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. Mean. It is the Arithmetic Average of data values:
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Measures of Central Location
Mean, Median, Mode
Measures of Variation
Range, Variance and Standard Deviation
Measures of Association
Covariance and Correlation
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+
·
·
·
+
x
x
x
n
=
x
å
x
=
i
2
n
i
=
Sample Mean
i
1
n
n
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5
Mean = 6
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
Median = 5
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
No Mode
Mode = 9
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
Population
Variance
Population
Standard
Deviation
Sample
Variance
Sample
Standard
Deviation
The Range

x
x
La
rgest
Smallest
Range = 12  7 = 5
Range = 12  7 = 5
7 8 9 10 11 12
7 8 9 10 11 12
Variance
)
2

m
å
(X
2
s
=
i
N
(
)
2

å
X
X
2
=
i
s

n
1
For the Population: use N in the denominator.
For the Sample : use n  1 in the denominator.
Standard Deviation
(
)
2

m
å
X
s
=
i
N
(
)
2

å
X
X
=
i
s

n
1
For the Population: use N in the denominator.
For the Sample : use n  1 in the denominator.
Sample Standard Deviation
(
)
2
For the Sample : use n  1 in the denominator.

å
X
X
s
=
i

n
1
Data:10 12 14 15 17 18 18 24
n = 8 Mean =16
s =
Sample Standard Deviation= 4.2426
Comparing Standard Deviations
Data A
Mean = 15.5
s = 3.338
11 12 13 14 15 16 17 18 19 20 21
Data B
Mean = 15.5
s = .9258
11 12 13 14 15 16 17 18 19 20 21
Data C
Mean = 15.5
s = 4.57
11 12 13 14 15 16 17 18 19 20 21
Coefficient of Variation:
Stock A: CV = 10%
Stock B: CV = 5%
RightSkewed
LeftSkewed
Symmetric
Mean
Median
Mode
Mean
=
Median
=
Mode
Mode
Median
Mean