Transcript Slide 1

QBA 260
Business Statistics
Chapter 5
Deviating from the Average

Compare two datasets (reported in
inches)




Dataset 1: 48, 46, 46, 49, 49, 47, 47, 46, 49
Dataset 2: 47, 51, 49, 53, 47, 45, 37, 38, 60
What is the average of each dataset?
How are they different?
Comparing Datasets
Histogram - dataset 2
10
Number within each bin
Number within each bin
Histogram - Dataset 1
8
6
4
2
0
10
20
30
40
50
Mean
Bin Categories
60
10
8
6
4
2
0
10
20
30
40
Bin Categories
50
Mean
Mean (Average) for both datasets is 47.4 inches
Variances differ: Dataset 1 = 1.78 sq. inches, Dataset 2 = 51.03 sq. inches
60
Variance and Deviations


Variance is a measure of how much the data
points in a sample deviate from the sample
mean
For data:


x1, x2, x3,…, xn
A deviation is the difference between a data value and
a central value such as the average
Deviation X i  X
Deviations
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Dataset 1: 48,
46, 46, 49, 49,
47, 47, 46, 49
Average = 47.4
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Deviations
The average
deviation is
(always) zero
Xi
X-Bar
Deviation
48
47.4
0.56
46
47.4
-1.44
46
47.4
-1.44
49
47.4
1.56
49
47.4
1.56
47
47.4
-0.44
47
47.4
-0.44
46
47.4
-1.44
49
47.4
1.56
47.4
0.00
(Average)
(Sum)
Deviations
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
Dataset 2: 47,
51, 49, 53, 47,
45, 37, 38, 60
Average = 47.4


Deviations
The average
deviation is
(always) zero
Xi
X-Bar
Deviation
47
47.4
-0.44
51
47.4
3.56
49
47.4
1.56
53
47.4
5.56
47
47.4
-0.44
45
47.4
-2.44
37
47.4
-10.44
38
47.4
-9.44
60
47.4
12.56
47.4
0.00
(Average)
(Sum)
Populations and Samples

Population variance
N
2 

2
(
X

X
)
 i
i 1
N
Sample variance
N
s 
2
2
(
X

X
)
 i
i 1
N 1
Sample Variance – Dataset 1
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Sample variance
Xi
X-Bar
Deviation
Sq Dev
48
47.4
0.56
0.31
46
47.4
-1.44
2.09
46
47.4
-1.44
2.09
49
47.4
1.56
2.42
49
47.4
1.56
2.42
47
47.4
-0.44
0.20
47
47.4
-0.44
0.20
46
47.4
-1.44
2.09
49
47.4
1.56
2.42
47.4
0.00
14.22
(Average)
(Sum)
(Sum)
N
s 
2
(Xi  X )
2
i 1
N 1
14 .22
s 
N 1
2
14.22
s 
 1.78
9 1
2
Sq Inches
Sample Variance – Dataset 2
Sample variance

N
s2 
(X
i 1
i
 X)
N 1
408 .22
s 
N 1
2
2
Xi
X-Bar
Deviation
Sq Dev
47
47.4
-0.44
0.20
51
47.4
3.56
12.64
49
47.4
1.56
2.42
53
47.4
5.56
30.86
47
47.4
-0.44
0.20
45
47.4
-2.44
5.98
37
47.4
-10.44
109.09
38
47.4
-9.44
89.20
60
47.4
12.56
157.64
47.4
0.00
408.22
(Average)
(Sum)
(Sum)
408 .22
s 
 51.03 Sq Inches
9 1
2
Variance and
Standard Deviation
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The calculated variance is in a different unit
of measure than the original dataset
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Original Dataset measured in inches
Variance measured in inches squared (due to the
use of squared deviations before we average
them)
To accommodate this, we calculate the
standard deviation which is the square root of
the Variance
Standard Deviation

Square root of the variance
N

2
(
X

X
)
 i
i 1
N
N
s
2
(
X

X
)
 i
i 1
N 1
Excel Functions
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VARP(…) Population Variance
VARPA(…) Population Variance (includes
cells that contain true/false data)
VAR(…) Sample Variance
VARA(…) Sample Variance (includes
cells that contain true/false data)
Excel Functions
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STDEVP(…) Population Standard Deviation
STDEVPA(…) Population Standard
Deviation (includes cells that contain
true/false data)
STDEV(…) Sample Standard Deviation
STDEVA(…) Sample Standard Deviation
(includes cells that contain true/false data)
Average Absolute Deviation
Xi
X-Bar
Deviation
ABS
48
47.4
0.56
0.56
46
47.4
-1.44
1.44
46
47.4
-1.44
1.44
49
47.4
1.56
1.56
49
47.4
1.56
1.56
47
47.4
-0.44
0.44
47
47.4
-0.44
0.44
46
47.4
-1.44
1.44
49
47.4
1.56
1.56
47.4
0.00
10.44
(Average)
(Sum)
(Sum)
Absolute Deviation = 10.44/9 = 1.16
EXCEL: AVEDEV