Download Summarizing and Describing Numerical Data

Describing Data:
Summary Measures
Measures of Central Location
Mean, Median, Mode
Measures of Variation
Range, Variance and Standard Deviation
Measures of Association
Covariance and Correlation
© 1999 Prentice-Hall, Inc.
Chap. 3 - 1
Mean
•It is the Arithmetic Average of data values:
x=
Sample Mean
n
 xi
i =1
n
xi + x 2 +    + xn
=
n
•The Most Common Measure of Central Tendency
•Affected by Extreme Values (Outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
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0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Chap. 3 - 2
Median
•Important Measure of Central Tendency
•In an ordered array, the median is the
“middle” number.
•If n is odd, the median is the middle number.
•If n is even, the median is the average of the 2
middle numbers.
•Not Affected by Extreme Values
0 1 2 3 4 5 6 7 8 9 10
Median = 5
© 1999 Prentice-Hall, Inc.
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
Chap. 3 - 3
Mode
•A Measure of Central Tendency
•Value that Occurs Most Often
•Not Affected by Extreme Values
•There May Not be a Mode
•There May be Several Modes
•Used for Either Numerical or Categorical Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
© 1999 Prentice-Hall, Inc.
0 1 2 3 4 5 6
No Mode
Chap. 3 - 4
Measures Of Variability
Variation
Variance
Range
Population
Variance
Sample
Variance
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Standard Deviation
Population
Standard
Deviation
Sample
Standard
Deviation
Coefficient of
Variation
S
CV = 
X

  100%

Chap. 3 - 5
The Range
• Measure of Variation
• Difference Between Largest & Smallest
Observations:
Range =
x Largest - x Smallest
• Ignores How Data Are Distributed:
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
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9
10
11
12
7
8
9
10
11
12
Chap. 3 - 6
Variance
•Important Measure of Variation
•Shows Variation About the Mean:
2
)
m

(X
2
i
•For the Population: s =
N
•For the Sample:
 (X i - X )
s =
n -1
2
For the Population: use N in the
denominator.
© 1999 Prentice-Hall, Inc.
2
For the Sample : use n - 1
in the denominator.
Chap. 3 - 7
Standard Deviation
•Most Important Measure of Variation
•Shows Variation About the Mean:
2
(
)
m
 Xi
•For the Population:
s=
•For the Sample:
s =
For the Population: use N in the
denominator.
© 1999 Prentice-Hall, Inc.
N
 (X i
- X
n -1
)2
For the Sample : use n - 1
in the denominator.
Chap. 3 - 8
Sample Standard Deviation
(
- X)

X
i
s =
n-1
Data:
Xi :
10
2
12
n=8
s=
For the Sample : use n - 1
in the denominator.
14
15
17 18 18 24
Mean =16
(10 - 16)2 + (12 - 16)2 + (14 - 16)2 + (15 - 16)2 + (17 - 16)2 + (18 - 16)2 + (24 - 16)2
8-1
Sample Standard Deviation= 4.2426
© 1999 Prentice-Hall, Inc.
Chap. 3 - 9
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
© 1999 Prentice-Hall, Inc.
Mean = 15.5
s = 4.57
Chap. 3 - 10
Coefficient of Variation
•Measure of Relative Variation
•Always a %
•Shows Variation Relative to Mean
•Used to Compare 2 or More Groups
•Formula ( for Sample):
S 
CV =    100%
X 
© 1999 Prentice-Hall, Inc.
Chap. 3 - 11
Comparing Coefficient of Variation
Stock A: Average Price last year = $50
Standard Deviation = $5
Stock B: Average Price last year = $100
Standard Deviation = $5
S 
CV =    100%
X 
© 1999 Prentice-Hall, Inc.
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
Chap. 3 - 12
Shape
•
•
Describes How Data Are Distributed
Measures of Shape:
Symmetric or skewed
Left-Skewed
Symmetric
Mean Median Mod
e
Mean = Median = Mode
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Right-Skewed
Mode Median Mean
Chap. 3 - 13