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3-1
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3-2
When you have completed this chapter, you will be able to:
1.
Calculate the arithmetic mean, the weighted mean, the median,
the mode, and the geometric mean of a given data set.
2.
Identify the relative positions of the arithmetic mean, median
and mode for both symmetric and skewed distributions.
3.
Point out the proper uses and common misuses of each
measure.
4.
Explain your choice of the measure of central tendency of
data.
5.
Explain the result of your analysis.
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Five Measures of
Central Tendency
arithmetic mean
weighted mean
Average price of a house in
Ottawa (2000) was $126 000
The average income of two
parent families with children in
Canada was $65,847 in 1995 and
$72,910 in 1999. (StatCan)
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3-3
mode
median
geometric mean
The average price of a
house in Toronto in 1996
was $238,511 (StatCan)
My grade point average
for last semester was 4.0
Arithmetic Mean
3-4
…is the most widely used measure of location.
It is calculated by summing the values and
dividing by the number of values
It requires the interval scale
All values are used
It is unique
The sum of the deviations from the mean is 0
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Population Mean
Formula

x

 
N
… is the population mean
(pronounced mu)
N
… is the total number of observations
x
… is a particular value

… indicates the operation of adding
(sigma)
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3-5
Terminology
Parameter
…is a measurable characteristic of a
Population
Statistic
…is a measurable characteristic of a
Sample
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3-6
Population Mean
Formula
The Kiers family
owns four cars.
The following is
the current mileage
on each of the four
cars:
56,000 23,000
42,000 73,000
3-7
x

 
N
Find the mean
mileage for the cars.
56000 + 23000 + 42000 + 73000
=
4
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= 48 500
Sample Mean
Formula
3-8
x

x 
n
x
…is the sample mean (read “x bar”)
n
… is the number of sample observations
x
… is a particular value

… indicates the operation of adding
(sigma)
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3-9
A sample of five executives received the
following bonuses last year ($000):
14.0
15.0
17.0
16.0
15.0
Determine the average bonus given last year:
Formula
x

x 
n
14 + 15 + 17 + 16 + 15
=
5
= 77 / 5
= 15.4
The average bonus given last year was $15 400
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Properties of an
Arithmetic Mean
…Every set of interval-level and ratiolevel data has a mean
… All the values are included in
computing the mean
…A set of data has a unique mean
…The mean is affected by unusually
large or small data values
…The arithmetic mean is the
only measure of central tendency where
the sum of the deviations
of each value from the mean is zero!
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 10
Arithmetic Mean
as a Balance Point
3 - 11
Illustrate the mean of the values 3, 8 and 4.
= 15 / 3
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=5
3 - 12
Determining
5
the Mean
in Excel
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Using
3 - 13
See
Click on DATA
ANALYSIS
See…
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Using
3 - 14
See
Highlight DESCRIPTIVE STATISTICS
…Click OK
See…
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Using
3 - 15
INPUT NEEDS
See
A3:A42
See…
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Using
3 - 16
See Solution
Alternate solution…
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Using
3 - 17
CLICK ON
CLICK ON PASTE FUCTION
See
See…
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Using
3 - 18
SCROLL DOWN TO STATISTICAL
See…
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Using
3 - 19
HIGHLIGHT AVERAGE IN RIGHT MENU
See
See…
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Using
See
The mean (average) is placed
in the cell on the worksheet where
your cursor was when you began.
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3 - 20
3 - 21
Weighted Mean
The weighted mean of a set of numbers
x1, x2, ... xn,
with corresponding weights w1, w2, ...,wn,
is computed from the following formula:
w1 x1  w2 x2  ...  wn xn
w 
w1  w2  w3  ...  wn xn
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 22
During a one hour period on a hot Saturday
afternoon cabana boy Chris served fifty drinks.
He sold:
…five drinks for $0.50
…fifteen for $0.75
…fifteen for $0.90
…fifteen for $1.10
Compute:
- the weighted mean of the price of the drinks -
μw
5($0.50)  15($0.75) 15($0.90) 15($1.15)

5  15  15  15
$44.50
 $ 0 . 89

50
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The Median
3 - 23
The Median is the midpoint of the
values after they have been ordered
from the smallest to the largest
There are as many values
above the median as below it in the data array
For an even set of values,
the median will be the
arithmetic average of the two middle numbers
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 24
The ages for a sample of five college students are:
21, 25, 19, 20, 22
Arranging the data in ascending order gives:
19, 20, 21, 22, 25
Thus the median is 21
The heights of four basketball players, in inches, are:
76, 73, 80, 75
Arranging the data in ascending order gives:
73, 75, 76, 80
Thus the median is 75.5
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Properties of the Median
3 - 25
 There is a unique median for each data set
 It is not affected by extremely large or small
values and is therefore a valuable measure of
central tendency when such values occur
 It can be computed for ratio-level,
interval-level, and ordinal-level data
 It can be computed for an
open-ended frequency distribution
if the median
does not lie in an open-ended class
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The Mode
The Mode
is the value of the observation
that appears most frequently used
The exam scores for ten students are:
81, 93, 84, 75, 68, 87, 81, 75, 81, 87
The score of 81 occurs the most often
…it is the Mode!
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 26
Geometric Mean
3 - 27
The Geometric Mean (GM) of a
set of n numbers is defined as the
nth root of the product of the n numbers.
The geometric mean is used to average
percents, indexes, and relatives.
The formula is:
GM  n ( x 1 )( x 2 )( x 3 ). .. ( xn )
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Geometric Mean
The interest rate on three bonds was
5, 21, and 4 percent
The Geometric Mean is:
GM 
3
( 5 )( 21 )( 4 )  7 . 49
The arithmetic mean is (5+21+4)/3 =10.0
The GM gives a more conservative profit figure
because it is not heavily weighted
by the rate of 21percent
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3 - 28
3 - 29
Geometric Mean
continued…
Another use of the geometric mean is to determine the
percent increase in sales,
production or other business or economic series
from one time period to another.
The formula is:
GM = n
(Value at end of period)
(Value at beginning of period)
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-1
Geometric Mean
3 - 30
continued…
The total number of females enrolled in American
colleges increased from
755,000 in 1992 to 835,000 in 2000.
GM  8
835 ,000
- 1  .0127
755 ,000
i.e. the Geometric Mean rate of increase is 1.27%.
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 31
Determining the
Median, Mode or
Geometric Mean
in Excel
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Using
3 - 32
Click
DATA ANALYSIS
See…
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Using
3 - 33
Highlight
DESCRIPTIVE
STATISTICS
INPUT NEEDS
SUMMARY
STATISTICS
See SOLUTION
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Using
3 - 34
Solution
The
geometric mean
doesn’t show up in
summary
statistics!
Alternate solution…
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Using
3 - 35
CLICK ON
CLICK ON PASTE FUCTION
See
See…
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Using
3 - 36
SCROLL DOWN
to
STATISTICAL
See…
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Using
See
3 - 37
HIGHLIGHT
MEDIAN
IN RIGHT MENU
OR…
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Using
See
3 - 38
HIGHLIGHT
GEOMETRIC MEAN
IN RIGHT MENU
OR…
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Using
See
3 - 39
HIGHLIGHT
MODE
IN RIGHT MENU
See…
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Using
3 - 40
The calculated values are placed in the cell on the
worksheet where your cursor was when you began
New
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The following table shows the expenditures of
Canadians in 15 countries they visited in 1999
Countries Visited
Australia
Cuba
Dominican Rep.
France
Germany
Hong Kong
Ireland
Italy
Japan
Mexico
Netherlands
Spain
Switzerland
United Kingdom
United States
3 - 41
Expenditures ($Cdn millions)
227
265
122
506
183
Source:
138
Statistics Canada,
114
Tourism
283
and the
150
Centre for
557
Education
107
Statistics
105
91
1009
8401
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
3 - 42
Is the mean or median expenditure a
more accurate
reflection of the “average” Canadian
out-of-country expenditure?
What happens to the values of the mean and median
when you remove the United States expenditures
from the sample?
…if you remove both the UK and US from the sample?
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Using
3 - 43
The mean is strongly affected by the inclusion of
these two OUTLIERS
… therefore, the median is a more appropriate measure
of “average” in this case
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The Mean
of Grouped Data
The mean of a sample of data organized
in a frequency distribution is
computed by the following formula:
x
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
fx


N
3 - 44
The Mean
of Grouped Data
3 - 45
A sample of ten movie theatres in a metropolitan
area tallied the total number of movies showing
last week.
Compute the
mean number of movies showing per theatre.
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The Mean
fx

x
of Grouped Data
N
Continued…
Class
(f)(x)
Midpoint
Movies
Showing
Frequency
1 to under 3
1
2
2
3 to under 5
2
4
8
5 to under 7
3
6
18
7 to under 9
1
8
8
9 to under 11
3
10
30
Total
10
f
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3 - 46
66
The Mean
fx

x
of Grouped Data
N
Movies
Showing
Frequency
Total
10
f
Formula
Continued…
Class
(f)(x)
Midpoint
66
Xf

X
n
 66
10
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3 - 47
= 6.6
The Mean
fx

x
of Grouped Data
N
3 - 48
Determine the average student study time
Hours
Studying
Frequency
(f)(x)
f
Class
Midpoint
10 to under 15
5
12.5
62.5
15 to under 20
12
17.5
210
20 to under 25
6
22.5
61027.5
= 20.33

30
32.5
135
25
to under 30
Formula
x
30 to under 35
Total

5fx

N
2
30
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137.5
65
610
Finding the Median of
Grouped Data
3 - 49
To determine the median class for Grouped Data:
1. Construct a cumulative frequency distribution
2. Divide the total number of data values by 2
3. Determine which class will contain this value
E.g.
If n = 50, 50/2 = 25,
then determine which class
will contain the 25th value
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Finding the Median of
Grouped Data
3 - 50
Estimate the median value within chosen class…
N CF
2
Median =
L+
(i)
f
L … is the lower limit of the median class
CF … is the cumulative frequency as you
enter the median class
f … is the frequency of the median class
i … is the class interval or size
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
Finding the Median of
Grouped Data
Movies
Showing
Frequency
Cumulative
f
f
1 to under 3
1
1
3 to under
5
Median
class
L 5 to under 7
2
CF 3
7 to under 9
1
7
3
10
9 to under 11
Total
3
i=2
10
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
f
6
3 - 51
N - CF
L+ 2
(i )
f
10 - 3
=5+ 2
2
3
= 6.33
The Mode of Grouped Data
3 - 52
The mode for grouped data is approximated by the
midpoint of the class with the largest class frequency
Movies
Showing
Frequency
f
Class
Midpoint
1 to under 3
1
2
3 to under 5
2
4
This is
considered
to be
5 to under 7
3
6
BiModal
7 to under 9
1
8
9 to under 11
3
10
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
The Mode of Grouped Data
Approximate the Mode of this distribution
Hours
Studying
Frequency
f
Class
Midpoint
10 to under 15
5
12.5
15 to under 20
12
17.5
20The
to under 25
22.520,
modal class6is 15 to under
25 to underapproximately
30
5
17.5 27.5
30 to under 35
2
Total
30
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32.5
3 - 53
Symmetric Distribution
zero skewness
mode = median = mean
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3 - 54
Right Skewed Distribution
Mean and Median are to the right of the Mode
Positively skewed
Mode<
Median<
Mean
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3 - 55
Left Skewed Distribution
Mean and Median are to the left of the Mode
Negatively skewed
< Mode
< Median
Mean
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3 - 56
Test your learning…
www.mcgrawhill.ca/college/lind
Online Learning Centre
for quizzes
extra content
data sets
searchable glossary
access to Statistics Canada’s E-Stat data
…and much more!
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3 - 57
3 - 58
This completes Chapter 3
Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.