Download Sec 3.2 Measures of Variation

Refer to Ex 3-18 on page 123-124
Record the info for Brand A in a column.
Allow 3 adjacent other columns to be
added.
Do the same for Brand B.
Test on Chapter 3
•Friday Sept
•You are expected to
provide you own
calculator on the test.
th
27 .
3-2 Measures of Variation
How Can We Measure Variability?
Range
Variance
Standard
Deviation
Coefficient
of Variation
Chebyshev’s
Empirical
Theorem
Rule (Normal)
Bluman, Chapter 3
3
Measures of Variation: Range

The range is the difference between the
highest and lowest values in a data set.
R  Highest  Lowest
Bluman, Chapter 3
4
Chapter 3
Data Description
Section 3-2
Example 3-18/19
Page #123
Bluman, Chapter 3
5
Example 3-18/19: Outdoor Paint
Two experimental brands of outdoor paint are
tested to see how long each will last before
fading. Six cans of each brand constitute a
small population. The results (in months) are
shown. Find the mean and range of each group.
Brand A
Brand B
10
35
60
45
50
30
30
35
40
40
20
25
Bluman, Chapter 3
6
Example 3-18: Outdoor Paint
X 210

Brand A
Brand B


 35
Brand A:
N
6
10
35
60
45
R  60  10  50
50
30
30
35
40
40
X


20
25
Brand B:
210

 35
N
6
R  45  25  20
The average for both brands is the same, but the range
for Brand A is much greater than the range for Brand B.
Which brand would you buy?
Bluman, Chapter 3
7
Measures of Variation: Variance &
Standard Deviation

The variance is the average of the
squares of the distance each value is
from the mean.

The standard deviation is the square
root of the variance.

The standard deviation is a measure of
how spread out your data are.
Bluman, Chapter 3
8
•Uses of the Variance and
Standard Deviation
To determine the spread of the data.
 To determine the consistency of a
variable.
 To determine the number of data values
that fall within a specified interval in a
distribution (Chebyshev’s Theorem).
 Used in inferential statistics.

Bluman, Chapter 3
9
Measures of Variation:
Variance & Standard Deviation
(Population Theoretical Model)

The population variance is


2
X  


2
N
The population standard deviation is

 X  
2
N
Bluman, Chapter 3
10
Chapter 3
Data Description
Section 3-2
Example 3-21
Page #125
Bluman, Chapter 3
11
Example 3-21: Outdoor Paint
Find the variance and standard deviation for the
data set for Brand A paint. 10, 60, 50, 30, 40, 20
Months, X
10
60
50
30
40
20
µ
35
35
35
35
35
35
X - µ (X -
-25
25
15
-5
5
-15
µ)2
625
625
225
25
25
225
1750

2
X  


n
1750

6
 291.7
1750

6
 17.1
Bluman, Chapter 3
2
Population
Variance
Population
Standard
Deviation
12
Measures of Variation:
Variance & Standard Deviation
(Sample Theoretical Model)

The sample variance is
s

2
X X



2
n 1
The sample standard deviation is
s
 X  X 
2
n 1
Bluman, Chapter 3
13
Measures of Variation:
Variance & Standard Deviation
(Sample Computational Model)

Is mathematically equivalent to the
theoretical formula.

Saves time when calculating by hand

Does not use the mean

Is more accurate when the mean has
been rounded.
Bluman, Chapter 3
14
Measures of Variation:
Variance & Standard Deviation
(Sample Computational Model)

The sample variance is
n X    X 
2
s 
2

2
n  n  1
The sample standard deviation is
s s
2
Bluman, Chapter 3
15
Chapter 3
Data Description
Section 3-2
Example 3-23
Page #129
Bluman, Chapter 3
16
Example 3-23: European Auto Sales
Find the variance and standard deviation for the
amount of European auto sales for a sample of 6
years. The data are in millions of dollars.
11.2, 11.9, 12.0, 12.8, 13.4, 14.3
X
11.2
11.9
12.9
12.8
13.4
14.3
75.6
X
2
125.44
141.61
166.41
163.84
179.56
204.49
958.94
n X    X 
2
s 
2
s 
2
2
n  n  1
6  958.94    75.6 
2
6  5

s 2  1.28
s  1.13

s 2  6  958.94  75.62 /  6  5
Bluman, Chapter 3
17
Finding Variance and Standard
deviation of Grouped Data

Find the variance and the standard
deviation for the frequency distribution of
the data in the next slide. The data
represents the number of miles that 20
runners ran during one week.
A
B
Class
Frequency
(f)
5.5-10.5
1
10.5-15.5
2
15.5-20.5
3
20.5-25.5
5
25.5-30.5
4
30.5-35.5
3
35.5-40.5
2
A
B
C
D
E
Class
Frequency
(f)
Midpoint
(Xm)
f•Xm
f•Xm2
5.5-10.5
1
10.5-15.5
2
15.5-20.5
3
20.5-25.5
5
25.5-30.5
4
30.5-35.5
3
35.5-40.5
2
s
2
f X



2
m

 f  X

2
m
n

n 1

13,310  490  20

 68.7
20  1
2
Example 3-23

Find the sample variance and standard
deviation for the amount of European auto
sales for a sample of 6 years shown. The
data are in millions of dollars.
11.2, 11.9, 12.0, 12.8, 13.4, 14.3
Example 3-24

Find he variance and the standard
deviation for the frequency distribution of
the data in Example 2-7. the data
represent
Uses for standard deviation.
1.
2.
3.
4.
Spread of data
Consistency
Determine the number (or %) of data
within an interval.
Many other used to be discussed fully
second semester!
Measures of Variation:
Coefficient of Variation
The coefficient of variation is the
standard deviation divided by the
mean, expressed as a percentage.
s
CVAR  100%
X
Use CVAR to compare standard
deviations when the units are different.
Bluman, Chapter 3
25
Chapter 3
Data Description
Section 3-2
Example 3-25
Page #132
Bluman, Chapter 3
26
Example 3-25: Sales of Automobiles
The mean of the number of sales of cars over a
3-month period is 87, and the standard
deviation is 5. The mean of the commissions is
$5225, and the standard deviation is $773.
Compare the variations of the two.
5
CVar  100%  5.7%
87
Sales
773
CVar 
100%  14.8%
5225
Commissions
Commissions are more variable than sales.
Bluman, Chapter 3
27
Measures of Variation:
Range Rule of Thumb
The Range Rule of Thumb
approximates the standard deviation
as
Range
s
4
when the distribution is unimodal and
approximately symmetric.
Bluman, Chapter 3
28
Measures of Variation:
Range Rule of Thumb
Use X  2 s to approximate the lowest
value and X  2 s to approximate the
highest value in a data set.
Example: X  10, Range  12
12
LOW

10

2
3

4


s
3
4
HIGH  10  2  3  16
Bluman, Chapter 3
29
Using Symbols
Let 𝑥 =20 and s=3, calculator the following:
1.
2.
3.
4.
𝑥+s=
𝑥 − 2s =
𝑥 + 3s =
𝑥 ± 2.5s =
Bluman, Chapter 3
30
Measures of Variation:
Chebyshev’s Theorem
The proportion of values from
any data set that fall within k
standard deviations of the mean
𝟏
will be at least 𝟏 − 𝟐,
𝐤
• where k >1
• k is not necessarily an integer.
Bluman, Chapter 3
31
Measures of Variation:
Chebyshev’s Theorem
# of
Minimum Proportion
standard
within k standard
deviations, k
deviations
2
3
4
1-1/4=3/4
1-1/9=8/9
1-1/16=15/16
Bluman, Chapter 3
Minimum Percentage
within k standard
deviations
75%
88.89%
93.75%
32
Measures of Variation:
Chebyshev’s Theorem
# of
standard
deviations, k
Minimum Proportion
within k standard
deviations
Minimum Percentage
within k standard
deviations
1.5
2.5
3.5
Bluman, Chapter 3
33
Measures of Variation:
Chebyshev’s Theorem
Bluman, Chapter 3
34
Chapter 3
Data Description
Section 3-2
Example 3-27
Page #135
Bluman, Chapter 3
35
Example 3-27: Prices of Homes
The mean price of houses in a certain
neighborhood is $50,000, and the standard
deviation is $10,000. Find the price range for
which at least 75% of the houses will sell.
Chebyshev’s Theorem states that at least 75% of
a data set will fall within 2 standard deviations of
the mean.
50,000 – 2(10,000) = 30,000
50,000 + 2(10,000) = 70,000
At least 75% of all homes sold in the area will have a
price range from $30,000 and $75,000.
Bluman, Chapter 3
36
Chapter 3
Data Description
Section 3-2
Example 3-28
Page #135
Bluman, Chapter 3
37
Example 3-28: Travel Allowances
A survey of local companies found that the mean
amount of travel allowance for executives was
$0.25 per mile. The standard deviation was 0.02.
Using Chebyshev’s theorem, find the minimum
percentage of the data values that will fall
between $0.20 and $0.30.
.30  .25 / .02  2.5
.25  .20  / .02  2.5
1  1/ k  1  1/ 2.5
 0.84
2
2
k  2.5
At least 84% of the data values will fall between
$0.20 and $0.30.
Bluman, Chapter 3
38
The Empirical Rule





The empirical rule is only valid for bell-shaped (normal)
distributions. The following statements are true.
Approximately 68% of the data values fall within one
standard deviation of the mean.
Approximately 95% of the data values fall within two
standard deviations of the mean.
Approximately 99.7% of the data values fall within three
standard deviations of the mean.
The empirical rule will be revisited later in the chapter on
normal probabilities.
Measures of Variation:
Empirical Rule (Normal)
Bluman, Chapter 3
41
Measures of Variation:
Empirical Rule (Normal)
The percentage of values from a data set that
fall within k standard deviations of the mean in
a normal (bell-shaped) distribution is listed
below.
# of standard Proportion within k standard
deviations, k
deviations
1
68%
2
95%
3
99.7%
Bluman, Chapter 3
42
Homework
Section 3-2 Page 137
 1-6 all, 7-17 every other odd,
 19, 21
 29-41 every other odd

Bluman, Chapter 3
45
Application of Empirical Rule

Given a data set comprised of 5057
measurements that is bell-shaped with a mean
of 177. It has a standard deviation of 55. What
percentage of the data should lie between 67
and 287?
Bluman, Chapter 3
46