
observations. Therefore, M = 32.5.
Step 3: The median of the bottom half of the data is the
first quartile, Q1.
20, 24, 27, 28, 29, 30, 32
The median of these seven observations is 28.
Therefore, Q1 = 28. The median of the top half of the
data is the third quartile, Q3. Therefore, Q3 = 38.
3-30
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Interpretation:
• 25% of the speeds are less than or equal to the first
quartile, 28 miles per hour, and 75% of the speeds
are greater than 28 miles per hour.
• 50% of the speeds are less than or equal to the
second quartile, 32.5 miles per hour, and 50% of the
speeds are greater than 32.5 miles per hour.
• 75% of the speeds are less than or equal to the third
quartile, 38 miles per hour, and 25% of the speeds
are greater than 38 miles per hour.
3-31
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Objective 4
• Determine and Interpret the Interquartile
Range
3-32
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The interquartile range, IQR, is the range of
the middle 50% of the observations in a data set.
That is, the IQR is the difference between the
third and first quartiles and is found using the
formula
IQR = Q3 – Q1
3-33
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EXAMPLE
Determining and Interpreting the
Interquartile Range
Determine and interpret the interquartile range of the
speed data.
Q1 = 28
Q3 = 38
IQR  Q3  Q1
 38  28
 10
The range of the middle 50% of the speed of cars
traveling through the construction zone is 10 miles per
hour.
3-34
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Suppose a 15th car travels through the construction zone at
100 miles per hour. How does this value impact the mean,
median, standard deviation, and interquartile range?
Mean
Median
Standard deviation
IQR
3-35
Without 15th car
32.1 mph
32.5 mph
6.2 mph
10 mph
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With 15th car
36.7 mph
33 mph
18.5 mph
11 mph
Objective 5
• Check a Set of Data for Outliers
3-36
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Checking for Outliers by Using Quartiles
Step 1 Determine the first and third quartiles of the
data.
Step 2 Compute the interquartile range.
Step 3 Determine the fences. Fences serve as
cutoff points for determining outliers.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q3 + 1.5(IQR)
Step 4 If a data value is less than the lower fence
or greater than the upper fence, it is
considered an outlier.
3-37
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EXAMPLE
Determining and Interpreting the
Interquartile Range
Check the speed data for outliers.
Step 1: The first and third quartiles are Q1 = 28 mph
and Q3 = 38 mph.
Step 2: The interquartile range is 10 mph.
Step 3: The fences are
Lower Fence = Q1 – 1.5(IQR) = 28 – 1.5(10) = 13 mph
Upper Fence = Q3 + 1.5(IQR) = 38 + 1.5(10) = 53 mph
Step 4: There are no values less than 13 mph or greater
than 53 mph. Therefore, there are no outliers.
3-38
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Section
3.5
The Five-Number
Summary and
Boxplots
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Compute the five-number summary
2. Draw and interpret boxplots
3-40
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Objective 1
•
3-41
Compute the Five-Number Summary
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The five-number summary of a set of data
consists of the smallest data value, Q1, the
median, Q3, and the largest data value. We
organize the five-number summary as follows:
3-42
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EXAMPLE
Obtaining the Five-Number Summary
Every six months, the United States Federal
Reserve Board conducts a survey of credit card
plans in the U.S. The following data are the
interest rates charged by 10 credit card issuers
randomly selected for the July 2005 survey.
Determine the five-number summary of the data.
3-43
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE
Obtaining the Five-Number Summary
Institution
Pulaski Bank and Trust Company
Rainier Pacific Savings Bank
Wells Fargo Bank NA
Firstbank of Colorado
Lafayette Ambassador Bank
Infibank
United Bank, Inc.
First National Bank of The Mid-Cities
Bank of Louisiana
Bar Harbor Bank and Trust Company
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm
3-44
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Rate
6.5%
12.0%
14.4%
14.4%
14.3%
13.0%
13.3%
13.9%
9.9%
14.5%
EXAMPLE
Obtaining the Five-Number Summary
First, we write the data in ascending order:
6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%,
14.3%, 14.4%, 14.4%, 14.5%
The smallest number is 6.5%. The largest
number is 14.5%. The first quartile is 12.0%.
The second quartile is 13.6%. The third
quartile is 14.4%.
Five-number Summary:
6.5%
3-45
12.0%
13.6%
14.4%
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
14.5%
Objective 2
• Draw and Interpret Boxplots
3-46
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Drawing a Boxplot
Step 1 Determine the lower and upper fences.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q3 + 1.5(IQR)
where IQR = Q3 – Q1
Step 2 Draw a number line long enough to include
the maximum and minimum values. Insert
vertical lines at Q1, M, and Q3. Enclose
these vertical lines in a box.
Step 3 Label the lower and upper fences.
3-47
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Drawing a Boxplot
Step 4 Draw a line from Q1 to the smallest data
value that is larger than the lower fence.
Draw a line from Q3 to the largest data
value that is smaller than the upper fence.
These lines are called whiskers.
Step 5 Any data values less than the lower fence or
greater than the upper fence are outliers and
are marked with an asterisk (*).
3-48
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EXAMPLE
Obtaining the Five-Number Summary
Every six months, the United States Federal
Reserve Board conducts a survey of credit card
plans in the U.S. The following data are the
interest rates charged by 10 credit card issuers
randomly selected for the July 2005 survey.
Construct a boxplot of the data.
3-49
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE
Obtaining the Five-Number Summary
Institution
Pulaski Bank and Trust Company
Rainier Pacific Savings Bank
Wells Fargo Bank NA
Firstbank of Colorado
Lafayette Ambassador Bank
Infibank
United Bank, Inc.
First National Bank of The Mid-Cities
Bank of Louisiana
Bar Harbor Bank and Trust Company
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm
3-50
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Rate
6.5%
12.0%
14.4%
14.4%
14.3%
13.0%
13.3%
13.9%
9.9%
14.5%
Step 1: The interquartile range (IQR) is 14.4% - 12% =
2.4%. The lower and upper fences are:
Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4%
Upper Fence = Q3 + 1.5(IQR) = 14.4 + 1.5(2.4) = 18.0%
Step 2:
*
3-51
[
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
]
Use a boxplot and quartiles to describe the shape of a
distribution.
The interest rate boxplot indicates that the distribution
is skewed left.
3-52
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