Statistical Techniques in Business and Economics 12e Chapter 03
Statistical Techniques in Business and Economics 12e Chapter 03
Statistical Techniques in Business and Economics 12e Chapter 03
Three
McGraw-Hill/Irwin
2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
3- 1
Chapter Three
Describing Data: Numerical Measures
GOALS
When 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
3- 2
FOUR
Compute and interpret the range, the mean deviation, the
variance, and the standard deviation of ungrouped data.
Describing Data: Numerical Measures
FIVE
Explain 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
Chapter Three
3- 3
Characteristics of the Mean
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.
The Arithmetic Mean
is the most widely used
measure of location and
shows the central value of the
data.
The major characteristics of the mean are:
Average
Joe
3- 4
Population Mean
N
X
=
where
is the population mean
N is the total number of observations.
X is a particular value.
E indicates the operation of adding.
For ungrouped data, the
Population Mean is the
sum of all the population
values divided by the total
number of population
values:
3- 5
Example 1
500 , 48
4
000 , 73 ... 000 , 56
=
+ +
= =
N
X
=
+ +
=
E
=
n
X X
s
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 . 2 30 . 5
2
= = = s s
3- 35
Chebyshevs theorem: For any set of
observations, the minimum proportion of the values
that lie within k standard deviations of the mean is at
least:
where k is any constant greater than 1.
2
1
1
k
Chebyshevs theorem
3- 36
Empirical Rule: For any symmetrical, bell-shaped
distribution:
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 3s of
the mean
Interpretation and Uses of the
Standard Deviation
3- 37
Bell -Shaped Curve showing the relationship between and . o
3o
2o 1o +1o +2o
+ 3o
68%
95%
99.7%
Interpretation and Uses of the Standard Deviation
3- 38
The Mean of Grouped Data
n
Xf
X
E
=
The Mean of a sample of data
organized in a frequency
distribution is computed by the
following formula:
3- 39
Example 12
A sample of ten
movie theaters
in a large
metropolitan
area tallied the
total number of
movies showing
last week.
Compute the
mean number of
movies
showing.
Movies
showing
frequency
f
class
midpoint
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 8
9 up to
11
3 10 30
Total 10 66
6 . 6
10
66
= =
E
=
n
X
X
3- 40
The Median of Grouped Data
) (
2
i
f
CF
n
L Median
+ =
where L is the lower limit of the median class, CF is the
cumulative frequency preceding the median class, f is
the frequency of the median class, and i is the median
class interval.
The Median of a sample of data organized in a
frequency distribution is computed by:
3- 41
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. For
example, if n=50, 50/2 = 25, then determine which
class will contain the 25
th
value.
3- 42
Example 12 continued
Movies
showing
Frequency Cumulative
Frequency
1 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
3- 43
Example 12 continued
33 . 6 ) 2 (
3
3
2
10
5 ) (
2
=
+ =
+ = i
f
CF
n
L Median
From the table, L=5, n=10, f=3, i=2, CF=3
3- 44
The Mode of Grouped Data
The modes in example 12 are 6 and 10
and so is bimodal.
The Mode for grouped data is
approximated by the midpoint of the
class with the largest class frequency.
3- 45