Download Example

Basic Statistics
for Business and Economics
Chapter 3
Describing Data
Fifth Edition
Numerical Measures
Douglas William Samuel
Irwin/McGraw-Hill
Topics Covered
• Measures of location; arithmetic mean,
weighted mean, median, mode, and
geometric mean
• Dispersion; range, deviation, variance, and
Standard deviation
Measures of Location
• Arithmetic mean ; shows the central values
* Population mean; the sum of all values in the
population divided by the number of values in the
population. The formula;
X


N
µ
is the population mean
N is the total number of observations
X is a particular value.
 indicates the operation of adding.
Example
The Kiers
family owns
four cars. The
following is the
current mileage
on each of the
four cars.
56,000
42,000
23,000
73,000
Find the mean mileage for the cars.
X


N
56,000  ...  73,000

 48,500
4
Measures of location
• Arithmetic mean
** Sample Mean; Is the sum of all sampled values
divided by the total number of the sampled
values.
X
X
n
X is the sample mean, read X bar
n is the total number of values in
the sample

Example
A sample of
five
executives
received the
following
bonus last
year ($000):
14.0,
15.0,
17.0,
16.0,
15.0
Find the mean bonus for the executives
X 14 .0  ...  15 .0 77
X 


 15 .4
n
5
5
Measures of location
Weighted Mean;
occurs when there are several observations of
the same value.
( w1 X 1  w2 X 2  ...  wn X n )
Xw 
( w1  w2  ...wn )
( wX )
X
W
Example
The carter construction company pays its hourly employees
$16.50, $17.50, or $18.50 per hour. There are 26 hourly
employees, 14 are paid at the $16.50 rate, 10 at $17.50
rate, and 2 at the $18.50 rate.
What is the mean hourly rate paid the 26 employees?
( wX )
X 
W
14($16.50)  10($17.50)  2($18.50) $443
X

 $17.04
14  10  2
26
Measures of location
• The Median
The midpoint of the values after they have been
ranked from the smallest to the largest or the
vise versa. Example of Odd numbers
Prices ordered
From low to high
Prices ordered
From high to low
$ 60,000
65,000
70,000
80,000
275,000
$275,000
80,000
70,000
65,000
60,000
Measures of location
• The Mode
The value of observation that appears most
frequently. Example
The annual salaries of quality-control managers in selected states in
The USA are shown below. What is the Modal annual salary?
State
Salary
Arizona
$35,000
California
49,100
Colorado
60,000
New jersey
65,000
Tennessee
60,000
State
Salary
Illinois
$58,000
Louisiana 60,000
Maryland 60,000
Ohio
50,000
Texas
71,400
Measures of location
• The Mode
- We can determine the mode for all levels of datanominal, ordinal, interval, and ratio
- sometimes there is no mode ; $18,19,21,23,24.
- We can determine the mode from charts. Example
Which bath oil consumers prefer?? Lamoure
400
350
300
250
200
150
100
50
0
Amor
Lamoure
Soothing
Smell Nice
Far out
Measures of location
• The Geometric Mean
used in finding the average of percentages, ratios,
and growth rate. The formula;
GM  n ( X 1)( X 2)( X 3)...( Xn)
The Geometric Mean
The difference between Arithmetic Mean & Geometric Mean
Example;
The interest rate on three bonds were 5, 21,4 percent.
**The arithmetic mean is (5+21+4)/3 =10.0.
***The geometric mean is GM  3 (5)(21)(4)  7.49
The GM gives a more conservative profit figure
because it is not heavily weighted by the rate
of 21percent.
The Geometric Mean
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;
GM  n
(Value at end of period)
1
(Value at beginning of period)
The Geometric Mean
•
Example; During the decade of 1990s, Las Vegas was the
fastest- growing metropolitan in the United States. The
population increased from 852,737 in 1990 to 1,563,282 in
2000. This is an increase of 710,545 or (710,545 ÷ 852,737 =
83%) an 83% increase over the period of 10 years. What is
the average annual increase?
(Value at end of period)
GM 
1
(Value at beginning of period)
n
1,563,282
GM  10
 1  .0625 
 6.25%
852,737
• The population of Las Vegas increased at a rate of 6.25%
per year from 1990 to 2000.
Measures of Dispersion
• Dispersion; Is the Mean variation in the set of data
• Why we study Dispersion?
A measure of location , such as the mean, or the median,
only describes the center of the data, but it does not tell us
anything about the spread of data. Example; if you would
like to cross a river and you are not a good swimmer, if
you know that the average depth of the river is 3 feet in
depth, you might decide to cross it, but if you have been
told that the variation is between one feet to 6 feet, you
might decide not to cross.
** Before making a decision about crossing the river, you
want information on both the typical depth and the
dispersion in the depth of the river.
*** Same thing if you want to buy stocks.
Measures of Dispersion
- Range
- Deviation
- Variance
- Standard Deviation
Measures of Dispersion
•
Range = largest value – Smallest value
Example;
The weight of containers being shipped to Ireland
000 pounds
95
103 105
110
104
105
112
90
• What is the range of the weight?
• Compute the arithmetic mean weight.
1- The range = 112- 90 = 22 pound
2- Arithmetic Mean = 95+103+….90 = 824 = 103
8
8
Measures of Dispersion
• Deviation
The absolute values of the deviations from the
arithmetic mean. Formula
MD =
X-X
n
X
X
n
II
is the value of each observation
is the arithmetic mean
is the number of observations
indicates the absolute values
•The drawbacks of deviation is the use of absolute
values because it’s hard to work with. From this point
came the impertinence of Standard deviation.
Measures of Dispersion
•
Deviation
Example;
The weights of a sample of crates containing books for
the bookstore (in pounds ) are:
103, 97, 101, 106, 103
• Find the mean deviation.
1- Arithmetic mean = 103+97+101+106+103 = 102
5
2-
MD 
X X

103  102  ...  103  102
n
1 5 1 4  5

 2.4
5
5
Measures of Dispersion
Variance & Standard Deviation
They are also based on the deviations from the mean,
however, instead of using the absolute value of the
deviation, the variance and the standard deviation square
the deviation. Formula
Population Variance
2

(X
)
 =
N

σ² variance is called sigma squared
X is the value of observation in the
population
µ is the arithmetic mean of population
N is the number of observations
Measures of Dispersion
Variance & Standard Deviation
Population Standard Deviation
σ =
2
Variance & Standard Deviation
Example
The number of traffic citation during the last five
months in Beaufort County , South Carolina , is
38,26,13,41,22.
• What is the population variance?
• What is the standard deviation?
Number
(x)
38
26
13
41
22
140
X- µ
+10
-2
-15
+13
-6
0
(x- µ)²
100
4
225
169
36
534
µ = 140 = 28
5
σ² = 534 = 106.8
5
σ = √106.8 = 10.3 Citation
Variance & Standard Deviation
Sample Variance
s2 =
(X - X)2
n-1
Sample Standard deviation
s s
2
Variance & Standard Deviation
Example
The hourly wages for the sample of a part-time
employees at Fruit Packers, Inc are; $12,
20,16,18, and 19
• What is the sample variance?
• What is the standard deviation?
Number
(x)
X- µ
12
20
16
18
19
85
-5
3
-1
+1
+2
0
(x- µ)²
25
9
1
1
4
40
x = 85 = $17
5
S² = 40 = 10
5-1
σ = √10 = 3.16
Coefficient of Variation
Definition
The coefficient of variation is an attribute of a
distribution: its standard deviation divided by its
mean . Formula
σ
CV =
µ
Coefficient of variation is a measure of how much
variation exists in relation to the mean, so it
represents a highly useful—and easy—concept in
data analysis.
Coefficient of Variation
From previous example; find the coefficient variation
(x)
38
26
13
41
22
140
X- µ
+10
-2
-15
+13
-6
0
(x- µ)²
100
4
225
169
36
534
µ = 140 = 28
5
σ² = 534 = 106.8
5
σ = √106.8 = 10.3 Citation
CV
=σ /µ
= 10.3 = .37
28