chapter 03 describing data: numerical measures
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Chapter
Three
McGraw-Hill/Irwin 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
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Chapter Three
Describing Data: Numerical Measures
GOALSWhen you have completed this chapter, you will be able to:
ONE
Calculate the arithmetic mean, median, mode, weighted
mean, and the geometric mean.TWO
Explain the characteristics, uses, advantages, and
disadvantages of each measure of location.
THREE
Identify the position of the arithmetic mean, median,
and mode for both a symmetrical and a skewed
distribution.
Goals
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FOUR
Compute and interpret the range, the mean deviation, the
variance, and the standard deviation of ungrouped data.
Describing Data: Numerical Measures
FIVEExplain the characteristics, uses, advantages, and
disadvantages of each measure of dispersion.
SIX
Understand Chebyshevs theorem and the Empirical Rule as
they relate to a set of observations.
Goals
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There are two numerical ways
of describing data, namely
measure of location and
measure of dispersion.
Measure of Locationare often referred to as average.An Average shows the central value of the data.
Measure of dispersionoften called the variation or
spread in the data.
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Characteristics of the Mean
It is calculated bysumming the values
and dividing by thenumber of values.
It requires the interval scale.
All values are used.It is unique.
The sum of the deviations from the mean is 0.
The Arithmetic Meanis the most widely used
measure of location andshows the central value of the
data.
The major characteristics of the mean are:Average
Joe
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Population Mean
N
X
where
is the population mean
N is the total number of observations.
X is a particular value.
indicates the operation of adding.
For ungrouped data, the
Population Meanis the
sum of all the populationvalues divided by the total
number of population
values:
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Example 1
500,484
000,73...000,56
N
X
Find the mean mileage for the cars.
A Parameteris a measurable characteristic of apopulation.
The Kiers
family owns
four cars. The
following is thecurrent mileage
on each of the
four cars.
56,000
23,000
42,000
73,000
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Sample Mean
nXX
where nis the total number ofvalues in the sample.
For ungrouped data, the sample mean is
the sum of all the sample values divided
by the number of sample values:
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Example 2
4.155
77
5
0.15...0.14
n
XX
A statisticis a measurable characteristic of a sample.
A sample offive
executives
received the
followingbonus last
year ($000):
14.0,
15.0,
17.0,16.0,
15.0
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Properties of the Arithmetic Mean
Every set of interval-level and ratio-level 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 locationwhere the sum of the deviations of each value from
the mean is zero.
Propertiesof the Arithmetic Mean3- 10
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Example 3
0)54()58()53()( XX
Consider the set of values: 3, 8, and 4.
The meanis 5. Illustrating the fifthproperty
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Weighted Mean
)21
)2211
...(
...(
n
nnw
www
XwXwXwX
The Weighted Meanof a set of
numbersX1,X2, ...,Xn, withcorresponding weights w1, w2,
...,wn, is computed from the
following formula:
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Example 4
89.0$50
50.44$
1515155)15.1($15)90.0($15)75.0($15)50.0($5
wX
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, and fifteen for $1.10.
Compute the weighted mean of
the price of the drinks.
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The Median
There are as manyvalues above themedian 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 and isfound at the (n+1)/2 ranked observation.
The Medianis the
midpoint of the values afterthey have been ordered from
the smallest to the largest.
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The ages for a sample of five college students are:
21, 25, 19, 20, 22.
Arranging the datain ascending ordergives:
19, 20, 21, 22, 25.
Thus the median is21.
The median (continued)
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Example 5
Arranging the data in
ascending order gives:
73, 75, 76, 80
Thus the median is 75.5.
The heights of four basketball players, in inches,
are: 76, 73, 80, 75.
The median is found
at the (n+1)/2 =(4+1)/2 =2.5thdata
point.
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Properties of the Median
There is a unique median for each data set.
It is not affected by extremely large or smallvalues and is therefore a valuable measure of
location when such values occur.It can be computed for ratio-level, interval-level, and ordinal-level data.
Properties of the Median
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The Mode: Example 6
Example 6:The exam scores for ten students are:
81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the scoreof 81 occurs the most often, it is the mode.
Data can have more than one mode. If it has twomodes, it is referred to as bimodal, three modes,
trimodal, and the like.
TheModeis another measure of location andrepresents the value of the observation that appears
most frequently.
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Symmetric distribution: A distribution having thesame shape on either side of the center
Skewed distribution: One whose shapes on eitherside of the center differ; a nonsymmetrical distribution.
Can be positively or negatively skewed, or bimodal
The Relative Positions of the Mean, Median, and Mode
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The Relative Positions of the Mean, Median, and Mode:
Symmetric Distribution
Zero skewness Mean
=Median
=Mode
Mode
Median
Mean
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The Relative Positions of the Mean, Median, and Mode:
Right Skewed Distribution
Positively skewed: Mean and median are to the right of the
mode.
Mean>Median>Mode
Mode
Median
Mean
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Negatively Skewed:Mean and Median are to the left of the Mode.
Mean
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Geometric Mean
GM X X X X nn ( )( )( )...( )1 2 3
The geometric mean is used to
average percents, indexes, and
relatives.
The Geometric Mean(GM) of a set of n numbers
is defined as the nthroot
of the product of the n
numbers. The formula is:
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Example 7
The interest rate on three bonds were 5, 21, and 4
percent.
The arithmetic mean is (5+21+4)/3 =10.0.The geometric mean is
49.7)4)(21)(5(3
GM
The GMgives a more conservative
profit figure because it is notheavily weighted by the rate of
21percent.
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Geometric Mean continued
1period)ofbeginningat(Value
period)ofendatValue( nGM
Another use of the
geometric mean is to
determine the percentincrease in sales,
production or other
business or economic
series from one time
period to another.
Grow th in Sales 1999-2004
0
10
20
30
40
50
1999 2000 2001 2002 2003 2004
Year
SalesinMillions($)
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Example 8
0127.1000,755
000,8358 GM
The total number of females enrolled in American
colleges increased from 755,000 in 1992 to 835,000 in
2000. That is, the geometric mean rate of increase is
1.27%.
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Dispersionrefers to the
spread orvariability in
the data.
Measures of dispersion include the following: range,
mean deviation, variance, and standard
deviation.
Range = Largest valueSmallest value
Measures of Dispersion
0
5
10
15
20
25
30
0 2 4 6 8 10 12
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The following represents the current years Return on
Equity of the 25 companies in an investors portfolio.
-8.1 3.2 5.9 8.1 12.3
-5.1 4.1 6.3 9.2 13.3
-3.1 4.6 7.9 9.5 14.0
-1.4 4.8 7.9 9.7 15.0
1.2 5.7 8.0 10.3 22.1
Example 9
Highest value: 22.1 Lowest value: -8.1
Range = Highest valuelowest value
= 22.1-(-8.1)
= 30.2
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Mean
DeviationThe arithmetic
mean of the
absolute values
of thedeviations from
the arithmetic
mean.
The main features of the
mean deviation are:
All values are used in the
calculation.
It is not unduly influenced by
large or small values.
The absolute values are
difficult to manipulate.
Mean Deviation
MD =X - X
n
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The weights of a sample of crates containing booksfor the bookstore (in pounds ) are:
103, 97, 101, 106, 103Find the mean deviation.
X = 102
The mean deviation is:
4.25
14151
5
102103...102103
n
XX
MD
Example 10
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Variance: the
arithmetic meanof the squared
deviations from
the mean.
Standard deviation: The squareroot of the variance.
Variance and standard Deviation
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Population Varianceformula:
(X - )2N
=
X is the value of an observation in the populationm is the arithmetic mean of the population
N is the number of observations in the population
Population Standard Deviationformula:
2Variance and standard deviation
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(-8.1-6.62)2+ (-5.1-6.62)2 + ... + (22.1-6.62)225
= 42.227
= 6.498
In Example 9, the variance and standard deviation are:
(X - )2
N =
Example 9 continued
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Sample variance (s2)
s2=(X - X)2
n-1
Sample standard deviation (s)
2ss
Sample variance and standard deviation
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40.75
37
n
XX
30.515
2.21
15
4.76...4.77
1
222
2
n
XXs
Example 11
The hourly wages earned by a sample of five students are:
$7, $5, $11, $8, $6.
Find the sample variance and standard deviation.
30.230.52 ss
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Chebyshevs theorem:For any set of
observations, the minimum proportion of the valuesthat lie withinkstandard deviations of the mean is atleast:
where k is any constant greater than 1.
211k
Chebyshevs theorem
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Empirical Rule: For any symmetrical, bell-shapeddistribution:
About 68% of the observations will lie within 1s
the mean
About 95% of the observations will lie within 2s of
the mean
Virtually all the observations will be within 3sof
the mean
Interpretation and Uses of the
Standard Deviation
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Bell -Shaped Curve showing the relationship between and .
3
1 1
3
68%
95%
99.7%
Interpretation and Uses of the Standard Deviation
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The Mean of Grouped Data
n
Xf
X
The Meanof a sample of dataorganized in a frequency
distribution is computed by the
following formula:
X is the midpoint of each class
f is the frequency in each classnis the total number of frequencies
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Example 12
A sample of ten
movie theaters
in a largemetropolitan
area tallied the
total number of
movies showing
last week.
Compute the
mean number ofmovies
showing.
Moviesshowing
frequencyf
classmidpoint
X
(f)(X)
1 up to 3 1 2 2
3 up to 5 2 4 8
5 up to 7 3 6 18
7 up to 9 1 8 89 up to
113 10 30
Total 10 66
6.610
66
n
XX
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The Median of Grouped Data
)(2 if
CFn
LMedian
whereLis the lower limit of the median class,
CFis the cumulative frequency preceding the median
class,f is the frequency of the median class,
andiis the median class interval.
The Medianof a sample of data organized in afrequency distribution is computed by:
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Finding the Median Class
To determine the median class for grouped
data
Construct a cumulative frequency distribution.
Divide the total number of data values by 2.
Determine which class will contain this value. Forexample, if n=50, 50/2 = 25, then determine which
class will contain the 25thvalue.
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Example 12 continued
Movies
showing
Frequency Cumulative
Frequency1 up to 3 1 1
3 up to 5 2 3
5 up to 7 3 6
7 up to 9 1 7
9 up to 11 3 10
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Example 12 continued
33.6)2(3
32
10
5)(2
if
CF
n
LMedian
From the table,L=5, n=10,f=3, i=2, CF=3
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The Mode of Grouped Data
The modes in example 12 are 6 and 10
and so is bimodal.
The Modefor grouped data isapproximated by the midpoint of the
class with the largest class frequency.
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