CHAPTER 3 part A.pptx business statistics

CHAPTER 3
NUMERICAL DESCRIPTIVE
MEASURES (Part A)
Prem Mann, Introductory Statistics, 8/E
Copyright © 2013 John Wiley & Sons. All rights reserved.

MEASURES OF CENTRAL TENDENCY FOR
UNGROUPED DATA
 Mean
 Median
 Mode
 Relationships among the Mean, Median, and Mode

Mean
The mean for ungrouped data is obtained by dividing the
sum of all values by the number of values in the data set. Thus,
Mean for population data:
Mean for sample data:
where is the sum of all values; N is the population size; n
is the sample size; is the population mean; and is the
sample mean.
N
x



n
x
x


x
 x

Example 3-1
Table 3.1 lists the total cash donations (rounded to millions of
dollars) given by eight U.S. companies during the year 2010
(Source: Based on U.S. Internal Revenue Service data
analyzed by The Chronicle of Philanthropy and USA TODAY).

Table 3.1 Cash Donations in 2010 by Eight U.S. Companies
Find the mean of cash donations made by these eight
companies.

Example 3-1: Solution
Thus, these eight companies donated an average of $139.5 million
in 2010 for charitable purposes.
8
7
6
5
4
3
2
1 x
x
x
x
x
x
x
x
x 








million
n
x
x 5
.
139
$
5
.
139
8
1116





1116
63
26
315
21
63
110
199
319 









Example 3-2
The following are the ages (in years) of all eight employees of a
small company:
53 32 61 27 39 44 49 57
Find the mean age of these employees.

years
25
.
45
8
362




N
x

The population mean is
Thus, the mean age of all eight employees of this company
is 45.25 years, or 45 years and 3 months.

Median
 Definition
 The median is the value of the middle term in a data set that
has been ranked in increasing order.
 The calculation of the median consists of the following two
steps:
1. Rank the data set in increasing order.
2. Find the middle term. The value of this term is the median.

Example 3-3
Refer to the data on the number of homes foreclosed in seven
states given in Table 3.2 of Example 3.3. Those values are
listed below.
173,175 49,723 20,352 10,824 40,911 18,038 61,848
Find the median for these data.

First, we rank the given data in increasing order as follows:
10,824 18,038 20,352 40,911 49,723 61,848 173,175
Since there are seven homes in this data set and the middle
term is the fourth term,
Thus, the median number of homes foreclosed in these seven
states was 40,911 in 2010.

Example 3-4
 Table 3.3 gives the total compensations (in millions of
dollars) for the year 2010 of the 12 highest-paid CEOs of
U.S. companies.

Table 3.4 Total Compensations of 12 Highest-Paid CEOs for the Year 2010
Find the median for
these data.

 First we rank the given total compensations of the 12 CESs as
follows:
 21.6 21.7 22.9 25.2 26.5 28.0 28.2 32.6 32.9 70.1 76.1 84.5
 There are 12 values in this data set. Because there are an
even number of values in the data set, the median is given by
the average of the two middle values.

 The two middle values are the sixth and seventh in the
arranged data, and these two values are 28.0 and 28.2.
 Thus, the median for the 2010 compensations of these 12
CEOs is $28.1 million.
million
1
.
28
$
1
.
28
2
2
.
56
2
2
.
28
0
.
28
Median 





Mode
 Definition
 The mode is the value that occurs with the highest frequency
in a data set.

Example 3-5
 The following data give the speeds (in miles per hour) of
eight cars that were stopped on I-95 for speeding violations.
77 82 74 81 79 84 74 78
Find the mode.

 In this data set, 74 occurs twice and each of the remaining
values occurs only once. Because 74 occurs with the highest
frequency, it is the mode. Therefore,
Mode = 74 miles per hour

Mode
 A major shortcoming of the mode is that a data set may
have none or may have more than one mode, whereas it
will have only one mean and only one median.
 Unimodal: A data set with only one mode.
 Bimodal: A data set with two modes.
 Multimodal: A data set with more than two modes.

Example 3-6 (Data set with no mode)
 Last year’s incomes of five randomly selected families were
$76,150, $95,750, $124,985, $87,490, and $53,740.
 Find the mode.

 Because each value in this data set occurs only once, this data
set contains no mode.

Example 3-7 (Data set with two modes)
A small company has 12 employees. Their commuting times
(rounded to the nearest minute) from home to work are 23,
36, 12, 23, 47, 32, 8, 12, 26, 31, 18, and 28, respectively.
Find the mode for these data.

In the given data on the commuting times of the 12
employees, each of the values 12 and 23 occurs twice, and
each of the remaining values occurs only once. Therefore, that
data set has two modes: 12 and 23 minutes.

Example 3-8 (Data set with three modes)
The ages of 10 randomly selected students from a class are 21,
19, 27, 22, 29, 19, 25, 21, 22 and 30 years, respectively.
Find the mode.

This data set has three modes: 19, 21 and 22. Each of these
three values occurs with a (highest) frequency of 2.

Mode
One advantage of the mode is that it can be calculated for
both kinds of data - quantitative and qualitative - whereas the
mean and median can be calculated for only quantitative data.

MEASURES OF DISPERSION FOR UNGROUPED
DATA
 Range
 Variance and Standard Deviation
 Population Parameters and Sample Statistics

Range
Finding the Range for Ungrouped Data
Range = Largest value – Smallest Value

Example 3-11
 Table 3.4 gives the total areas in square miles of the four
western South-Central states of the United States.
 Find the range for this data set.

Table 3.4

Range = Largest value – Smallest Value
= 267,277 – 49,651
= 217,626 square miles
Thus, the total areas of these four states are spread over a range of
217,626 square miles.

Variance and Standard Deviation
 The standard deviation is the most-used measure of
dispersion.
 The value of the standard deviation tells how closely the
values of a data set are clustered around the mean.
 In general, a lower value of the standard deviation for a data
set indicates that the values of that data set are spread over
a relatively smaller range around the mean.

 In contrast, a larger value of the standard deviation for a
data set indicates that the values of that data set are
spread over a relatively larger range around the mean.
 The standard deviation is obtained by taking the positive
square root of the variance.

 The variance calculated for population data is denoted by σ²
(read as sigma squared), and the variance calculated for
sample data is denoted by s².
 The standard deviation calculated for population data is
denoted by σ, and the standard deviation calculated for
sample data is denoted by s.
 Consequently, the standard deviation calculated for
population data is denoted by σ, and the standard deviation
calculated for sample data is denoted by s.

Basic Formulas for the Variance and Standard Deviation for
Ungrouped Data
where σ² is the population variance, s² is the sample variance,
σ is the population standard deviation, and s is the sample
standard deviation.
   
   
1
and
1
and
2
2
2
2
2
2














n
x
x
s
N
x
n
x
x
s
N
x





Table 3.5

Short-cut Formulas for the Variance and Standard Deviation
for Ungrouped Data
σ is the population standard deviation, and s is the sample
standard deviation.
   
   
1
and
1
and
2
2
2
2
2
2
2
2
2
2










 
 
 
 
n
n
x
x
s
N
N
x
x
n
n
x
x
s
N
N
x
x



Example 3-12
Until about 2009, airline passengers were not charged for checked
baggage. Around 2009, however, many U.S. airlines started
charging a fee for bags. According to the Bureau of Transportation
Statistics, U.S. airlines collected more than $3 billion in baggage fee
revenue in 2010. The following table lists the baggage fee revenues
of six U.S. airlines for the year 2010. (Note that Delta’s revenue
reflects a merger with Northwest. Also note that since then United
and Continental have merged; and American filed for bankruptcy
and may merge with another airline.)
Find the variance and standard deviation for these data.

Example 3-12

Let x denote the 2010 baggage fee revenue (in millions of
dollars) of an airline. The values of Σx and Σx2
are calculated
in Table 3.6.

Step 1. Calculate Σx
The sum of values in the first column of Table 3.6 gives
2,854.
Step 2. Find Σx2
The results of this step are shown in the second column of
Table 3.6, which is 1,746,098.

Step 3. Determine the variance
   
06666
.
709
,
77
5
667
.
552
,
357
,
1
098
,
746
,
1
1
6
6
854
,
2
098
,
746
,
1
1
2
2
2
2









 
n
n
x
x
s

Step 4. Obtain the standard deviation
The standard deviation is obtained by taking the (positive) square
root of the variance:
Thus, the standard deviation of the 2010 baggage fee revenues of
these six airlines is $278.76 million.
 
million
n
n
x
x
s
76
.
278
$
7634601
.
278
06666
.
709
,
77
1
2
2






 

Two Observations
1. The values of the variance and the standard deviation are
never negative.
2. The measurement units of variance are always the square
of the measurement units of the original data.

Example 3-13
Following are the 2011 earnings (in thousands of dollars)
before taxes for all six employees of a small company.
88.50 108.40 65.50 52.50 79.80 54.60
Calculate the variance and standard deviation for these data.

Let x denote the 2011 earnings before taxes of an employee
of this company. The values of ∑x and ∑x2
are calculated in
Table 3.7.

 
2
2
2
2
(449.30)
35,978.51
6 388.90
6
388.90 $19.721 thousand $19,721
x
x
N
N




  
  


Thus, the standard deviation of the 2011 earnings of all six
employees of this company is $19,721.

Warning
Note that ∑x2
is not the same as (∑x)2
. The value of ∑x2
is
obtained by squaring the x values and then adding them. The
value of (∑x)2
is obtained by squaring the value of ∑x.

Population Parameters and Sample Statistics
 A numerical measure such as the mean, median, mode,
range, variance, or standard deviation calculated for a
population data set is called a population parameter, or
simply a parameter.
 A summary measure calculated for a sample data set is
called a sample statistic, or simply a statistic.

MEAN, VARIANCE AND STANDARD DEVIATION
FOR GROUPED DATA
 Mean for Grouped Data
 Variance and Standard Deviation for Grouped Data

Calculating Mean for Grouped Data
Mean for population data:
Mean for sample data:
where m is the midpoint and f is the frequency of a class.
Mean for Grouped Data
N
mf



n
mf
x



Example 3-14
Table 3.8 gives the frequency distribution of the daily
commuting times (in minutes) from home to work for all 25
employees of a company.
Calculate the mean of the daily commuting times.

Example 3-14

minutes
21.40




25
535
N
mf

Thus, the employees of this company spend an average of
21.40 minutes a day commuting from home to work.

Example 3-15
Table 3.10 gives the frequency distribution of the number of
orders received each day during the past 50 days at the office
of a mail-order company.
Calculate the mean.

Example 3-15

orders
16.64




50
832
n
mf
x
Thus, this mail-order company received an average of
16.64 orders per day during these 50 days.

Variance and Standard Deviation for Grouped Data
Basic Formulas for the Variance and Standard Deviation for
Grouped Data
and m is the midpoint of a class. In either case, the standard
deviation is obtained by taking the positive square root of the
variance.
   
1
2
2
2
2







n
x
m
f
s
N
m
f
and



Short-Cut Formulas for the Variance and Standard Deviation
for Grouped Data
and m is the midpoint of a class.
 
1
)
(
2
2
2
2
2
2







 
n
n
mf
f
m
s
N
N
mf
f
m
and


Short-cut Formulas for the Variance and Standard Deviation for
Grouped Data
The standard deviation is obtained by taking the positive
square root of the variance.
Population standard deviation:
Sample standard deviation: 2
s
s 
2

 

Example 3-16
The following data, reproduced from Table 3.8 of Example 3-14,
give the frequency distribution of the daily commuting times (in
minutes) from home to work for all 25 employees of a company.
Calculate the variance and standard deviation.

Example 3-16

minutes
62
.
11
04
.
135
04
.
135
25
3376
25
25
)
535
(
825
,
14
)
(
2
2
2
2
2









 



N
N
mf
f
m
Thus, the standard deviation of the daily commuting times for
these employees is 11.62 minutes.

Example 3-17
The following data, reproduced from Table 3.10 of Example 3-
15, give the frequency distribution of the number of orders
received each day during the past 50 days at the office of a
mail-order company.
Calculate the variance and standard deviation.

Example 3-17

orders
75
.
2
5820
.
7
5820
.
7
1
50
50
)
832
(
216
,
14
1
)
(
2
2
2
2
2










 
s
s
n
n
mf
f
m
s
Thus, the standard deviation of the number of orders received at the
office of this mail-order company during the past 50 days is 2.75.

CHAPTER 3 part A.pptx business statistics

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