Chapter 03 NCC STAT

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.
Chapter
Numerically
Summarizing Data
3

Section
Measures of
Central Tendency
3.1

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.3-3
Objectives
1. Determine the arithmetic mean of a variable from
raw data
2. Determine the median of a variable from raw data
3. Explain what it means for a statistic to be resistant
4. Determine the mode of a variable from raw data

Objective 1
• Determine the Arithmetic Mean of a Variable
from Raw Data

The arithmetic mean of a variable is computed
by adding all the values of the variable in the
data set and dividing by the number of
observations.

The population arithmetic mean, μ
(pronounced “mew”), is computed using all the
individuals in a population.
The population mean is a parameter.

The sample arithmetic mean, (pronounced
“x-bar”), is computed using sample data.
The sample mean is a statistic.
x

If x1, x2, …, xN are the N observations of a
variable from a population, then the population
mean, µ, is

If x1, x2, …, xn are the n observations of a
variable from a sample, then the sample mean,
, is x

EXAMPLE Computing a Population Mean and a Sample
Mean
The following data represent the travel times (in minutes)
to work for all seven employees of a start-up web
development company.
23, 36, 23, 18, 5, 26, 43
(a) Compute the population mean of this data.
(b)Then take a simple random sample of n = 3 employees.
Compute the sample mean. Obtain a second simple
random sample of n = 3 employees. Again compute the
sample mean.

Mean
1 2 7...
7
ix
N
x x x
 
  


23 36 23 18 5 26 43
7
     

174
7

24.9 minutes
(a)

Mean
(b) Obtain a simple random sample of size n = 3 from the
population of seven employees. Use this simple random sample
to determine a sample mean. Find a second simple random
sample and determine the sample mean.
1 2 3 4 5 6 7
23, 36, 23, 18, 5, 26, 43
5 36 26
3
22.3
x
 


36 23 26
3
28.3
x
 



Objective 2
• Determine the Median of a Variable from Raw
Data

The median of a variable is the value that lies in
the middle of the data when arranged in
ascending order.
We use M to represent the median.

Steps in Finding the Median of a Data Set
Step 1 Arrange the data in ascending order.
Step 2 Determine the number of observations, n.
Step 3 Determine the observation in the middle
of the data set.

Steps in Finding the Median of a Data Set
• If the number of observations is odd, then the
median is the data value exactly in the middle
of the data set. That is, the median is the
observation that lies in then (n + 1)/2 position.
• If the number of observations is even, then the
median is the mean of the two middle
observations in the data set. That is, the median
is the mean of the observations that lie in the
n/2 position and the n/2 + 1 position.

EXAMPLE Computing a Median of a Data Set with an Odd
Number of Observations
The following data represent the travel times (in minutes) to
work for all seven employees of a start-up web development
company.
23, 36, 23, 18, 5, 26, 43
Determine the median of this data.
Step 1: 5, 18, 23, 23, 26, 36, 43
Step 2: There are n = 7 observations.
1 7 1
4
2 2
n  
 Step 3: M = 23
5, 18, 23, 23, 26, 36, 43

EXAMPLE Computing a Median of a Data Set with an
Even Number of Observations
Suppose the start-up company hires a new employee. The travel
time of the new employee is 70 minutes. Determine the median
of the “new” data set.
23, 36, 23, 18, 5, 26, 43, 70
Step 1: 5, 18, 23, 23, 26, 36, 43, 70
Step 2: There are n = 8 observations.
1 8 1
4.5
2 2
n  
 Step 3:
5, 18, 23, 23, 26, 36, 43, 70
23 26
24.5 minutes
2
M

 
24.5M



Objective 3
• Explain What it Means for a Statistic to Be
Resistant

EXAMPLE Computing a Median of a Data Set with an
Even Number of Observations
The following data represent the travel times (in minutes) to
work for all seven employees of a start-up web development
company.
23, 36, 23, 18, 5, 26, 43
Suppose a new employee is hired who has a 130 minute
commute. How does this impact the value of the mean and
median?
Mean before new hire: 24.9 minutes
Median before new hire: 23 minutes
Mean after new hire: 38 minutes
Median after new hire: 24.5 minutes

A numerical summary of data is said to be
resistant if extreme values (very large or
small) relative to the data do not affect its
value substantially.

EXAMPLE Describing the Shape of the Distribution
The following data represent the asking price of homes
for sale in Lincoln, NE.
Source: http://www.homeseekers.com
79,995 128,950 149,900 189,900
99,899 130,950 151,350 203,950
105,200 131,800 154,900 217,500
111,000 132,300 159,900 260,000
120,000 134,950 163,300 284,900
121,700 135,500 165,000 299,900
125,950 138,500 174,850 309,900
126,900 147,500 180,000 349,900

Find the mean and median. Use the mean and
median to identify the shape of the distribution.
Verify your result by drawing a histogram of the
data.

Find the mean and median. Use the mean and
median to identify the shape of the distribution.
Verify your result by drawing a histogram of the
data.
The mean asking price is $168,320 and the
median asking price is $148,700. Therefore,
we would conjecture that the distribution is
skewed right.

350000300000250000200000150000100000
12
10
8
6
4
2
0
Asking Price
Frequency
Asking Price of Homes in Lincoln, NE

Objective 4
• Determine the Mode of a Variable from Raw
Data

The mode of a variable is the most frequent
observation of the variable that occurs in the data
set.
A set of data can have no mode, one mode, or
more than one mode.
If no observation occurs more than once, we say
the data have no mode.

EXAMPLE Finding the Mode of a Data Set
The data on the next slide represent the Vice
Presidents of the United States and their state of
birth. Find the mode.

Joe Biden Pennsylvani
a

The mode is
New York.

Tally data to determine
most frequent observation

Section
Measures of
Dispersion
3.2

Objectives
1. Determine the range of a variable from raw data
2. Determine the standard deviation of a variable from
raw data
3. Determine the variance of a variable from raw data
4. Use the Empirical Rule to describe data that are bell
shaped
5. Use Chebyshev’s Inequality to describe any data set

To order food at a McDonald’s restaurant, one must choose
from multiple lines, while at Wendy’s Restaurant, one
enters a single line. The following data represent the wait
time (in minutes) in line for a simple random sample of 30
customers at each restaurant during the lunch hour. For
each sample, answer the following:
(a) What was the mean wait time?
(b) Draw a histogram of each restaurant’s wait time.
(c ) Which restaurant’s wait time appears more dispersed?
Which line would you prefer to wait in? Why?

1.50 0.79 1.01 1.66 0.94 0.67
2.53 1.20 1.46 0.89 0.95 0.90
1.88 2.94 1.40 1.33 1.20 0.84
3.99 1.90 1.00 1.54 0.99 0.35
0.90 1.23 0.92 1.09 1.72 2.00
3.50 0.00 0.38 0.43 1.82 3.04
0.00 0.26 0.14 0.60 2.33 2.54
1.97 0.71 2.22 4.54 0.80 0.50
0.00 0.28 0.44 1.38 0.92 1.17
3.08 2.75 0.36 3.10 2.19 0.23
Wait Time at Wendy’s
Wait Time at McDonald’s

(a) The mean wait time in each line is 1.39
minutes.

(b)

Objective 1
• Determine the Range of a Variable from Raw
Data

The range, R, of a variable is the difference
between the largest data value and the smallest data
values. That is,
Range = R = Largest Data Value – Smallest Data Value

EXAMPLE Finding the Range of a Set of Data
The following data represent the travel times (in
minutes) to work for all seven employees of a
start-up web development company.
23, 36, 23, 18, 5, 26, 43
Find the range.
Range = 43 – 5
= 38 minutes

Objective 2
• Determine the Standard Deviation of a
Variable from Raw Data

The population standard deviation of a
variable is the square root of the sum of squared
deviations about the population mean divided by
the number of observations in the population, N.
That is, it is the square root of the mean of the
squared deviations about the population mean.
The population standard deviation is
symbolically represented by σ (lowercase Greek
sigma).

where x1, x2, . . . , xN are the N observations in
the population and μ is the population mean.
 
x1   2
 x2   2
L  xN   2
N

xi   2

N

A formula that is equivalent to the one on the
previous slide, called the computational
formula, for determining the population
standard deviation is
 
xi
2

xi 
2
N

N

EXAMPLE Computing a Population Standard
Deviation
minutes) to work for all seven employees of a start-
up web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population standard deviation of this
data.

xi μ xi – μ (xi – μ)2
23 24.85714 -1.85714 3.44898
36 24.85714 11.14286 124.1633
23 24.85714 -1.85714 3.44898
18 24.85714 -6.85714 47.02041
5 24.85714 -19.8571 394.3061
26 24.85714 1.142857 1.306122
43 24.85714 18.14286 329.1633
902.8571 
2
ix  
 
xi   2

N

902.8571
7
11.36 minutes

xi (xi )2
23 529
36 1296
23 529
18 324
5 25
26 676
43 1849
Σ xi = 174 Σ (xi)2 = 5228
 
xi
2

xi 
2
N
N

5228 
174 2
7
7
11.36 minutes
Using the computational formula, yields the same
result.

The sample standard deviation, s, of a variable is
the square root of the sum of squared deviations
about the sample mean divided by n – 1, where n
is the sample size.
where x1, x2, . . . , xn are the n observations in the
sample and is the sample mean.
s 
xi  x 2

n 1

x1  x 2
 x2  x 2
L  xn  x 2
n 1
x

A formula that is equivalent to the one on the
previous slide, called the computational
formula, for determining the sample standard
deviation is
s 
xi
2

xi 
2
n

n 1

We call n - 1 the degrees of freedom because
the first n - 1 observations have freedom to be
whatever value they wish, but the nth value has
no freedom. It must be whatever value forces
the sum of the deviations about the mean to
equal zero.

EXAMPLE Computing a Sample Standard
Deviation
Here are the results of a random sample taken
from the travel times (in minutes) to work for all
seven employees of a start-up web development
company:
5, 26, 36
Find the sample standard deviation.

xi
5 22.33333 -17.333 300.432889
26 22.33333 3.667 13.446889
36 22.33333 13.667 186.786889
500.66667xi
 x 
2

s 
xi  x 2

n 1

500.66667
2
15.82 minutes
x xi  x xi  x 2

xi (xi )2
5 25
26 676
36 1296
Σ xi = 67 Σ (xi)2 = 1997
 
xi
2

xi 
2
n
n 1

1997 
67 2
3
2
15.82 minutes
Using the computational formula, yields the same
result.

EXAMPLE Comparing Standard Deviations
Determine the standard deviation waiting time
for Wendy’s and McDonald’s. Which is
larger? Why?

1.50 0.79 1.01 1.66 0.94 0.67
2.53 1.20 1.46 0.89 0.95 0.90
1.88 2.94 1.40 1.33 1.20 0.84
3.99 1.90 1.00 1.54 0.99 0.35
0.90 1.23 0.92 1.09 1.72 2.00
3.50 0.00 0.38 0.43 1.82 3.04
0.00 0.26 0.14 0.60 2.33 2.54
1.97 0.71 2.22 4.54 0.80 0.50
0.00 0.28 0.44 1.38 0.92 1.17
3.08 2.75 0.36 3.10 2.19 0.23
Wait Time at Wendy’s
Wait Time at McDonald’s

EXAMPLE Comparing Standard Deviations
Sample standard deviation for Wendy’s:
0.738 minutes
Sample standard deviation for McDonald’s:
1.265 minutes
Recall from earlier that the data is more dispersed
for McDonald’s resulting in a larger standard
deviation.

Objective 3
• Determine the Variance of a Variable from
Raw Data

The variance of a variable is the square of the
standard deviation. The population variance is
σ2 and the sample variance is s2.

EXAMPLE Computing a Population Variance
minutes) to work for all seven employees of a start-up
web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population and sample variance of this
data.

EXAMPLE Computing a Population Variance
Recall that the population standard deviation (from
slide #49) is σ = 11.36 so the population variance is
σ2 = 129.05 minutes
and that the sample standard deviation (from slide
#55) is s = 15.82, so the sample variance is
s2 = 250.27 minutes

Objective 4
• Use the Empirical Rule to Describe Data That
Are Bell Shaped

The Empirical Rule
If a distribution is roughly bell shaped, then
• Approximately 68% of the data will lie within
1 standard deviation of the mean. That is,
approximately 68% of the data lie between
μ – 1σ and μ + 1σ.
• Approximately 95% of the data will lie within
2 standard deviations of the mean. That is,
approximately 95% of the data lie between
μ – 2σ and μ + 2σ.

The Empirical Rule
If a distribution is roughly bell shaped, then
• Approximately 99.7% of the data will lie
within 3 standard deviations of the mean. That
is, approximately 99.7% of the data lie
between μ – 3σ and μ + 3σ.
Note: We can also use the Empirical Rule based
on sample data with used in place of μ and s
used in place of σ.
x

EXAMPLE Using the Empirical Rule
The following data represent the serum HDL
cholesterol of the 54 female patients of a family
doctor.
41 48 43 38 35 37 44 44 44
62 75 77 58 82 39 85 55 54
67 69 69 70 65 72 74 74 74
60 60 60 61 62 63 64 64 64
54 54 55 56 56 56 57 58 59
45 47 47 48 48 50 52 52 53

(a) Compute the population mean and standard
deviation.
(b) Draw a histogram to verify the data is bell-shaped.
(c) Determine the percentage of all patients that have
serum HDL within 3 standard deviations of the
mean according to the Empirical Rule.
(d) Determine the percentage of all patients that have
serum HDL between 34 and 69.1 according to the
Empirical Rule.
(e) Determine the actual percentage of patients that
have serum HDL between 34 and 69.1.

(a) Using a TI-83 plus graphing calculator, we find
(b)
7.11and4.57  

22.3 34.0 45.7 57.4 69.1 80.8 92.5
(e) 45 out of the 54 or 83.3% of the patients have a serum HDL
between 34.0 and 69.1.
(c) According to the Empirical Rule, 99.7% of the all patients that
have serum HDL within 3 standard deviations of the mean.
(d) 13.5% + 34% + 34% = 81.5% of all patients will have a serum
HDL between 34.0 and 69.1 according to the Empirical Rule.

Objective 5
• Use Chebyshev’s Inequality to Describe Any
Set of Data

Chebyshev’s Inequality
For any data set or distribution, at least
1
1
k2



100% of the observations lie within k
Note: We can also use Chebyshev’s Inequality
based on sample data.
standard deviations of the mean, where k is any
number greater than 1. That is, at least
of the data lie between μ – kσ
and μ + kσ for k > 1.
1
1
k2



100%

EXAMPLE Using Chebyshev’s Theorem
Using the data from the previous example, use
Chebyshev’s Theorem to
(a) determine the percentage of patients that have serum
HDL within 3 standard deviations of the mean.
(b) determine the actual percentage of patients that have
serum HDL between 34 and 80.8 (within 3 SD of mean).
52/54 ≈ 0.96 ≈ 96%
1
1
32



100%  88.9%

Section
Measures of
Central Tendency
and
Dispersion from
Grouped Data
3.3

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

Objective 1
• Approximate the Mean of a Variable from
Grouped Data

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.

Approximate the Mean of a Variable from a
Frequency Distribution
 
xi fi
fi

x1 f1  x2 f2  ... xn fn
f1  f2  ... fn
Sample MeanPopulation Mean
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
x 
xi fi
fi

x1 f1  x2 f2  ... xn fn
f1  f2  ... fn

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
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.
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
EXAMPLE Approximating the Mean from a Relative

Time Frequency xi xi fi
0 0 0 0
1 - 5 130 3 390
6 - 10 250 8 2000
11 - 15 230 13 2990
16 - 20 180 18 3240
21 - 25 100 23 2300
26 – 30 60 28 1680
31 – 35 50 33 1650
xi
fi  14,250fi  1000
x 
xi fi
fi

14,250
1000
 14.25

Objective 2
• Compute the Weighted Mean

The weighted mean, , 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 
wi xi
wi

w1x1  w2 x2  ... wn xn
w1  w2  ... wn
where w is the weight of the ith observation
xi is the value of the ith observation
xw

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.
xw 
1($1.25)  2($3.25) 1.5($5.40)
1 2 1.5

$15.85
4.5
 $3.52

Objective 3
• Approximate the Standard Deviation of a
Variable from Grouped Data

Approximate the Standard Deviation of a
Variable from a Frequency Distribution
 
xi   2
fi
fi
Sample
Standard Deviation
Population
Standard Deviation
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
s 
xi  x 2
fi
fi 1

xi
2
fi 
xi f 
2
fi

fi
An algebraically equivalent formula for the
population standard deviation is

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
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.
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
EXAMPLE Approximating the Mean from a Relative

Time
Frequ
ency xi
0 0 0 0 0
1 - 5 130 3 –11.25 16,453.125
6 - 10 250 8 –6.25 9765.625
11 - 15 230 13 –1.25 359.375
16 - 20 180 18 3.75 2531.25
21 - 25 100 23 8.75 7656.25
26 – 30 60 28 13.75 11,343.75
31 – 35 50 33 18.75 17,578.125
xi
 x 
2
fi  65,687.5fi  1000
s2

xi
 x 
2
fi
fi 1

65,687.5
1000 1
 65.8
s  s2
 65.8
 8.1 hours
xi
 x 
2
fi
xi
 x

Section
Measures of
Position and
Outliers
3.4

Objectives
1. Determine and interpret z-scores
2. Interpret percentiles
3. Determine and interpret quartiles
4. Determine and interpret the interquartile
range
5. Check a set of data for outliers

Objective 1
• Determine and Interpret z-scores

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:
z 
x  

Sample z-scorePopulation z-score
The z-score is unitless. It has mean 0 and standard
deviation 1.
z 
x  x
s

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

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.
zkg 
83 69.1
2.8
 4.96
zcp 
76  63.7
2.7
 4.56

Objective 2
• Interpret Percentiles

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.

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.

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%.

Objective 3
• Determine and Interpret Quartiles

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.

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.

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.
EXAMPLE Finding and Interpreting Quartiles

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.

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.

Objective 4
• Determine and Interpret the Interquartile
Range

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

EXAMPLE Determining and Interpreting the
Interquartile Range
Determine and interpret the interquartile range of the
speed data.
Q1 = 28 Q3 = 38
The range of the middle 50% of the speed of cars
traveling through the construction zone is 10 miles per
hour.
IQR  Q3
 Q1
 38  28
 10

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?
Without 15th car With 15th car
Mean 32.1 mph 36.7 mph
Median 32.5 mph 33 mph
Standard deviation 6.2 mph 18.5 mph
IQR 10 mph 11 mph

Objective 5
• Check a Set of Data for Outliers

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.

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.

Section
The Five-Number
Summary and
Boxplots
3.5

Objectives
1. Compute the five-number summary
2. Draw and interpret boxplots

Objective 1
• Compute the Five-Number Summary

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:

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.

Institution Rate
Pulaski Bank and Trust Company 6.5%
Rainier Pacific Savings Bank 12.0%
Wells Fargo Bank NA 14.4%
Firstbank of Colorado 14.4%
Lafayette Ambassador Bank 14.3%
Infibank 13.0%
United Bank, Inc. 13.3%
First National Bank of The Mid-Cities 13.9%
Bank of Louisiana 9.9%
Bar Harbor Bank and Trust Company 14.5%
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

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% 12.0% 13.6% 14.4% 14.5%

Objective 2
• Draw and Interpret Boxplots

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.

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 (*).

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.

Institution Rate
Pulaski Bank and Trust Company 6.5%
Rainier Pacific Savings Bank 12.0%
Wells Fargo Bank NA 14.4%
Firstbank of Colorado 14.4%
Lafayette Ambassador Bank 14.3%
Infibank 13.0%
United Bank, Inc. 13.3%
First National Bank of The Mid-Cities 13.9%
Bank of Louisiana 9.9%
Bar Harbor Bank and Trust Company 14.5%
Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

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:
[ ]*

The interest rate boxplot indicates that the distribution
is skewed left.
Use a boxplot and quartiles to describe the shape of a
distribution.

Chapter 03 NCC STAT

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Chapter 03 NCC STAT